Embed Size (px)
Citation preview
Determinants in K-theory and operator algebras
by
Joseph Migler
B.S., University of California, Santa Barbara, 2010
M.A., University of Colorado, Boulder, 2012
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Department of Mathematics
2015
This thesis entitled:Determinants in K-theory and operator algebras
written by Joseph Miglerhas been approved for the Department of Mathematics
Professor Alexander Gorokhovsky
Professor Carla Farsi
Professor Judith Packer
Professor Arlan Ramsay
Professor Bahram Rangipour
Date
The final copy of this thesis has been examined by the signatories, and we find that both thecontent and the form meet acceptable presentation standards of scholarly work in the above
mentioned discipline.
iii
Migler, Joseph (Ph.D., Mathematics)
Determinants in K-theory and operator algebras
Thesis directed by Professor Alexander Gorokhovsky
A determinant in algebraic K-theory is associated to any two Fredholm operators which
commute modulo the trace class. This invariant is defined in terms of the Fredholm determinant,
which itself extends the usual notion of matrix determinant. On the other hand, one may consider
a homologically defined invariant known as joint torsion. This thesis answers in the affirmative a
conjecture of R. Carey and J. Pincus, namely that these two invariants agree.
The strategy is to analyze how joint torsion transforms under an action by certain groups
of pairs of invertible operators. This allows one to reduce the calculation to the finite dimensional
setting, where joint torsion is shown to be trivial. The equality implies that joint torsion has
continuity properties, satisfies the expected Steinberg relations, and depends only on the images
of the operators modulo trace class. Moreover, we show that the determinant invariant of two
commuting operators can be computed in terms of finite dimensional data.
The second main goal of this thesis is to investigate how joint torsion behaves under the
functional calculus. We study the extent to which the functional calculus commutes, modulo
operator ideals, with projections in a finitely summable Fredholm module. As an application, we
recover in particular some results of R. Carey and J. Pincus on determinants of Toeplitz operators
and Tate tame symbols. In addition, we obtain variational formulas and explicit integral formulas
for joint torsion.
iv
Acknowledgements
I am indebted to Alexander Gorokhovsky, whom it is my honor to have as an advisor. His
endless patience and mathematical insight are a source of inspiration for me. I am grateful for his
guidance, for countless enlightening conversations, and for his generosity in sharing both his time
and knowledge. His help was never more than a hallway away.
I wish to thank Carla Farsi, Judith Packer, Arlan Ramsay, and Bahram Rangipour for serving
on my dissertation committee. I would also like to thank Carla Farsi for her encouragement and
guidance during several classes I taught as a graduate student. I wish to thank Judith Packer for
organizing many fascinating seminars which I had the pleasure of attending. I would like to thank
Arlan Ramsay for his careful reading of this dissertation. I wish to express my gratitude to Bahram
Rangipour for generously sharing his time with me during his stay in Boulder.
I am grateful for the help of many mathematicians, in particular Richard Carey, Guillermo
Cortinas, Raul Curto, Jorg Eschmeier, Karl Gustafson, Nigel Higson, Matthias Lesch, Jens Kaad,
Jerry Kaminker, Ryszard Nest, Joel Pincus, Raphael Ponge, Dan Voiculescu, and Mariusz Wodz-
icki. Their comments on the subject of this thesis and related topics have greatly enhanced my
understanding.
Finally I wish to thank my family – this degree is as much theirs as it is mine. I would like to
thank my parents for their support at every stage of my education. They taught me integrity and
most everything I know today. I am especially grateful for their encouragement during my years
of studying mathematics. I would like to thank my sisters and their families for the joy they have
shared with me and for their help throughout my life.
Contents
Chapter
1 Introduction 1
2 Preliminaries 8
2.1 Fredholm theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.1 Schatten classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.2 Fredholm determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.3 Fredholm index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Algebraic K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1 Algebraic K0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.2 Algebraic K1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.3 Algebraic K2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.4 The determinant invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Toeplitz operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.1 Index and spectral theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.2 Commutators of Toeplitz operators . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Joint torsion 24
3.1 Background: The work of Carey and Pincus . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Commuting operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.1 Algebraic torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
vi
3.2.2 Joint torsion of commuting operators . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.3 The case of two commuting operators . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Almost commuting operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3.1 Perturbation vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3.2 Joint torsion of two almost commuting operators . . . . . . . . . . . . . . . . 35
4 Equality 36
4.1 The finite dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.1.1 Torsion of a double complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.1.2 Joint torsion in finite dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Factorization results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2.1 Perturbation vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2.2 Joint torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3.1 Group actions on commuting squares . . . . . . . . . . . . . . . . . . . . . . . 51
4.3.2 A proof of equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.4.1 Properties of the determinant invariant . . . . . . . . . . . . . . . . . . . . . 53
4.4.2 Properties of joint torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4.3 Joint torsion and Hilbert-Schmidt operators . . . . . . . . . . . . . . . . . . . 59
5 Explicit formulas for joint torsion 62
5.1 Transformation rules for joint torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.1.1 Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.1.2 Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.1.3 Joint torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.1.4 Positivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2 Fredholm modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
vii
5.3 Toeplitz operators and tame symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.3.1 H∞ symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.3.2 L∞ symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3.3 An integral formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Bibliography 85
Appendix
A The existence of perturbations 89
Chapter 1
Introduction
Let A and B be two invertible operators on a complex separable Hilbert space H that com-
mute modulo the trace class L1(H), which consists of compact operators with summable singular
values. The multiplicative commutator ABA−1B−1 is an invertible determinant class operator,
that is it differs from the identity by a trace class operator, and therefore has a nonzero Fredholm
determinant. The assignment (A,B) 7→ det(ABA−1B−1) is bimultiplicative and skew-symmetric.
Moreover, det(ABA−1B−1) = det(ABA−1B−1) for any invertible trace class perturbations A and
B of A and B, respectively.
L. Brown observed in [8] that this is a special case of a more general phenomenon. Indeed,
any two bounded Fredholm operators A and B that commute modulo trace class have invertible
commuting images a and b, respectively, in the quotient B/L1. Here B = B(H) is the algebra
of bounded linear operators on H, and L1 = L1(H) is the ideal of trace class operators. As we
shall see in Section 2.2, these images determine a canonical element a, b, known as a Steinberg
symbol, in the second algebraic K-group K2(B/L1). Consequently, there is a determinant invariant
d(a, b) = det ∂a, b ∈ C×. Brown showed that when A and B are invertible, this invariant recovers
the classical determinant of a multiplicative commutator, that is, d(a, b) = det(ABA−1B−1). This
immediately yields the remarkable fact that the determinant of the multiplicative commutator
depends only on the Steinberg symbol in K-theory. See also the paper of J. W. Helton and
R. Howe [32] for work in this area around the same time.
As an example, consider two non-vanishing smooth functions f and g on the unit circle. Then
2
one may form the Toeplitz operators Tf and Tg, which are compressions of multiplication operators
by f and g on L2(S1) to the Hardy space H2(S1) (Section 2.3). Since f and g are non-vanishing
and continuous, it turns out that Tf and Tg are Fredholm. Moreover, since f and g are smooth,
we shall see that Tf and Tg commute with each other modulo trace class. In [15] R. Carey and J.
Pincus obtain an integral formula for the determinant invariant of Tf and Tg:
d(Tf + L1, Tg + L1) = exp1
2πi
(∫S1
log f d(log g)− log g(p)
∫S1
d(log f)
). (1.1)
The integrals are taken counterclockwise starting at any point p ∈ S1. If h(eiθ) = |h(eiθ)|eiφ(θ) for
a continuous real-valued function φ, then we take log h(eiθ) = log |h| + iφ(θ). Any other choice of
log h will differ from this one by a multiple of 2πi and hence will leave the quantity in the formula
unaffected.
Now let A and B be commuting operators with images a and b in B/L1. In [13] Carey and
Pincus expressed the determinant invariant d(a − z1, b − z2) in terms of multiplicative Lefschetz
numbers of the form
det(B − z2)|ker(A−z1)
det(B − z2)|coker (A−z1)
det(A− z1)|coker (B−z2)
det(A− z1)|ker(B−z2)(1.2)
whenever a− z1 and a− z2 are invertible and (z1, z2) /∈ σT (A,B), the Taylor joint spectrum of A
and B. However, the determinant invariant is defined even if (z1, z2) ∈ σT (A,B). Thus, Carey and
Pincus introduce an invariant τ(A,B), known as joint torsion, for any two commuting Fredholm
operators A and B. This generalizes the multiplicative Lefschetz number in (1.2). By replacing A
by exp(A) and B by I, joint torsion recovers the Fredholm index of A, so the subject of this thesis
may be seen as a type of multiplicative index theory.
In order to define joint torsion, Carey and Pincus use the notion of algebraic torsion from
homological algebra. For any finite length exact sequence (V•, d•) of finite dimensional vector
spaces, the algebraic torsion τ(V•, d•) is a canonical generator of the determinant line
detV• =(
ΛdimVnVn
)∗⊗(
ΛdimVn−1Vn−1
)⊗ · · ·
Section 3.2.1 reviews the details of this construction. For example, the torsion of an automorphism
of a finite dimensional vector space is its determinant. As another example, J.-M. Bismut, H. Gillet,
3
and C. Soule have shown in [6] that the Ray-Singer analytic torsion can be calculated as the norm
of τ(V•, d•).
J. Kaad has generalized the notion of joint torsion to n ≥ 2 commuting operators [37].
Moreover, he shows that joint torsion is multiplicative, satisfies cocycle identities, and is trivial
under appropriate Fredholm assumptions. He has also investigated the relationship between joint
torsion on the one hand, and determinant functors and K-theory of triangulated categories on
the other. In the case of n = 2 commuting operators, he has shown in [36] that the determinant
invariant coincides with the Connes-Karoubi multiplicative character [20]. Furthermore, Kaad has
constructed a product in relative K-theory and investigated the relative Chern character with
values in continuous cyclic homology [35]. He uses these results to calculate the multiplicative
character applied to Loday products of exponentials. See also Section 3.2 below for a discussion of
this commuting case.
Carey and Pincus extended their definition of joint torsion, in a different direction, to two
almost commuting Fredholm operators [16]. Thus, let A and B be Fredholm operators with trace
class commutator, and moreover, assume the existence of operators C and D such that AB = CD,
A − D ∈ L1, and B − C ∈ L1. They then proceed as before, defining τ(A,B,C,D) in terms of
short exact sequences of Koszul complexes. However, the result is no longer a scalar, but rather an
element of a certain determinant line. To obtain a scalar, Carey and Pincus associate a perturbation
vector σA,A′ to each pair of Fredholm operators A and A′ with A − A′ ∈ L1. See Section 3.3 for
details on the constructions. More recently, J. Kaad and R. Nest have generalized the notion of
perturbation vectors to finite rank perturbations of Fredholm complexes [39].
Perturbation vectors can be seen as a generalization to the singular setting of the classical
perturbation determinant det(A−1A′). Carey and Pincus have shown in [16] that perturbation
vectors form a non-vanishing section of a Quillen determinant line bundle [48]. Moreover, they
have applied joint torsion to Toeplitz operators, especially problems where standard techniques
only apply to symbols with zero winding number. They prove Szego-type limit theorems on the
asymptotic behavior of determinants of Toeplitz operators whose symbols have nonzero winding
4
number [16]. See Section 3.1 for a discussion of this and other related work by Carey and Pincus.
In the case when f, g ∈ H∞(S1), the joint torsion τ(Tf , Tg) is the product of tame symbols [15],
and can be expressed in terms of Deligne cohomology [23]. In particular, the determinant invariant
d(Tf + L1, Tg + L1) is equal to the joint torsion τ(Tf , Tg) when f and g are smooth functions in
H∞(S1). More generally, Carey and Pincus state in [16, Section 8, p. 345]:
The existence of the identification map (in the existence theorem for the pertur-
bation vector) has uncovered a basic problem – which deserves to be stated as a
question or perhaps as a conjecture:
Let a, b be commuting units in B(H)/L1(H). Let A,B,C,D be elements in B(H)
so that AB = CD and let A,D denote lifts of a and B,C denote lifts of b. In what
generality is it true that det ∂a, b = τ(A,B,C,D)?
This is quickly seen to be true for invertible operators A, B, C, and D [16, Section 8]. Carey
and Pincus proved in [13] that this is also true for commuting Fredholm operators A and B with
acyclic Koszul complex K•(A,B). We will see later that this follows from (1.2). More generally,
they have shown in unpublished work that d(a, b) = τ(A,B,B,A) for any commuting Fredholm
operators A and B (see [15, Theorem 2]). One of the main results of this thesis is Theorem 4.3.3,
which answers the above question in full generality.
Let us take a moment to outline the strategy, which is carried out in Chapter 4. Since the
determinant invariant is trivial in finite dimensions, the first order of business is to show that this
true for joint torsion as well. This is done in Section 4.1, closely following the work of Kaad [37].
Our main tool is the double complex (4.4), which is composed of the homology spaces of modified
Koszul complexes. The resulting vertical and horizontal torsion vectors agree, up the sign of a
permutation which appears in the definition of joint torsion. By picking generators carefully, we
will see that these vectors comprise the joint torsion, which is consequently trivial.
Then in Section 4.2 we consider multiplicative properties of perturbation vectors and joint
torsion with respect to invertible operators. This will allow us to investigate the transformation
5
of joint torsion under certain group actions in Section 4.3. More specifically, consider a quadruple
(A,B,C,D) where A, B, C, and D are Fredholm operators such that
A−D ∈ L1, B − C ∈ L1, and AB = CD.
Let U and V be invertible operators such that U − V ∈ L1 and the commutator [U,B] ∈ L1 as
well. The set of all such pairs (U, V ) forms a group which acts on quadruples by
(A,B,C,D) • (U, V ) = (U−1A,B,U−1CU, V −1D).
We will see that joint torsion transforms according to the rule
τ((A,B,C,D) • (U, V )) = d(U−1 + L1, B + L1) · τ(A,B,C,D).
A similar formula holds for pairs of invertibles that commute with the first operator A modulo L1.
These group actions will allow us to reduce the general case to operators A,B,C,D in the coset of
the identity modulo finite rank operators. The proof of Theorem 4.3.3 then follows quickly from
the finite dimensional case.
In Section 4.4 we record a number of consequences of this equality. On the one hand, joint
torsion is a finite dimensional object, at least for commuting operators. The determinant invariant
on the other hand is defined in terms of the infinite Fredholm determinant. Hence it is somewhat
surprising that the determinant invariant turns out to be expressible in terms of finite dimensional
data. Moreover, we will see that joint torsion enjoys continuity properties and satisfies the usual
Sternberg relations (e.g. it is multiplicative and skew-symmetric). Finally, we will use these results
to investigate the behavior of joint torsion with respect to Hilbert-Schmidt operators.
The second main goal of this thesis is to study the transformation of joint torsion under the
functional calculus. J. Kaad and R. Nest have investigated local indices of commuting tuples of
operators under the holomorphic functional calculus [38], and they obtain a global index theorem
originally due to J. Eschmeier and M. Putinar [29]. In Section 5.1, we establish a multiplicative
analogue of such transformation rules: the joint torsion τ(f(A), B) is given by∏λ∈σ(A) | f(λ)=0
τ(A− λ,B)ordλ(f) · τ(q(A), B).
6
Here q(A) is an invertible operator, so the second factor may be viewed as a type of multiplicative
Lefschetz number. In addition, we investigate variational formulas for joint torsion (Corollaries
4.4.5, 4.4.9, and 5.1.15).
In [26], T. Ehrhardt generalizes the Helton-Howe-Pincus formula on determinants of expo-
nentials (Proposition 2.1.13) by showing that
eAeB − eA+B ∈ L1 (1.3)
whenever [A,B] ∈ L1, and moreover,
det(eAeBe−A−B
)= e
12
tr[A,B].
Now let P : L2(S1) → H2(S1) be the orthogonal projection onto the Hardy space (Section 2.3).
Let φ ∈ L∞(S1). With A = T(I−P )φ and B = TPφ, and under suitable regularity assumptions on
φ, one may use (1.3) to show that
Teφ − eTφ ∈ L1. (1.4)
We investigate the following question: to what extent does (1.4) hold with the exponential
replaced by other functions? In Section 5.2 we consider entire functions in the more general setting
of finitely summable Fredholm modules, before specializing to Toeplitz operators in Section 5.3. In
particular, Theorem 5.3.12 establishes (1.4) when
(1) f is holomorphic on a neighborhood of σ(Tφ), or
(2) φ is real-valued and f is C∞ on φ(S1).
Along the way, we investigate the functional calculus modulo ideals of compact operators. Under
suitable assumptions on f , we find
(1) f(A)− f(A′) ∈ Lp if A−A′ ∈ Lp (Proposition 5.1.6)
(2) [f(A), B] ∈ Lp if [A,B] ∈ Lp (see Proposition 5.1.2)
(3) Tf(φ) − f(Tφ) ∈ Lp in a 2p-summable Fredholm module (Proposition 5.2.7)
7
Thus we obtain functional calculi on the Calkin-type algebra B/Lp of bounded operators B modulo
the Schatten ideal Lp of compact operators with p-summable singular values. Result (2) is due
to A. Connes [19]. See also J. W. Helton and R. Howe’s work [32] for the self-adjoint case. We
also obtain expressions for the trace and estimates on the Lp-norms of operators as above, and we
apply these results to obtain the integral formula (1.1) for the joint torsion τ(Tf , Tg) of Toeplitz
operators Tf and Tg.
In Section 5.3 we obtain explicit formulas for determinants of Toeplitz operators. First,
we recall the notion of a symbol in arithmetic [58], which is a bimultiplicative map c(·, ·) on the
multiplicative group of a field such that c(a, 1 − a) = 1. One example is Brown’s determinant
invariant described above. Another example is the Tate tame symbol on a field of meromorphic
functions. This symbol, denoted ca(f, g), is defined as a weighted ratio of two functions f and g
evaluated at a point a (Definition 5.3.3). It turns out that the tame symbol is closely related to
the determinant invariant. Indeed, Theorem 5.3.11 expresses the joint torsion of Toeplitz operators
in terms of their tame symbols. This extends a result due to R. Carey and J. Pincus [15]: if
f, g ∈ H∞(S1), then the joint torsion of Tf and Tg is the product of tame symbols
τ(Tf , Tg) =∏|a|<1
ca(f, g).
Let us conclude by noting that J. Kaad and R. Nest have recently investigated the local behavior of
joint torsion transition numbers associated to commuting tuples of operators [40]. They generalize
the above Carey-Pincus formula in a different direction and extend the notion of tame symbol to
the setting of transversal functions on a complex analytic curve.
Chapter 2
Preliminaries
In this chapter we introduce the basic objects of study in this thesis and collect a number of
results that will prove useful later. First we define the Fredholm determinant, a generalization of
the matrix determinant to infinite dimensions. Next we review the Fredholm index, which may be
viewed as a measure of uniqueness and existence of solutions to vector equations. Then in Section
2.2 we discuss algebraic K-theory and use the Fredholm determinant to define one of main objects
of study in this thesis, the determinant invariant. Section 2.3 reviews Toeplitz operators, which
provide a rich source of examples, and indeed motivation, for our work.
2.1 Fredholm theory
2.1.1 Schatten classes
Let us begin by reviewing some classes of compact operators which will be important for this
thesis, namely the Schatten ideals Lp of p-summable operators. Our reference for this section is
[51].
Fix a separable complex Hilbert space H. Let B denote the algebra of bounded operators
on H and let K be the ideal of compact operators. If K ∈ K, then by the spectral theorem for
compact self-adjoint operators, K∗K is diagonalizable with respect to a basis of eigenvectors. The
eigenvalues µk (listed according to algebraic multiplicity and known as the singular values of K)
are non-negative and have no accumulation points, except possibly at zero.
9
Definition 2.1.1. Let 0 < p <∞. A compact operator K is in the Schatten p-class Lp if
∑µp/2 <∞.
Elements of L1 are known as trace class operators, and elements of L2 are known as Hilbert-
Schmidt operators. Notice that every Lp contains the finite rank operators.
Proposition 2.1.2.
(1) The Schatten classes Lp are two-sided ideals of B.
(2) If p ≥ 1, then ‖K‖p =(∑
µp/2)1/p
defines a norm on Lp.
(3) If A ∈ Lp and B ∈ B, then ‖AB‖p ≤ ‖A‖p‖B‖.
(4) (Lp, ‖ · ‖p) is a Banach space.
Example 2.1.3.
(1) Let ei∞i=0 be an orthonormal basis for H. Let S be the unilateral shift Sei = ei+1. Then
S∗e0 = 0
S∗ei = ei−1, i > 0
The self-commutator [S∗, S] is evidently the rank one projection onto the span of e0, and
hence is in every Schatten class. More generally, we will see below that commutators
of Toeplitz operators are in the trace class, under conditions on the smoothness of their
symbols.
(2) Let X be a locally compact Hausdorff space with a positive Borel measure. Assume that
L2(X) is separable. If k ∈ L2(X ×X), then the operator K defined on L2(X) by
(Kf)(x) =
∫Xk(x, y)f(y) dy
is a Hilbert-Schmidt operator with ‖K‖2 = ‖k‖L2.
10
(3) A pseudodifferential operator T on a closed n-dimensional manifold M is trace class if its
order is less than −n. More generally, T ∈ Lp if ord T < −n/p.
Remark 2.1.4. In light of the example above, it may be of interest in the future to investigate the
subject of this thesis in the context of the ideals
Lp− =⋃q<p
Lp.
Definition 2.1.5. Let L ∈ L1 be a trace class operator on a Hilbert space, say with orthonormal
basis ei. The trace of L is defined to be
trL =∑〈Lei, ei〉.
Notice that in finite dimensions this recovers the usual matrix trace.
Proposition 2.1.6.
(1) If L ∈ L1, then trL is well-defined, that is, its defining sum converges and does not depend
on the choice of basis.
(2) The trace is a continuous linear functional on (L1, ‖ · ‖1).
(3) If A ∈ L1 and B ∈ B, then tr(AB) = tr(BA).
(4) If A,B ∈ L2, then tr(AB) = tr(BA).
Example 2.1.7. The trace of a pseudodifferential operator T ∈ L1 from Example 2.1.3(3) is given
by integrating its Schwarz kernel along the diagonal (see e.g. [43, Section 4.1]:
trT =
∫MKT (x, x) dx.
2.1.2 Fredholm determinant
The goal of this section is to describe the Fredholm determinant, which extends the usual
matrix determinant to the coset I + L1. Our reference for this section and the next is [62].
11
In order to define the Fredholm determinant, we will need the notion of exterior power of a
vector space H. The k-th exterior power ΛkH is the vector space spanned by elements of the form
v1 ∧ · · · ∧ vk, vi ∈ H
subject to the relation
vσ(1) ∧ · · · ∧ vσ(k) = (−1)sign(σ)v1 ∧ · · · ∧ vk
for any permutation σ on the set 1, · · · , k. See [51, Section 1.5] for more details. By definition,
Λ0H = C. If H is finite dimensional, then ΛkH is evidently trivial for k > dimH. An inner product
〈·, ·〉 on H induces an inner product on ΛkH given by
〈v1 ∧ · · · ∧ vk, w1 ∧ · · · ∧ wk〉 = det(〈vi, wj〉).
If H is a Hilbert space, then ΛkH is as well. Moreover, any operator T ∈ B(H) induces an operator
ΛkT ∈ B(ΛkH) by
ΛkT (v1 ∧ · · · ∧ vk) = Tv1 ∧ · · · ∧ Tvk.
Definition 2.1.8. Let L ∈ L1(H). The Fredholm determinant of I + L is given by
det(I + L) =∞∑k=0
tr(
ΛkL).
One readily checks that ΛkL ∈ L1(ΛkH) and that the above series converges. In fact, we
have the following:
Proposition 2.1.9. The function z 7→ det(I + zL) is entire with
| det(I + zL)| ≤ e|z|‖L‖1 .
Corollary 2.1.10. The map L 7→ det(I + L) is continuous on L1.
In finite dimensions, the Fredholm determinant recovers the usual determinant of a matrix.
Indeed, let V be a finite dimensional vector space with basis ei. Let L be a linear transformation
12
on V , and let T = U−1LU be its Jordan canonical form, with eigenvalues ai, repeated according
to algebraic multiplicities. Then
ΛkT (ei1 ∧ · · · ∧ eik) = ai1 · · · aik(ei1 ∧ · · · ∧ eik)
and consequently,
tr(
ΛkL)
= tr(
ΛkT)
=∑
i1<···<ik
ai1 · · · aik .
Hence the Fredholm determinant of I + L is
∑k
∑i1<···<ik
ai1 · · · aik .
On the other hand, one calculates the determinant as
det(I + L) =∏i
(1 + ai)
which agrees with the Fredholm determinant above.
Remark 2.1.11. Indeed, an equivalent and perhaps more familiar definition of the Fredholm de-
terminant is given by
det(I + L) =∏i
(1 + λi)
where λi are the eigenvalues of L, repeated according to multiplicity.
Let G denote the set of of invertible operators in the coset I + L1, with the operation of
composition. That G is closed under taking inverses is immediate from the identity
(I + L)−1 = I − L(I + L)−1.
Proposition 2.1.12.
(1) If L ∈ L1, then eL ∈ G and
det(eL)
= etrL.
(2) If A ∈ I + L1 and X is invertible, then X−1AX ∈ I + L1 and
det(X−1AX) = detA.
13
(3) If A,B ∈ I + L1, then AB ∈ I + L1 and
det(AB) = detA · detB.
(4) Let A ∈ I + L1. Then detA 6= 0 if and only if A ∈ G.
Proposition 2.1.13 (Helton-Howe-Pincus). If A and B are operators such that [A,B] ∈ L1, then
det(eAeBe−Ae−B) = etr[A,B].
2.1.3 Fredholm index
On the coset I+L1 considered above, or more generally I+K, one has the famous Fredholm
alternative, one formulation of which may be stated as follows: if K is compact, then I + K is
either invertible or has nontrivial kernel and cokernel. This may be viewed as a manifestation of
the fact that I +K is a Fredholm operator with index zero, which we describe next.
Definition 2.1.14. An operator A on a Hilbert space is known as a Fredholm operator if its kernel
and cokernel are finite dimensional. In this case, the index of A is the difference of these two
dimensions:
indA = dim kerA− dim cokerA.
Proposition 2.1.15 (Atkinson’s criterion). The following are equivalent:
(1) A is Fredholm.
(2) There exist operators Q and R such that I −AQ and I −RA are compact.
(3) There exist operators Q and R such that I −AQ and I −RA have finite rank.
Property (2) above may be rephrased as saying that A is invertible in the Calkin algebra
B/K. Property (3) asserts that the same is true even modulo finite rank, and hence modulo any
operator ideal.
Proposition 2.1.16.
14
(1) The set of Fredholm operators is open in B with respect to the operator norm.
(2) The map A 7→ indA on the set of Fredholm operators is continuous, and hence, locally
constant. In fact, the collections
Uk = A | indA = k
are precisely the connected components of the set of Fredholm operators in B.
(3) If A and B are Fredholm operators, then AB and A⊕B are Fredholm with
ind(AB) = ind(A⊕B) = indA+ indB.
(4) If A is Fredholm and K is compact, then A+K is Fredholm with
ind(A+K) = indA.
2.2 Algebraic K-theory
For any unital ringR and ideal J , there are algebraicK-groupsKi(R), Ki(R/J), andKi(R, J)
that fit into Quillen’s long exact sequence
· · · → Ki+1(R/J)∂−→ Ki(R, J)→ Ki(R)→ Ki(R/J)
∂−→ . . .
For the purposes of this thesis, we will mainly be interested in the groups K1(R, J) and K2(R/J),
as well as the boundary map between them. This section reviews the relevant definitions, closely
following [50].
2.2.1 Algebraic K0
For the sake of completeness, let us begin with a discussion of K0. Recall that for a ring R,
a projective R-module can be characterized as an R-module that embeds as a direct summand in a
free R-module. The set of isomorphism classes of finitely generated projective R-modules forms a
semigroup Proj(R) under the operation of direct sum. In fact, the assignment R→ Proj(R) defines
15
a covariant functor from rings to abelian semigroups since a ring homomorphism ϕ : R → R′
induces a map
ϕ : Proj(R)→ Proj(R′), [M ] 7→ [R′ ⊗ϕM ].
Then one canonically obtains a group K0(R) via the Grothendieck construction. Indeed, for any
abelian semigroup (S,+), let G(S) be the abelian group generated by [x], x ∈ S, subject to the
relations
[x] + [y] = [z] if x+ y = z in S.
This construction S → G(S) defines a covariant functor since a homomorphism of semigroups
ϕ : S → S′ defines a group homomorphism
ϕ : G(S)→ G(S′), [x] 7→ [φ(x)].
Definition 2.2.1. K0(R) = G(Proj(R)).
Example 2.2.2. Finitely generated projective modules over a field F are simply finite dimensional
F -vector spaces. Up to isomorphism, these are classified by the dimension – a non-negative integer.
Hence K0(F ) = Z.
On the other hand, one may also consider the topological K-theory of a compact topological
space X. The group K0(X) is defined to be the Grothendieck group of the semigroup of finite rank
vector bundles over X. By the Serre-Swan Theorem [52], the category of finite rank vector bundles
over X is naturally equivalent to the category of finitely generated projective modules over C(X).
Hence we find
K0(C(X)) = K0(X).
In higher degrees, however, topological and algebraic K-theory differ in general. For example, one
may define topological K-theory on the category of C∗-algebras. The algebraic group K1, which
we describe next, turns out to be a quotient of the topological group Ktop1 .
16
2.2.2 Algebraic K1
Let R be a unital ring and let GLn(R) be the group of invertible n × n matrices over R.
Define the group GL(R) to be the direct limit of the groups GLn(R) under the embedding
GLn(R) → GLn+1(R), a 7→
a 0
0 1
.
Let En(R) ⊂ GLn(R) be the subgroup of elementary matrices, which are generated by matrices
that differ from the identity by at most one off-diagonal entry. Then one has an embedding
En(R) → En+1(R) as above, and hence a direct limit E(R) ⊂ GL(R).
Proposition 2.2.3 (see Proposition 2.1.4 in [50]). The commutator subgroup [GL(R), GL(R)] =
E(R). In particular, E(R) is normal in GL(R).
Definition 2.2.4. For a unital ring R, define
K1(R) =GL(R)
E(R).
By the preceding proposition, K1(R) coincides with the abelianization GL(R)ab of GL(R).
The assignment R → K1(R) defines a covariant functor from the category of unital rings to
the category of abelian groups. Indeed, a morphism R → S induces a map GLn(R) → GLn(S),
and hence a map on the direct limits GL(R)→ GL(S) as well as their abelianizations GL(R)ab →
GL(S)ab.
Example 2.2.5 (see Section 2.2 of [50]). If R is a commutative ring, the determinant GLn(R)→
R× extends to a surjection GL(R)→ R× and hence to a surjection
det : K1(R)→ R×.
If R is a field, this is in fact an isomorphism.
Let GL(R, J) denote the kernel of the group homomorphism GL(R) → GL(R/J) induced
by the quotient map R → R/J . Denote by E(R, J) the subgroup of elementary matrices gen-
erated by matrices that differ from the identity by at most one off-diagonal element of J . Let
17
[GL(R), E(R, J)] ⊆ GL(R, J) denote the subgroup generated by all elements of the form
ghg−1h−1, g ∈ GL(R), h ∈ E(R, J).
This subgroup is normal in GL(R, J), and we make the following definition:
Definition 2.2.6. K1(R, J) = GL(R, J)/[GL(R), E(R, J)].
2.2.3 Algebraic K2
The Steinberg group Stn(R) is the group with generators xij(a) for a ∈ R, i 6= j, 1 ≤ i, j ≤ n,
and relations
xij(a)xij(b) = xij(a+ b)
xij(a)xkl(b)xij(a)−1xkl(b)−1 = 1, j 6= k, i 6= l
xij(a)xjk(b)xij(a)−1xjk(b)−1 = xik(ab), i, j, k distinct
Since the generators of the group of elementary matrices En(R) satisfy these same relations, we
have a map Stn(R)→ En(R), and hence a natural map φ on the inductive limits St(R) and E(R).
Definition 2.2.7. K2(R) = ker(φ : St(R)→ E(R)).
In fact, St(R) is the universal central extension of E(R), and K2(R) is isomorphic to the
second homology group H2(E(R),Z) [50, Theorem 4.2.7 and Corollary 4.2.10].
Next we define specific elements in K2(R) known as Steinberg symbols, which turn out to be
quite useful. For example, K2 of a field is generated by its Steinberg symbols. For any a ∈ R× and
i 6= j, define elements wij(a), hij(a) ∈ St(R) by
wij(a) = xij(a)xji(−a−1)xij(a), hij(a) = wij(a)wij(−1).
Then we have
φ(w12(a)) =
0 a
−a−1 0
and φ(h12(a)) =
a 0
0 a−1
.
18
Definition 2.2.8. For commuting units a, b ∈ R, the Steinberg symbol a, b ∈ K2(R) is
a, b = h12(a)h13(b)h12(a)−1h13(b)−1.
Proposition 2.2.9 (see Lemma 4.2.14 and Theorem 4.2.17 of [50]). For any units a, a′, and b in
R such that b commutes with a and a′, the following identities hold:
(1) a, b = b, a−1
(2) aa′, b = a, ba′, b
(3) a, 1− a = 1, whenever 1− a is invertible
(4) a,−a = a, 1 = 1
The boundary map ∂ : K2(R/J) −→ K1(R, J) is defined as follows. Any element u ∈ K2(R/J)
can be expressed in terms of generators xij(uk) ∈ St(R/J). We obtain an element r ∈ St(R) by
lifting each uk ∈ R/J to an element rk ∈ R. Then φ(r) ∈ GL(R, J), and we define ∂(u) to be the
image of φ(r) in K1(R, J). One then checks that this is independent of the choice of lifts.
2.2.4 The determinant invariant
In the case when R = B, the ring of bounded linear operators on a Hilbert space, and
J = L1, the ideal of trace class operators, the Fredholm determinant induces a surjective group
homomorphism
det : K1(B,L1)→ C×.
Indeed, let g ∈ GL(B) and let h ∈ E(B,L1). Then h is represented by a determinant class operator,
and by Proposition 2.1.12,
det(ghg−1h−1) = det(ghg−1) det(h−1).
The first factor is deth, again by Proposition 2.1.12, so det(ghg−1h−1) = 1 as desired. In fact,
K1(B,L1) = V ⊕ C×, where V is the additive group of a vector space of uncountable linear
dimension, and det is the projection onto the second factor [1].
19
Definition 2.2.10. For any commuting units a, b ∈ B/L1, the determinant invariant d(a, b) is the
nonzero number
d(a, b) = det ∂a, b.
In particular, d(a, b) satisfies the relations in Proposition 2.2.9.
In calculating the determinant invariant, a ∈ B/L1 is lifted to an operator A ∈ B, which is
necessarily Fredholm. Moreover, a−1 is lifted to a parametrix Q of A modulo trace class. Here,
and in the sequel, a parametrix modulo an ideal is an inverse modulo that ideal. Thus I −AQ and
I − QA are trace class operators. When a and b have invertible lifts, Brown made the following
observation in [8]:
Proposition 2.2.11. Let a and b be commuting units in B/L1. If a and b have invertible lifts
A ∈ B and B ∈ B, respectively, then d(a, b) = detABA−1B−1. In particular, the determinant of
the multiplicative commutator depends only on the Steinberg symbol a, b.
Proof. Since a−1 and b−1 can be lifted to A−1 and B−1, we have
d(a, b) = det ∂ h12(σ(A))h13(σ(B))h12(σ(A))−1h13(σ(B))−1
= det
A 0 0
0 A−1 0
0 0 I
B 0 0
0 I 0
0 0 B−1
A 0 0
0 A−1 0
0 0 I
−1
B 0 0
0 I 0
0 0 B−1
−1
= detABA−1B−1
Example 2.2.12. Suppose A = expα and B = expβ for operators α and β with [α, β] ∈ L1. Then
A and B are invertible and [A,B] ∈ L1. By the Helton-Howe-Pincus formula (Proposition 2.1.13),
d(a, b) = exp tr [α, β].
Lemma 2.2.13. For any commuting units ai and bi in B/L1, i = 1, 2, we have
d(a1 ⊕ a2, b1 ⊕ b2) = d(a1, b1) d(a2, b2).
20
Proof. In calculating the boundary map, ai is lifted to a Fredholm operator Ai, and a−1i is lifted to
any parametrix Qi of Ai modulo trace class. Then one may lift a1⊕a2 to A1⊕A2, and (a1⊕a2)−1
to Q1 ⊕ Q2. Similarly for bi, b1 ⊕ b2, and (b1 ⊕ b2)−1. The result then follows since determinants
are multiplicative over direct sums.
In fact, the determinant invariant can always be calculated in terms of a multiplicative
commutator. To see this, let a, b ∈ B/L1 be invertible commuting elements, and pick lifts A and B
in B of a and b, respectively. Let QA be an operator with index opposite that of A. For example, we
may take QA to be a parametrix for A, a unilateral shift, or the adjoint A∗. Pick QB similarly. Then
A⊕QA⊕I has index zero, so we may pick a finite rank operator FA such that A = A⊕QA⊕I+FA
is invertible. Form B = B ⊕ I ⊕ QB + FB similarly. Then [A, B] ∈ L1(H ⊕ H ⊕ H), and by the
preceding lemma, we immediately obtain the following:
Proposition 2.2.14. With A and B as above, one has
d(a, b) = det(ABA−1B−1).
Example 2.2.15. Let H be a Hilbert space with orthonormal basis ei∞i=0, and let S be the uni-
lateral shift Sei = ei+1. Let F2 and F3 be the rank one operators on H⊕H⊕H such that
F2(0, e0, 0) = (e0, 0, 0) and F3(0, 0, e0) = (e0, 0, 0).
Then S⊕S∗⊕I+F2 and S⊕I⊕S∗+F3 are invertible. By the preceding proposition, one calculates
d(S + L1, S + L1) = −1.
By replacing the first argument by S∗S, Proposition 2.2.9 (2) and (4) imply that d(S∗, S) = −1.
Similarly, d(S∗, S∗) = −1. By a similar technique, one may show more generally that for any
invertible a ∈ B/L1,
d(a, a) = (−1)ind a
where ind a is the index of any lift of a to B.
21
2.3 Toeplitz operators
A special class of operators known as Toeplitz operators will provide a rich source of examples
in this thesis. For example, their index and spectral properties are particularly well-understood
(Section 2.3.1). Furthermore, as we shall see in Section 2.3.2, Toeplitz operators are prototypical
examples of operators with trace class commutators. Our reference in this section is [24, Chapter
7].
To begin, consider the Hilbert space L2(S1) of square-integrable functions on the unit circle
S1 with respect to the Lebesgue measure. The trigonometric monomials
zn, −∞ < n <∞
form a basis for L2(S1). The Hardy space H2(S1) is the closed span of the basis vectors
zn, n ≥ 0.
Let P : L2(S1)→ H2(S1) be the orthogonal projection. The Hardy space can be characterized in
a number of other ways, for example, as the closure in L2(S1) of the boundary values of functions
that are continuous on the closed unit disk and holomorphic on the interior.
Definition 2.3.1. Let f ∈ L∞(S1). The Toeplitz operator Tf : H2(S1)→ H2(S1) is the compres-
sion to H2(S1) of multiplication by f :
Tf (φ) = P (fφ).
One may check that Toeplitz operators satisfy the following properties:
(1) ‖Tf‖ = ‖f‖∞
(2) Tλf+g = λTf + Tg
(3) T ∗f = Tf
Example 2.3.2. The Toeplitz operator Tz acts as a unilateral shift on H2(S1) since
Tz(zn) = zn+1.
22
2.3.1 Index and spectral theory
Let T be the C∗-algebra generated by the compact operators and Toeplitz operators Tf with
continuous symbol f . Equivalently, T is the C∗-algebra generated by the unilateral shift Tz. Thus
we have a C∗-algebra extension of C(S1) by the compact operators:
0→ K → T → C(S1)→ 0.
The following index theorem may be interpreted in terms of the boundary map of the resulting
six-term cyclic exact sequence in K-theory, although it long predates these developments.
Theorem 2.3.3 (F. Noether). A Toeplitz operator Tφ is Fredholm if and only if φ is continuous
and non-vanishing. In this case, the index of Tφ is opposite the winding number of φ:
indTφ = − 1
2πi
∫dφ
φ.
This formula can be verified, for example, by calculating the commutator of a Toeplitz opera-
tor with a shift operator, approximating continuous functions by polynomials, and using homotopy
invariance of the index. See [34] for more details.
We will need the following well-known results on the spectra of multiplication operators and
Toeplitz operators in Chapter 5.
Proposition 2.3.4.
(1) If φ ∈ L∞(S1), the spectrum of the multiplication operator φ ∈ B(L2) is the essential range
of φ.
(2) If φ ∈ C(S1), the essential spectrum of the Toeplitz operator Tφ ∈ B(H2) is the range
φ(S1). The spectrum of Tφ in addition consists of the connected components of C− φ(S1)
about which φ has nonzero winding number.
2.3.2 Commutators of Toeplitz operators
Lemma 2.3.5. If φ ∈ L∞(S1) is in the Sobolev space W12,2(S1) = H
12 (S1), then
(I − P )φP, Pφ(I − P ), [φ, P ] ∈ L2(L2(S1)).
23
with
‖[φ, P ]‖2 ≤ ‖φ‖W
12 ,2(S1)
.
Proof. Write φ =∑cnz
n. A straightforward calculation shows that (I − P )φP ∈ L2, with
‖(I − P )φP‖22 =∑n>0
n|cn|2,
By taking adjoints, Pφ(I − P ) ∈ L2 as well, with
‖Pφ(I − P )‖22 = −∑n<0
n|cn|2.
Hence [φ, P ] = (I − P )φP − Pφ(I − P ) ∈ L2, and
‖[φ, P ]‖22 =∑n 6=0
|n||cn|2.
In this case, Toeplitz operators have trace class commutators, and the Berger-Shaw formula
calculates this trace:
Theorem 2.3.6. If f, g ∈ L∞(S1) ∩W12,2(S1), then [Tf , Tg] ∈ L1. If f, g ∈ C1(S1), then
tr[Tf , Tg] =1
2πi
∫f dg.
Proof. First notice that [Tf , Tg] = Pg(I − P )fP − Pf(I − P )gP . Both terms are trace class since
they are products of two operators which are Hilbert-Schmidt by the preceding lemma. The trace
formula then follows by writing f and g in the basis einθ.
Example 2.3.7. For example, if f and g are smooth functions on the unit circle, then the Toeplitz
operators Tef and Teg are Fredholm and commute modulo L1. Then Tef = expTf and Teg = expTg
modulo L1 by Proposition 5.3.12, and one may use the Helton-Howe-Pincus formula and the Berger-
Shaw formula to show that
d(Tef + L1, Teg + L1) = exp1
2πi
∫f dg.
Chapter 3
Joint torsion
The goal of this chapter is to define an invariant known as joint torsion, and to do so in a
way that will be most convenient for our purposes. Moreover, we establish some basic results that
we will need in Chapter 4 to prove that joint torsion agrees with Brown’s determinant invariant.
Since joint torsion was introduced by Carey and Pincus, let us begin by discussing some of their
seminal work in this area.
3.1 Background: The work of Carey and Pincus
In a series of papers beginning with [11], Carey and Pincus carried out a study of almost
normal operators, that is, operators T with trace class self-commutator: [T, T ∗] ∈ L1. Notice that
we may write T = X + iY , for self-adjoint operators
X =T + T ∗
2and Y =
T − T ∗
2i
and [X,Y ] ∈ L1. Conversely, if X and Y are self-adjoint operators with [X,Y ] ∈ L1, then T =
X + iY is almost normal. For any polynomial p ∈ C[x, y], the operator p(X,Y ) is well-defined
modulo L1. (One may instead form p(T, T ∗) and carry out an equivalent analysis.) If q is another
polynomial, then the commutator [p(X,Y ), q(X,Y )] has a well-defined trace. Helton and Howe
showed in [32] that there exists a measure mT supported on σ(T ) such that
tr[p(X,Y ), q(X,Y )] =1
2πi
∫p, q dmT
25
where p, q denotes the Poisson bracket
∂p
∂x
∂q
∂y− ∂p
∂y
∂q
∂x.
Moreover, mT is absolutely continuous with respect to Lebesgue measure, and its Radon-Nikodym
derivative is the Pincus principal function [12]. In the case of the unilateral shift, for example, the
Pincus principal function is the characteristic function of the unit disk.
Brown’s determinant invariant provides a multiplicative analogue of the trace of commuta-
tors considered above via the Helton-Howe-Pincus formula. Thus Carey and Pincus turned their
attention to the determinant invariant, first for commuting operators A and B with trivial joint
Koszul homology. As we alluded to in the introduction, they use intricate linear algebra arguments
and a careful choice of parametrices for A and B in [13] to calculate the determinant invariant
d(A+ L1, B + L1) in terms of multiplicative Lefschetz numbers
det(B|kerA)
det(B|cokerA)
det(A|cokerB)
det(A|kerA).
The above expression is not necessarily defined when A and B have nontrivial joint Koszul
homology, even though the determinant invariant is defined. Carey and Pincus were thus led to
introduce a homological invariant known as joint torsion, which generalizes the above expression
to the singular case. In [15] they express the joint torsion of Toeplitz operators Tf and Tg with
f, g ∈ H∞ as a product of tame symbols
∏|a|<1
ca(f, g).
Here the tame symbol ca(f, g) of any meromorphic functions f and g is defined to be the value at
a of the function
(−1)dega(f)·dega(g) fdega(g)
gdega(f).
In [16] Carey and Pincus extend joint torsion to operators A and B with trace class com-
mutator. They use this invariant to obtain generalizations of the classical Szego limit theorems
on the asymptotic behavior of determinants of Toeplitz matrices. Let us begin by recalling the
26
groundbreaking work of Szego in [53]. Let Pn : H2(S1) → H2(S1) be the orthogonal projection
onto the first n basis vectors zk, 0 ≤ k < n. If φ ∈ L∞(S1) with Fourier coefficients φ(k), then
the truncated Toeplitz operator PnTφPn is expressed as a Toeplitz – that is, constant diagonal –
matrix
φ(0) φ(1) φ(2) · · · φ(n− 1)
φ(−1) φ(0) φ(1) · · · φ(n− 2)
φ(−2) φ(−1) φ(0) · · · φ(n− 3)
......
.... . .
...
φ(−n+ 1) φ(−n+ 2) φ(−n+ 3) · · · φ(0)
If φ > 0 and log φ ∈ L1(S1), then Szego showed that
limn→∞
PnTφPnPn−1TPn−1
= exp1
2πi
∫log φ.
Let G(φ) denote the above quantity. If in addition φ′ is Lipschitz continuous, then Szego showed
in [54] that
limn→∞
PnTφPnG(φ)n
= exp∞∑k=1
k log f(k) log f(−k)
in response to a question of Onsager and Yang on spontaneous magnetization in the Ising model
in statistical mechanics.
In [16] Carey and Pincus investigate the more general case when φ has nonzero winding
number. They found that quantities like Brown’s determinant invariant appeared quite naturally
in their analysis. For example,
PnTφPnd(Tφ + L1, Sn+1 + L1)
= det(TφS
∗n+1QφSn+1 + T ∗φS
n+1).
where S = Tz is the unilateral shift and Qφ is a parametrix for Tφ. To aid in these calculations,
Carey and Pincus extended their joint torsion to the case of Toeplitz operators, and more generally,
Fredholm operators that commute modulo the trace class. Joint torsion may be viewed heuristically
as a replacement for the logarithm in Szego’s limit theorem. This invariant is the subject of the
present chapter.
27
3.2 Commuting operators
3.2.1 Algebraic torsion
To define their new invariant, Carey and Pincus use the notion of algebraic torsion, which
we describe in this section. Let (V•, d•) be an exact sequence of finite dimensional vector spaces,
0 −→ Vndn−→ Vn−1
dn−1−−−→ Vn−2 −→ · · · −→ V0 → 0.
Denote by detVk the top exterior power ΛdimVkVk, and define the determinant line
detV• = detV ∗n ⊗ detVn−1 ⊗ detV ∗n−2 ⊗ . . .
For each k, pick a nonzero element tk ∈ Λrank dkVk such that dktk 6= 0. By exactness, dktk ∧ tk−1 is
a nonzero element of detVk−1.
Definition 3.2.1. The torsion τ(V•, d•) of the complex (V•, d•) is the volume element
τ(V•, d•) = (tn)∗ ⊗ dntn ∧ tn−1 ⊗ (tn−2 ∧ dn−1tn−1)∗ ⊗ dn−2tn−2 ∧ tn−3 ⊗ . . .
This is nonzero and is independent of the choices of the tk [41]. Hence τ(V•, d•) defines a
canonical generator of detV•.
For a finite dimensional vector space V , we will make frequent use of the isomorphism detV ∗⊗
detV ∼= C given by
(v∗1 ∧ v∗2 ∧ . . . )⊗ (w1 ∧ w2 ∧ . . . ) 7→ det (v∗i (wj)) . (3.1)
For finite dimensional vector spaces V and W , we also identify the line detV ⊗detW with detW ⊗
detV according to the rule
s⊗ t 7→ (−1)dimV ·dimW t⊗ s. (3.2)
See [40] for a formulation of torsion in terms of the Picard category of graded lines, among other
insights.
Example 3.2.2. The torsion of an automorphism of a finite dimensional vector space is its deter-
minant.
28
3.2.2 Joint torsion of commuting operators
For a collection of commuting operators, Carey and Pincus [15] and Kaad [37] have defined
invariants known as joint torsion. This section reviews their constructions. We begin with the
notion of a mapping cone from homological algebra. Thus, let (V•, dV• ) and (W•, d
W• ) be chain
complexes, i.e.
dVk−1dVk = dWk−1d
Wk = 0
and let f• : V• →W• be a morphism of chain complexes, i.e.
fk−1dVk = dWk fk.
Then the mapping cone (C(f)•, dC• ) is the chain complex with C(f)k = Vk−1 ⊕Wk and
dCk =
−dVk−1 0
fk−1 dWk
.
Now let A = (A1, . . . , An) be a collection of commuting operators on a vector space H. The
Koszul complex K•(A) is the chain complex with
Ki(A) = H⊗ ΛiCn
and differential di : Ki(A)→ Ki−1(A) given by
di =n∑k=1
Ak ⊗ ε∗k
where εk : ΛiCn → Λi+1Cn is the operation of exterior multiplication by the unit vector ek, and ε∗k
is its adjoint. The n-tuple A is said to be Fredholm if (K•(A), d•) has finite dimensional homology.
We also have maps ιk : ΛiCn → ΛiCn+1 induced by the inclusion
Cn → Cn+1, (a1, . . . , an) 7→ (a1, . . . , ak−1, 0, ak, . . . , an).
For any j ∈ 1, . . . , n, let j(A) = (A1, . . . , Aj , . . . , An). Since A = (A1, . . . , An) is commu-
tative, each Aj defines a morphism of chain complexes Aj := Aj ⊗ 1 : K•(j(A))→ K•(j(A)). The
29
mapping cone of this morphism is isomorphic to the Koszul complex K•(A) via the isomorphismι∗jε∗jι∗j
: ΛkCn ∼−→ Λk−1Cn−1 ⊕ ΛkCn−1.
We thus obtain a triangle of chain complexes
K•(j(A))→ K•(j(A))→ K•(A)→ K•(j(A))[1]
and hence a long exact sequence in homology
Ej : 0→ Hn(A)→ Hn−1(j(A)) −→ Hn−1(j(A))→ Hn−1(A)→ . . .
The torsion vector τ(Ej) ∈ detH•(A)⊗detH•(j(A))∗⊗detH•(j(A)). Here, and in the sequel,
we use the notation
detH•(A) = detHn(A)∗ ⊗ detHn−1(A)⊗ . . .
for a sequence of homology spaces H•(A). We regard τ(Ej) ∈ detH•(A) using the isomorphism in
(3.1) applied to the line detH•(j(A)). Carrying out the same process for some i 6= j yields an exact
sequence Ei. If i(A) and j(A) are Fredholm, then Ei and Ej consist of finite dimensional vector
spaces, so we can form two torsion vectors τ(Ei), τ(Ej) ∈ detH•(A). Then τ(Ei) ⊗ τ(Ej)∗ can be
identified with a nonzero scalar, which up to a sign is the joint torsion defined by Carey and Pincus
for n = 2 [15] and by Kaad for n ≥ 2 [37].
3.2.3 The case of two commuting operators
Let us describe this construction more explicitly in the case n = 2. Thus, let A1 = A and
A2 = B be two commuting Fredholm operators. Upon choosing bases for the exterior algebras, we
have Koszul complexes
K•(A) : H A−→ H, K•(B) : H B−→ H
K•(A,B) : H
(−BA
)−−−−→ H2 (A B )−−−−→ H
Then H0(A) = cokerA, H1(A) = kerA, H2(A,B) = kerA∩kerB, H0(A,B) = H/(AH+BH), and
H1(A,B) =(y, z) |Ay +Bz = 0(−Bx,Ax) |x ∈ H
.
30
The two exact sequences EA and EB are given by
EA : 0 −→ (kerA ∩ kerB)−ι−→ kerB
−A−−→ kerB−ι2∗−−−→ H1(A,B)→
π1∗−−→ cokerBA−→ cokerB
π−→ H/(AH+BH) −→ 0
EB : 0 −→ (kerA ∩ kerB)ι−→ kerA
−B−−→ kerA−ι1∗−−−→ H1(A,B)→
π2∗−−→ cokerAB−→ cokerA
π−→ H/(AH+BH) −→ 0
The map ιk∗ is induced by inclusion into the k-th coordinate, and πk∗ is induced by projection onto
the k-th coordinate. Here we have chosen signs to simplify formulas later.
Definition 3.2.3. The joint torsion τ(A,B) of A and B is the nonzero scalar
τ(A,B) = (−1)λ(A,B) τ(EA)⊗ τ(EB)∗.
Here λ(A,B) = sgn θ + γ, where θ is the permutation that maps the factors in (4.6) to (4.5)
and γ is given by (4.7). See Remark 4.1.3. Definition 3.3.4 deals with the more general situation,
i.e. when [A,B] is not necessarily zero, but we may write AB = CD. We note that our sign
conventions differ from those of [16]. In the above definition, τ(A,B) is identified with a scalar
according to (3.1) and (3.2).
Lemma 3.2.4. If Hi(A,B) = 0 for i = 0, 1, 2, then we have
τ(A,B) =detB|kerA
detB|cokerA
detA|cokerB
detA|kerB.
In particular, τ(A, I) = τ(I, A) = 1.
Proof. Since Hi(A,B) = 0, the exact sequence EA breaks up into the isomorphisms
−A|kerB : kerB → kerB and A|cokerB : cokerB → cokerB
and EB breaks up as
−B|kerA : kerA→ kerA and B|cokerA : cokerA→ cokerA.
31
Joint torsion is the alternating product of the torsion vectors of these isomorphisms, which are
simply the determinants by Example 3.2.2. The sign (−1)λ(A,B) is calculated directly.
Example 3.2.5. Joint torsion generalizes the Fredholm index. Indeed, by Lemma 3.2.4,
τ(A, expB) =det expB|kerA
det expB|cokerA
The ratio on the right hand side is seen to be exp tr(B|kerA−B|cokerA), which is the exponential of
the Lefschetz number of B as an endomorphism of the chain complex K•(A). By taking B = I, we
obtain exp indA.
3.3 Almost commuting operators
One might wish to calculate joint torsion for operators A and B that do not necessarily
commute with one another. The difficulty is that there is no longer a well-defined Koszul complex
K•(A,B). Carey and Pincus circumvent this problem in the case n = 2 by introducing auxiliary
operators C and D that are perturbations of the original operators A and B. Thus, consider two
Fredholm operators A and B with [A,B] ∈ L1. Furthermore, assume the existence of operators C
and D such that AB = CD,A −D ∈ L1, and B − C ∈ L1. (See Appendix A for a discussion of
the existence of such perturbations.) Thus, one has the following commutative diagram:
H C−−−−→ H
D
x xAH B−−−−→ H
By analogy with the commuting case, consider the mapping cone K•(A,B,C,D) of the
vertical chain map (A,D). Explicitly,
K•(A,B,C,D) : H
(−BD
)−−−−→ H2 (A C )−−−−→ H
This yields a triangle of modified Koszul complexes, and hence a long exact sequence in homology:
EA,D : 0 −→ (kerB ∩ kerD)−ι−→ kerB
−D−−→ kerC−ι2∗−−−→ H1(A,B,C,D)→
π1∗−−→ cokerBA−→ cokerC
π−→ H/(AH+ CH) −→ 0
32
The homology space H1(A,B,C,D) is given by
H1(A,B,C,D) =(y, z) |Ay + Cz = 0(−Bx,Dx) |x ∈ H
.
The map ι2∗ is induced by inclusion into the second coordinate, and π1∗ is induced by projection
onto the first coordinate. We have again picked signs to simplify formulas later. Likewise, there is
a mapping cone of the horizontal chain map (B,C), and hence a long exact sequence
EB,C : 0 −→ (kerB ∩ kerD)ι−→ kerD
−B−−→ kerA−ι1∗−−−→ H1(A,B,C,D)→
π2∗−−→ cokerDC−→ cokerA
π−→ H/(AH+ CH) −→ 0
However, τ(EA,D)⊗τ(EB,C)∗ ∈ detH•(A)⊗detH•(D)∗⊗detH•(B)⊗detH•(C)∗ is no longer
canonically identified with a scalar. To obtain a scalar, Carey and Pincus introduce canonical
generators of these determinant lines, known as perturbation vectors [16, Section 3]. We will find
it convenient to give a slightly different definition. Proposition 3.3.2 below verifies that these two
definitions agree.
3.3.1 Perturbation vectors
Let A and A′ be Fredholm operators such that A−A′ ∈ L1. First assume that A (and hence
also A′) has index zero. Let L and L′ be trace class operators such that
A+ L and A′ + L′ are invertible (3.3)
L(kerA) ∩ imA = L′(kerA′) ∩ imA′ = 0 (3.4)
Then L and L′ induce isomorphisms
πL = π L|kerA : kerA∼−→ cokerA
π′L′ = π′ L′|kerA′ : kerA′∼−→ cokerA′
where π and π′ are the quotient maps by imA and imA′, respectively. Let τ(πL) and τ(π′L′) be
the torsion vectors of the above isomorphisms. Let P : H → imA be the continuous projection
33
along L(kerA). Then ker((I − P )L) ∩ kerA = 0, and hence A defines an isomorphism
ker((I − P )L)→ imA
since A is assumed to have index zero. Let A† be the inverse of this map, followed by the inclusion
into H. Define
D(L) = det(I +A†PL).
Now in general, if the index of A and A′ is possibly nonzero, let Q be a Fredholm operator
with index negative that of A. Then A⊕Q and A′⊕Q are Fredholm operators with index zero on
H⊕H. Choose L and L′ as above, now for A⊕Q and A′ ⊕Q, respectively. Since (A⊕Q+ L)−1
is a parametrix for (A′ ⊕ Q + L′) modulo L1(H ⊕ H), we obtain an invertible determinant class
operator
Σ = (A⊕Q+ L)−1(A′ ⊕Q+ L′).
Definition 3.3.1. The perturbation vector σA,A′ ∈ detH(A)⊗ detH(A′)∗ is defined by
σA,A′ = D(L)D(L′)−1 det Σ · τ(πL)⊗ τ(π′L′)∗.
Proposition 3.3.2. The above definition agrees with that of Carey and Pincus [16, Section 3,
Equation 41]. In particular, σA,A′ is independent of the choices of Q, L, and L′.
Proof. First we note that this definition is exactly Carey and Pincus’s perturbation vector σA⊕Q,A′⊕Q
for A ⊕ Q and A′ ⊕ Q, provided that we choose L so that its range is linearly independent from
that of A⊕Q, and similarly for L′.
Now suppose that L and L′ only satisfy (3.3) and (3.4). Then (I − P )L and (I − P ′)L′ are
as in the preceding paragraph, where P is the projection onto im(A ⊕ Q) along L(ker(A ⊕ Q)).
Moreover, one calculates
(A⊕Q+ (I − P )L)−1 (A⊕Q+ L) = I + (A⊕Q)†PL
the determinant of which is D(L), and similarly for D(L′). Hence we find that this definition agrees
with Carey and Pincus’s definition of σA⊕Q,A′⊕Q. In particular, it is independent of the choices of
L and L′ by [16, Theorem 11].
34
Next we show that specific choices of L and L′ (and hence all choices) recover Carey and
Pincus’s perturbation vector σA,A′ . First assume indA ≥ 0. Write kerA = X0 ⊕ X1 with
dim cokerA = dimX0, and write imQ⊥ = X0 ⊕ X1 with dim kerQ = dim X0. Choose sub-
spaces X ′0, X′1 for A′ similarly. Pick isomorphisms L11 : X0 −→ imA⊥, L′11 : X ′0 −→ imA′⊥,
L22 : kerQ −→ X0, L21 : ker (A + L11) → im (Q + L22)⊥, and N : ker (A′ + L′11) → ker (A + L11).
Extend these operators by zero orthogonally to all of H. Define the operators
L =
L11 0
L21 L22
L′ =
L′11 0
L21N L22
Then A⊕Q+ L and A′ ⊕Q+ L′ are invertible. Moreover,
Σ = (A⊕Q+ L)−1(A′ ⊕Q+ L′)
=
(A+ L11)R L−121
0 (Q+ L22)L
A′ + L′11 0
L21N Q+ L22
so det Σ = det
((A+ L11)R(A′ + L′11) +N
). Here, (A + L11)R denotes the right inverse such that
(A+ L11)R(A+ L11) is the orthogonal projection onto X⊥1 and similarly for the left inverse. This
agrees with the corresponding term in Carey and Pincus’s definition. The factors associated with
Q in τ(πL) and τ(π′L′) cancel by the identity v∗(v) = 1. We are then left with the definition of
Carey and Pincus.
The case when indA ≤ 0 is similar. The last statement in the proposition follows from [16,
Theorem 11].
In particular, we have the following recipe for perturbation vectors in the case of index zero
operators:
Lemma 3.3.3. Let A and A′ be index zero Fredholm operators with A − A′ ∈ L1. Let π : H →
cokerA and π′ : H → cokerA′ be the quotient maps. Take L and L′ to be trace class operators
satisfying (3.3) and (3.4). Then we have
σA,A′ = det(A+ L)−1(A′ + L′) · τ(πL)⊗ τ(π′L′)∗.
35
For example, let ZA be a subspace complementary to imA, and let F : kerA → ZA be
any isomorphism. Then we may take L to be the extension of F by zero to all of H along any
complement of kerA. We may define L′ similarly.
3.3.2 Joint torsion of two almost commuting operators
Definition 3.3.4. Suppose A,B,C, and D are Fredholm operators with AB = CD,A −D ∈ L1,
and B − C ∈ L1. The joint torsion τ(A,B,C,D) is the nonzero scalar
τ(A,B,C,D) = (−1)λ(A,B,C,D)τ(EA,D)⊗ σA,D ⊗ σB,C ⊗ τ(EB,C)∗.
Here λ(A,B,C,D) = sgn θ+ γ, where θ is the permutation that maps the factors in (4.6) to
(4.5) and γ is given by (4.7). See Remark 4.1.3. Once again, we note that our sign conventions
differ from those of [16] and that τ(A,B,C,D) has been identified with a scalar according to (3.1)
and (3.2).
If A and B commute, then τ(A,B,B,A) = τ(A,B) since σA,A and σB,B are trivial. By
Proposition 3.3.2 and the exact sequences EA,D and EB,C , we find that the above definition agrees
with that of Carey and Pincus [16, Section 5, Equation 51], up to the evident difference in our sign
conventions. By the multiplicativity of the determinant, we have the following:
Lemma 3.3.5. For any Fredholm operators Ai, Bi, Ci, Di with AiBi = CiDi, Ai − Di ∈ L1, and
Bi − Ci ∈ L1, i = 1, 2, we have
τ(A1 ⊕A2, B1 ⊕B2, C1 ⊕ C2, D1 ⊕D2) = τ(A1, B1, C1, D1) · τ(A2, B2, C2, D2).
Remark 3.3.6. For n ≥ 2 almost commuting operators, one can define joint torsion by introducing
auxiliary operators as in Definition 3.3.4 and using the constructions in [39]. It is not known in
what generality this can be done. In fact, not every pair of almost commuting Fredholm operators
A and B have trace class perturbations D and C, respectively, such that AB = CD. See Appendix
A for a more detailed discussion.
Chapter 4
Equality
In this chapter we show that joint torsion is equal to the determinant invariant (Theorem
4.3.3). We begin by showing that joint torsion is trivial in finite dimensions. Next we establish
factorization results for joint torsion under actions by invertible operators in Section 4.2. Then
in Section 4.3, we will see that the proof of equality follows quickly from the preceding sections.
Finally, we obtain a number of consequences of Theorem 4.3.3, many of which will be used in the
following chapter. Most of the results of this chapter have appeared in the author’s work [45].
4.1 The finite dimensional case
In this section we show that joint torsion is trivial in a finite dimensional space, closely
following Kaad [37]. In fact, for commuting operators, this is a special case of [37, Theorem 4.3.3].
Later we will see that joint torsion is also trivial for operators in the coset I + L1. First consider
the exact sequence
0 −→ kerA −→ H A−→ H −→ cokerA −→ 0.
If H is finite dimensional, the torsion vector τ(A) of the above sequence is defined.
Lemma 4.1.1. If A and D are linear operators on a finite dimensional space H, then
σA,D = τ(A)⊗ τ(D)∗.
Proof. First we calculate the perturbation vector σA,D as
σA,D = det(A+ LA)−1(D + LD) · τ(πALA)⊗ τ(πDLD)∗.
37
where LA, LD, πA, and πD are as in Lemma 3.3.3.
On the other hand, choose any nonzero t0 and t1 with t0 ∈ det kerA and, for example,
t1 ∈ det (kerA)⊥. Then we calculate
τ(A) = (t0)∗ ⊗ t0 ∧ t1 ⊗ (LAt0 ∧At1)∗ ⊗ (πALAt0).
We move the fourth factor past the middle two factors according to (3.2) (applied twice) and obtain
τ(πALA). For the middle factors, we calculate
t0 ∧ t1 ⊗ (LAt0 ∧At1)∗ = (−1)dimH ((A+ LA)(t0 ∧ t1))∗ ⊗ (t0 ∧ t1)
= (−1)dimH det(A+ LA)−1
The same calculation for τ(D) completes the proof.
4.1.1 Torsion of a double complex
Now consider a double complex X•• consisting of finite dimensional vector spaces Xij , 0 ≤
i ≤ m, 0 ≤ j ≤ n, with exact rows (Xi•, h) and exact columns (X•j , v), where h : Xi• → Xi,•+1
and v : X•j → X•+1,j . Assume that the horizontal and vertical maps h and v anti-commute, i.e.
vh+ hv = 0.
For each i, one may form the torsion vector τ(Xi•) of row i and combine all these to obtain the
horizontal torsion vector
τh = τ(X0•)⊗ τ(X1•)∗ ⊗ . . .
One may also form the vertical torsion vector
τv = τ(X•0)⊗ τ(X•1)∗ ⊗ . . .
Both of these vectors are generators of the same determinant line of X••, and moreover by the work
of F. F. Knudsen and D. Mumford [41], they agree up to a sign.
38
One rather direct proof of this fact begins by noticing that each component Xi,j of the double
complex is isomorphic to
im(vh)⊕ ker v ∩ kerh
im(vh)⊕ ker v
ker v ∩ kerh⊕ kerh
ker v ∩ kerh⊕ ker(vh)
ker v + kerh⊕ Xi,j
ker(vh)(4.1)
Here, and below, the indices on v and h may be inferred, and hence are suppressed. We note of
course that some of these spaces may be trivial. The vertical map v induces isomorphisms
kerh
ker v ∩ kerh∼= im(vh)
ker(vh)
ker v + kerh∼=
ker v ∩ kerh
im(vh)
Xi,j
ker(vh)∼=
ker v
ker v ∩ kerh
Similarly, h induces isomorphisms
ker v
ker v ∩ kerh∼= im(vh)
ker(vh)
ker v + kerh∼=
ker v ∩ kerh
im(vh)
Xi,j
ker(vh)∼=
kerh
ker v ∩ kerh
For each Xi,j , one may pick representative subspaces for each of the six quotients in (4.1) above.
Thus:
X1i,j = im(vh) (4.2)
X2i,j is a complement of im(vh) in ker v ∩ kerh
X3i,j is a complement of ker v ∩ kerh in ker v
X4i,j is a complement of ker v ∩ kerh in kerh
X5i,j is a complement of ker v + kerh in ker(vh)
X6i,j is a complement of ker(vh) in Xi,j
These give an algebraic decomposition of Xi,j in which X1i,j , X
2i,j , X
3i,j span ker v and X4
i,j , X5i,j , X
6i,j
span a complement. Furthermore, X1i,j , X
2i,j , X
4i,j span kerh and X3
i,j , X5i,j , X
6i,j span a complement.
39
Next we note that given a decomposition of the spaces Xi,j on the n-th diagonal, i.e. i+j = n,
we may find a compatible decomposition for the (n+1)-st diagonal. Indeed, let i+ j = n+1. Then
we may take
X1i,j = v(X4
i−1,j) = h(X3i,j−1) (4.3)
X2i,j = v(X5
i−1,j) = h(X5i,j−1)
X3i,j = v(X6
i−1,j)
X4i,j = h(X6
i,j−1)
In (4.3), X5i−1,j and X5
i,j−1 have already been chosen in this way so that v(X5i−1,j) = h(X5
i,j−1).
Moreover, having chosen X5i−1,j+1, we may take X5
i,j to be a lift of of v(X5i−1,j+1) along h. Finally,
we may take X6i,j to be any complement of ker(vh) in Xi,j .
Thus, one may inductively pick generators for all of the spaces Xki,j in order to calculate both
the vertical and horizontal torsion according to Definition 3.2.1. The same generators are used,
only in a different order. Hence these torsion vectors are the same, up to the sign given by repeated
application of (3.2). The proof of the proposition below follows the same strategy.
4.1.2 Joint torsion in finite dimensions
Now let A, B, C, and D be operators on a finite dimensional vector space H such that
AB = CD. Consider the Koszul complexes K•(A), K•(B), K•(C), K•(D), and K•(A,B,C,D).
At the level of homology, we obtain the following bicomplex of finite dimensional vector spaces with
exact rows and columns. Write Hi = Hi(A,B,C,D) for the homology spaces.
40
0 0 0...y y y y−ι1∗
0 −−−−→ H2−ι−−−−→ kerB
−D−−−−→ kerC−ι2∗−−−−→ H1
π1∗−−−−→ · · ·yι yι yι yπ2∗
0 −−−−→ kerDι−−−−→ H D−−−−→ H π−−−−→ cokerD −−−−→ 0y−B yB yC yC
0 −−−−→ kerAι−−−−→ H −A−−−−→ H −π−−−−→ cokerA −−−−→ 0y−ι1∗ yπ yπ yπ
· · · −ι2∗−−−−→ H1π1∗−−−−→ cokerB
A−−−−→ cokerCπ−−−−→ H0 −−−−→ 0yπ2∗
y y y... 0 0 0
(4.4)
Here the upper right and lower left corners are identified. The above diagram therefore
consists of three exact rows and three exact columns. Two of the rows are the sequences from
Lemma 4.1.1 corresponding to D and A (up to a sign) and hence will be combined below to
form the perturbation vector σA,D. The other, seven term, row is the exact sequence EA,D in the
definition of joint torsion. Two of the columns correspond to B and C, and the other is the exact
sequence EB,C . Thus we obtain horizontal and vertical torsion vectors
τ(EA,D)∗ ⊗ τ(D)⊗ τ(A)∗
τ(EB,C)∗ ⊗ τ(B)⊗ τ(C)∗
The theorem below asserts that these agree, up to the signs in Definition 3.3.4 and Lemma 4.1.1.
We note that the above bicomplex arises from an odd homotopy exact bitriangle of Z2-graded
chain complexes that appears in [37, Equation 5.1]:
41
H −−−−→ H[1] −−−−→ K•(B) −−−−→ Hy y y yH[1] −−−−→ H −−−−→ K•(C)[1] −−−−→ H[1]y y y yK•(D) −−−−→ K•(A)[1] −−−−→ K•(A,B,C,D) −−−−→ K•(D)y y y yH −−−−→ H[1] −−−−→ K•(B) −−−−→ H
Here the vertical maps are induced by A and D, and the horizontal maps are induced by B and C.
The spaceH is given the grading with trivial odd part, and the notation X[1] denotes the Z2-graded
chain complex X with the grading reversed and the differential negated. Thus, the horizontal and
vertical arrows are odd chain maps which anticommute with the differential, and the squares are
anticommutative.
Theorem 4.1.2. If H is finite dimensional and A,B,C, and D are operators on H such that
AB = CD, then τ(A,B,C,D) = 1.
Proof. For convenience, let us temporarily rename the spaces in (4.4):
0 0 0...y y y y
0 −−−−→ X1,1 −−−−→ X1,2 −−−−→ X1,3 −−−−→ Y −−−−→ · · ·y y y y0 −−−−→ X2,1 −−−−→ X2,2 −−−−→ X2,3 −−−−→ X2,4 −−−−→ 0y y y y0 −−−−→ X3,1 −−−−→ X3,2 −−−−→ X3,3 −−−−→ X3,4 −−−−→ 0y y y y· · · −−−−→ Y −−−−→ X4,2 −−−−→ X4,3 −−−−→ X4,4 −−−−→ 0y y y y
... 0 0 0
42
As described in (4.2) and (4.3), we may pick generators
tkij ∈ detXki,j
for all of the spaces except the diagonal containing Y , that is, all spaces Xi,j with i+ j 6= 5. One
checks that we may also pick generators for subspaces
X1Y , X
3Y , X
4Y , X
6Y
X12,3, X
32,3, X
42,3, X
62,3
X13,2, X
33,2, X
43,2, X
63,2
In particular, we may pick all these generators to be compatible with one another in the sense of
(4.3). Moreover, there exist
x = t5Y ∈ detY 5, w = t523 ∈ detX52,3, z = t532 ∈ detX5
3,2
such that
v(z) = h(x), h(w) = v(x), h(z) = v(w)
for example by [37, Equations (4.5) and (4.6)]. Similarly we find that there exist compatible
generators
t2Y ∈ detY 2, t223 ∈ detX22,3, t232 ∈ detX2
3,2,
Having picked generators for the top exterior powers of all spaces in the bicomplex, we may
now use the columns to calculate τ(EB,C)⊗ σ∗B,C as:
(t611
)∗ ⊗ (vt611 ∧ t621
)⊗(t531 ∧ t631 ∧ vt612
)∗ ⊗ (vt531 ∧ vt631 ∧ ht613 ∧ t5Y)
(4.5)
⊗(ht632 ∧ vht613 ∧ vt5Y
)∗ ⊗ (vht632 ∧ ht633
)⊗(vht633
)∗⊗[ (ht611 ∧ t612
)∗ ⊗ (vht611 ∧ vt612 ∧ ht621 ∧ t522 ∧ t622
)⊗(ht631 ∧ t532 ∧ t632 ∧ vht621 ∧ vt522 ∧ vt622
)∗ ⊗ (vht631 ∧ vt532 ∧ vt632
) ]∗⊗(ht612 ∧ t513 ∧ t613
)∗ ⊗ (vht612 ∧ vt513 ∧ vt613 ∧ ht622 ∧ t523 ∧ t623
)⊗(ht632 ∧ t633 ∧ vht622 ∧ vt523 ∧ vt623
)∗ ⊗ (vht632 ∧ vt633
)
43
Likewise, we use the rows to calculate τ(EA,D)⊗ σA,D as:
(t611
)∗ ⊗ (ht611 ∧ t612
)⊗(t513 ∧ t613 ∧ ht612
)∗ ⊗ (vt531 ∧ ht613 ∧ vt631 ∧ t5Y)
(4.6)
⊗(vt632 ∧ hvt631 ∧ vt532
)∗ ⊗ (hvt632 ∧ vt633
)⊗(hvt633
)∗⊗[ (vt611 ∧ t621
)∗ ⊗ (hvt611 ∧ ht621 ∧ vt612 ∧ t522 ∧ t622
)⊗(vt613 ∧ t523 ∧ t623 ∧ hvt612 ∧ vt513 ∧ ht622
)∗ ⊗ (hvt613 ∧ vt5Y ∧ ht623
) ]∗⊗(vt621 ∧ t531 ∧ t631
)∗ ⊗ (vt632 ∧ t633 ∧ hvt622 ∧ vt523 ∧ ht632
)∗⊗(hvt621 ∧ vt522 ∧ ht631 ∧ vt622 ∧ t532 ∧ t632
)⊗(hvt632 ∧ ht633
)Note that we have accounted for the sign in the third row of (4.4) by switching the order of
the factors corresponding to X3,2 and X3,3. The vector (4.6) consists of the same factors as (4.5),
only in a different order. Let θ denote this permutation. In more detail, the spaces Xij appear in
a different order, and moreover the subspaces X3ij and X4
ij are switched. In addition hv = −vh, so
we find that (4.6) and (4.5) further differ by a factor of (−1)γ , where
γ = dimX12,2 + dimX1
2,3 + dimX12,4 + dimX1
3,2
+ dimX13,3 + dimX1
3,4 + dimX14,2 + dimX1
4,3 + dimX14,4 (4.7)
Setting λ(A,B,C,D) = sgn θ + γ, we have
τ(EA,D)⊗ σA,D = (−1)λ(A,B,C,D)τ(EB,C)⊗ σ∗B,C
so that τ(A,B,C,D) = 1 as desired.
Remark 4.1.3. The subspaces X62,2, X4
2,3, X33,2, and X1
3,3 all have the same dimension. One finds
that this dimension appears once in both γ and the permutation θ. Hence this dimension does not
appear in the factor (−1)λ(A,B,C,D). Since this is the only dimension in the double complex (4.4)
that may be infinite in general, we find that (−1)λ(A,B,C,D) is indeed well-defined, regardless of
whether or not H is finite dimensional.
44
Corollary 4.1.4. Suppose A,B,C, and D are operators on a Hilbert space H, each of which differs
from the identity by a finite rank operator. If AB = CD, then τ(A,B,C,D) = 1.
Proof. With respect to a decomposition H = H0 ⊕ V for some finite dimensional subspace V , the
operators A,B,C and D are of the form I ⊕ FA, I ⊕ FB, I ⊕ FC , and I ⊕ FD, respectively. In
particular, FAFB = FCFD. By Lemma 3.3.5, we have
τ(A,B,C,D) = τ(I, I, I, I) · τ(FA, FB, FC , FD)
The first factor is one, and the second factor is also one by the preceding proposition.
4.2 Factorization results
For an invertible operator U and finite dimensional subspace V of H, denote by U |V the
isomorphism U |V : V → U(V ). Let τ(U |V ) denote the torsion of this isomorphism. Also, for a
Fredholm operator T , let U |cokerT denote the isomorphism U |cokerT : cokerT → cokerUT given by
v + TH 7→ Uv + UTH. Let τ(U |cokerT ) denote the torsion of this isomorphism.
4.2.1 Perturbation vectors
Lemma 4.2.1. Let a and u be commuting units in B/L1, and let A,D ∈ B be lifts of a. If u has
an invertible lift U ∈ B, then
σA,U−1DU = d(a, u) · σA,D ⊗ τ(U−1|kerD
)⊗ τ
(U−1|cokerD
)∗.
Proof. First we note that A − U−1DU ∈ L1, so the perturbation vector σA,U−1DU is defined. Let
us begin by proving the lemma in the case when A, and hence also D, has index zero.
For T = A,D, choose a subspace ZT complementary to imT , and choose isomorphisms FT :
kerT → ZT . Define LT as in Lemma 3.3.3 and the subsequent paragraph. Let πT : H → cokerT
be the quotient map. The operator U−1FDU defines an isomorphism
U−1FDU : kerU−1DU = U−1(kerD)→ U−1(ZD) (4.8)
45
and U−1(ZD) is a subspace complementary to im(U−1DU) = U−1(imD). The torsion of the
isomorphism induced by (4.8) is identified as
τ(πU−1DUU−1FDU) = τ
(U−1|kerD
)∗ ⊗ τ(πDFD)⊗ τ(U−1|cokerD
).
Additionally, one has
det(A+ LA)−1(U−1DU + U−1LDU) = det(A+ LA)−1(D + LD)
· det(D + LD)−1U−1(D + LD)U
= det(A+ LA)−1(D + LD) · d(a, u)
Combining the two preceding equations, we calculate
σA,U−1DU = det(A+ LA)−1(U−1DU + U−1LDU) · τ(πAFA)⊗ τ(πU−1FDU)∗
= d(a, u) · det(A+ LA)−1(D + LD) · τ(πAFA)⊗ τ(πDFD)∗
⊗ τ(U−1|kerD
)⊗ τ
(U−1|cokerD
)∗= d(a, u) · σA,D ⊗ τ
(U−1|kerD
)⊗ τ
(U−1|cokerD
)∗In the second equality, we have used (3.2) several times.
Now if indA is not necessarily zero, let Q be any Fredholm operator with indQ = −indA.
Let q be the image of in B/L1 of Q. Let A = A⊕Q, D = D ⊕Q, and U = U ⊕ I. Then A and D
have index zero, A− D ∈ L1(H2), and [D, U ] ∈ L1(H2). Hence we calculate
σA,U−1DU = d(a, u) · σA,D ⊗ τ(U−1|ker D
)⊗ τ
(U−1|coker D
)∗= d(a, u) · d(q, 1) · σA,D ⊗ σS,S ⊗ τ
(U−1|kerD
)⊗ τ
(U−1|cokerD
)∗= d(a, u) · σA,D ⊗ τ
(U−1|kerD
)⊗ τ
(U−1|cokerD
)∗On the other hand,
σA,U−1DU = σA,U−1DU
so we have
σA,U−1DU = d(a, u) · σA,D ⊗ τ(U−1|kerD
)⊗ τ
(U−1|cokerD
)∗.
46
Lemma 4.2.2. Let a and u be units in B/L1 and let A,D ∈ B be lifts of a. If u has an invertible
lift U ∈ B, then
(1) σAU,DU = τ(U−1|kerA
)∗ ⊗ σA,D ⊗ τ (U−1|kerD
)(2) σUA,UD = τ (U |cokerA)⊗ σA,D ⊗ τ (U |cokerD)∗
Proof. First we note that AU −DU,UA−UD ∈ L1. Let us begin with the case when A has index
zero. With FA, FD, LA, LD as before, we have
det(AU + LAU)−1(DU + LDU) = det(A+ LA)−1(D + LD).
Furthermore, kerAU = U−1(kerA) and kerDU = U−1(kerD), and we calculate
τ(πAFAU) = τ(πAFA)⊗ τ(U−1|kerA)∗
τ(πDFDU) = τ(πDFD)⊗ τ(U−1|kerD)∗
Therefore σAU,DU = τ(U−1|kerA
)∗ ⊗ σA,D ⊗ τ (U−1|kerD
).
In general, if indA is not necessarily zero, let Q, A, D, U be as in Lemma 4.2.1. Then
σAU ,DU = τ(U−1|kerA
)∗ ⊗ σA,D ⊗ τ (U−1|kerD
).
On the other hand,
σAU ,DU = σAU,DU .
Therefore σAU,DU = τ(U−1|kerA
)∗⊗σA,D⊗ τ (U−1|kerD
). The second part is proved similarly.
Lemma 4.2.3. Let a be a unit in B/L1, and let A,D ∈ B be lifts of a. If U ∈ B is an invertible
lift of 1 ∈ B/L1, then
(1) σA,DU = σA,D ⊗ τ(U−1|kerD
)∗ · detU
(2) σA,UD = σA,D ⊗ τ (U |cokerD) · detU
Proof. First we note that U is an invertible determinant class operator by assumption. Moreover,
A−DU,A− UD ∈ L1, so the perturbation vectors are defined. The proof then proceeds as in the
previous lemma.
47
4.2.2 Joint torsion
In this section, we use the previous three lemmas to calculate the joint torsion of quadruples
(A,B,C,D) in terms of quadruples modified by an invertible operator. This will allow us to reduce
the calculation of joint torsion to a determinant invariant and a finite dimensional calculation,
which has already been dealt with in Section 4.1. In the following section, we will interpret the
proposition below as calculating the transformation of the joint torsion of quadruples (A,B,C,D)
under an action by a group of invertibles.
Proposition 4.2.4. Let a and b be commuting units in B/L1. Let A,D ∈ B be lifts of a, and let
B,C ∈ B be lifts of b such that AB = CD. Suppose u ∈ B/L1 has an invertible lift U ∈ B.
(1) If a and u commute, then
τ(A,BU,CU,U−1DU) = d(a, u) · τ(A,B,C,D).
(2) If b and u commute, then
τ(UA,B,UCU−1, UD) = d(u, b) · τ(A,B,C,D).
Proof. First we note that A(BU) = (CU)(U−1DU) and A − U−1DU ∈ L1, and BU − CU ∈ L1.
Hence the joint torsion in (1) is defined, and likewise for (2). Let us first prove (1). By Lemmas
4.2.1 and 4.2.2, we have the factorization
σA,U−1DU ⊗ σBU,CU = d(a, u) · σA,D ⊗ σB,C ⊗ τ(U−1|kerB
)∗ ⊗ τ (U−1|kerC
)⊗
⊗ τ(U−1|kerD
)⊗ τ
(U−1|cokerD
)∗(4.9)
One uses (3.2) to check that the torsion vectors can indeed be moved past each other with impunity.
To calculate τ(EA,D), we choose generators t0, . . . , t5 appropriately:
• t0 ∈ det(kerB ∩ kerD)
• t0 ∧ t1 ∈ det kerB
48
• Dt1 ∧ t2 ∈ det kerC
• ι2∗t2 ∧ t3 ∈ detH1(A,B,C,D)
• π1∗t3 ∧ t4 ∈ det cokerB
• At4 ∧ t5 ∈ det cokerC
• πt5 ∈ det (H/(AH+ CH))
Then τ(EA,D) = t∗0 ⊗ (t0 ∧ t1) ⊗ · · · ⊗ (At4 ∧ t5) ⊗ (πt5)∗. Here, H1 = H1(A,B,C,D) is the first
Koszul homology space, which we recall is expressed as:
H1 =(y, z) |Ay = −Cz(−Bx,Dx) |x ∈ H
.
The map ι2∗ is induced by inclusion into the second coordinate, and π1∗ is induced by projection
onto the first coordinate. Let v ∈ kerC be such that v /∈ D(kerC). Then ι2∗v = [(0, v)] 6= 0 in H1,
so we can take t2 to be the product of sufficiently many such vectors, say ∧ivi. On the other hand,
let w /∈ imB be such that Aw = Cu ∈ imC for some u. Then [(w,−u)] 6= 0 in H1 and π1∗[(w,−u)],
so we can take t3 to be the product of sufficiently many such vectors, say ∧j [(wj ,−uj)]. Of course,
τ(EA,D) is independent of these specific choices.
Now we would like to calculate the torsion vectors τ(EA,U−1DU ) and τ(EBU,CU ) in terms
of τ(EA,D) and τ(EB,C). The only potential difficulty is in the H1 position, so let us compare
H ′1 = H1(A,BU,CU,U−1DU) with the discussion of the previous paragraph. First,
H ′1 =(y, U−1z) |Ay = −Cz(−Bx,U−1Dx) |x ∈ H
.
The invertible operator I ⊕ U−1 on H ⊕ H induces an isomorphism from H1 onto H ′1, which we
denote by (I ⊕ U−1)|H1 . Moreover, we have
49
U−1(Dt1 ∧ t2) = U−1Dt1 ∧ U−1t2
∈ det kerCU
(I ⊕ U−1)|H1(ι2∗t2 ∧ t3) =(∧i[(0, U−1vi)]
)∧(∧j [(wj ,−U−1uj)]
)= ι2∗U
−1t2 ∧ (I ⊕ U−1)|H1t3
∈ detH ′1
π1∗(I ⊕ U−1)|H1t3 ∧ t4 = π1∗t3 ∧ t4
∈ det cokerBU
Hence, we calculate
τ(EA,U−1DU ) = (U−1t0)∗ ⊗ ((−U−1t0) ∧ U−1t1)⊗ (U−1t2 ∧ −U−1Dt1)∗⊗
⊗ ((I ⊕ U−1)|H1(−ι2∗)t2 ∧ t3)⊗ (t4 ∧ π1∗t3)∗ ⊗ (At4 ∧ t5)⊗ (πt5)∗
= τ(EA,D)⊗ τ((I ⊕ U−1)|H1)⊗ τ(U−1|kerB
)⊗ τ
(U−1|kerC
)∗⊗ (4.10)
⊗ τ(U−1|kerB∩kerD
)∗Similarly, we find
τ(EBU,CU ) = τ(EB,C)⊗ τ((I ⊕ U−1)|H1)⊗ τ(U−1|kerD
)⊗ τ
(U−1|cokerD
)∗⊗ (4.11)
⊗ τ(U−1|kerB∩kerD
)∗Notice that λ = λ(A,B,C,D) is the same as λ(A,BU,CU,U−1DU) in Definition 3.3.4 since
all the spaces involved have the same dimension. Combining (4.9), (4.10), and (4.11), we find
τ(A,BU,CU,U−1DU) = (−1)λτ(EA,U−1DU )⊗ τ(EBU,CU )∗ ⊗ σA,U−1DU ⊗ σBU,CU
= (−1)λd(a, u) · τ(EA,D)⊗ τ(EB,C)∗ ⊗ σA,D ⊗ σB,C
= d(a, u) · τ(A,B,C,D)
This completes the proof of part 1. The second part follows by a similar analysis.
50
Proposition 4.2.5. Let A,B,C, and D be Fredholm operators with AB = CD, A−D ∈ L1, and
B − C ∈ L1. For any invertible determinant class operator U , we have
τ(A,B,CU,U−1D) = τ(A,B,C,D).
Proof. First we note that I − U−1 = (U − I)U−1 ∈ L1, so A− U−1D ∈ L1 and B −CU ∈ L1. We
calculate perturbation vectors as in (4.9), this time using using Lemma 4.2.3:
σA,U−1D ⊗ σB,CU = σA,D ⊗ σB,C ⊗ τ(U−1|cokerD)∗ ⊗ τ(U |ker C)∗. (4.12)
Next, pick generators tk as in Proposition 4.2.4. As before, we find that
H1(A,B,CU,U−1D) =(y, U−1z) |Ay = −Cz(−Bx,U−1Dx) |x ∈ H
.
The invertible operator I ⊕ U−1 on H ⊕ H induces an isomorphism from H1(A,B,C,D) onto
H1(A,B,CU,U−1D), which we denote by (I ⊕ U−1)|H1 . Moreover, we have
U−1(Dt1 ∧ t2) = U−1Dt1 ∧ U−1t2
∈ det kerCU
(I ⊕ U−1)|H1(ι2∗t2 ∧ t3) = ι2∗U−1t2 ∧ (I ⊕ U−1)|H1t3
∈ detH1(A,B,CU,U−1D)
π1∗(I ⊕ U−1)|H1t3 ∧ t4 = π1∗t3 ∧ t4
∈ det cokerB
Hence we calculate
τ(EA,U−1D) = τ(EA,D)⊗ τ((I ⊕ U−1)|H1)⊗ τ(U−1|kerC
)∗ ⊗ τ (U−1|kerB∩kerD
)∗(4.13)
τ(EB,CU ) = τ(EB,C)⊗ τ((I ⊕ U−1)|H1)⊗ τ(U−1|cokerD
)∗ ⊗ τ (U−1|kerB∩kerD
)∗(4.14)
Combining equations (4.12), (4.13), and (4.14), we find that τ(A,B,CU,U−1D) = τ(A,B,C,D),
as desired.
51
4.3 Main results
4.3.1 Group actions on commuting squares
As usual, let π : B → B/L1 denote the projection. Denote by S the set of all quadruples of
Fredholm operators (A,B,C,D) on a fixed Hilbert space such that AB = CD, A −D ∈ L1, and
B − C ∈ L1.
Definition 4.3.1. For a, b ∈ B/L1, define the following sets:
S1,a = (A,B,C,D) ∈ S |π(A) = a
S2,b = (A,B,C,D) ∈ S |π(B) = b
and the following group:
Ga = (U, V ) ∈ B(H)× × B(H)× |π(U) = π(V ), [π(U), a] = 0
Then Ga acts on S1,a on the right:
(A,B,C,D) •1,a (U, V ) = (A,BU,CV, V −1DU)
and Gb acts on S2,b on the right:
(A,B,C,D) •2,b (U, V ) = (U−1A,B,U−1CV, V −1D)
Now we can rephrase our factorization results from the previous section in terms of these
group actions:
Theorem 4.3.2.
(1) If (U, V ) ∈ Ga and (A,B,C,D) ∈ S1,a, then
τ ((A,B,C,D) •1,a (U, V )) = d(a, π(U)) · τ(A,B,C,D).
(2) If (U, V ) ∈ Gb and (A,B,C,D) ∈ S2,b, then
τ ((A,B,C,D) •2,b (U, V )) = d(π(U)−1, b) · τ(A,B,C,D).
52
4.3.2 A proof of equality
We are now in a position to prove the main result of this chapter, namely that joint torsion
is equal to the determinant invariant:
Theorem 4.3.3. Let a and b be commuting units in B/L1. Let A,D ∈ B be lifts of a, and let
B,C ∈ B be lifts of b such that AB = CD. Then d(a, b) = τ(A,B,C,D).
Proof. First suppose A and B have index zero. Let a = π(A), b = π(B). Let U be an invertible
parametrix for B modulo finite rank operators. For example, we may take U = (B + F )−1 for any
appropriate finite rank operator F . Similarly, let V be an invertible parametrix for C modulo finite
rank operators. Let U ′ and V ′ be invertible finite rank perturbations of A and D, respectively.
Then (U, V ) ∈ Ga and (U ′, V ′) ∈ Gb, and we find
(A,B,C,D) •1,a (U, V ) •2,b (U ′, V ′)
= (U ′−1A,BU,U ′−1CV V ′, V ′−1V −1DU)
Notice that all four of the above operators are finite rank perturbations of the identity. Hence
τ((A,B,C,D) •1,a (U, V ) •2,b (U ′, V ′)
)= 1
by Theorem 4.1.2. On the other hand, the above theorem gives
τ((A,B,C,D) •1,a (U, V ) •2,b (U ′, V ′)
)= d(π(U ′)−1, π(BU)) · d(a, π(U)) · τ(A,B,C,D)
Notice that d(π(U ′)−1, π(BU)) = 1 and d(a, π(U)) = d(a, b)−1 since π(U) = b−1. By comparing
the two preceding equations, we find
τ(A,B,C,D) = d(a, b).
In general, if indA = n and indB = m, let Q be a Fredholm operator with index −n and let
R be a Fredholm operator with index −m. Then the operators A = A ⊕ Q ⊕ I, B = B ⊕ I ⊕ R,
53
C = C ⊕ I ⊕R, D = D ⊕Q⊕ I are index zero operators on H⊕H⊕H such that AB = CD. Let
a and b be the images of A and B, respectively, modulo trace class. By the above, we find that
τ(A, B, C, D) = d(a, b).
Moreover,
τ(A,B,C,D) = τ(A, B, C, D)
and
d(a, b) = d(a, b).
The result follows by combining the above three equations.
4.4 Applications
In the first two subsections, we apply Theorem 4.3.3 to derive some properties of the deter-
minant invariant and joint torsion. This provides new proofs of known results, namely Theorem
4.4.1 and Lemma 4.4.6 in the commutative case, as well as other results that appear to be new.
4.4.1 Properties of the determinant invariant
By Theorem 4.3.3, the determinant invariant enjoys the same properties as joint torsion. Al-
though the determinant invariant is defined in terms of infinite dimensional Fredholm determinants,
Theorem 4.3.3 guarantees that for commuting operators, it can actually be calculated in terms of
finite dimensional data.
Next, we note that in the case when the Koszul complex K•(A,B) is acyclic, we have the
following consequence of Theorem 4.3.3 and Lemma 3.2.4, which was first obtained in [13].
Theorem 4.4.1. If A and B are commuting Fredholm operators and the Koszul complex K•(A,B)
is acyclic, we have
d(a, b) =detB|kerA
detB|cokerA
detA|cokerB
detA|kerB.
54
There are analogues of Theorem 4.4.1 more generally in the case of almost commuting oper-
ators. Let a and b be commuting units in B/L1. Let A,D ∈ B be lifts of a, and let B,C ∈ B be
lifts of b such that AB = CD. For simplicity, assume that the Koszul complex K•(A,B,C,D) is
acyclic. Pick Fredholm operators Q and R such that indQ = −indA and indR = −indB. Then
A = A⊕Q⊕ I, B = B⊕ I ⊕R, C = C ⊕ I ⊕R, and D = D⊕Q⊕ I all have index zero. Moreover,
the new Koszul complex K•(A, B, C, D) is acyclic, and d(A,B) = d(A, B) = τ(A, B). Then we
find
τ(EA,D) = τ(B : ker D → ker A)⊗ τ(C : coker D → coker A)
τ(EB,C) = τ(D : ker B → ker C)⊗ τ(A : coker B → coker C)
For T = A, B, C, D, pick trace class operators LT and let πT be the quotient map as in
Lemma 3.3.3. Thus we have isomorphisms πTLT : kerT → cokerT , and we calculate
σA,D = det(A+ LA)−1(D + LD)τ(πALA)⊗ τ(πDLD)∗
σB,C = det(B + LB)−1(C + LC)τ(πBLC)⊗ τ(πBLC)∗
Corollary 4.4.2. In the situation above, d(a, b) is given by
(−1)λ(A,B,C,D) det(A+ LA)−1(D + LD)(B + LB)−1(C + LC)
·det(πDLD)−1(C|coker D)−1(πALA)(B|ker D)
det(πBLB)−1(A|coker B)−1(πCLC)(D|ker B)
4.4.2 Properties of joint torsion
Now let us record a number of properties of joint torsion that follow from Theorem 4.3.3.
First, we note that for any two Fredholm operators A and B with commuting images a and b,
respectively, in B/L1, we are justified in writing τ(a, b), regardless of the existence of perturbations
C,D required in the definition of joint torsion.
Next we study the continuity of joint torsion. Consider the space of almost commuting pairs
M = (A,B) |A and B are Fredholm and [A,B] ∈ L1
55
endowed with the metric
d((A1, B1), (A2, B2)) = ‖A1 −A2‖+ ‖B1 −B2‖+ ‖[A1, B1]− [A2, B2]‖1.
One may check that (M,d) is complete. In light of Theorem 4.3.3, joint torsion may be defined on
M by
τ(A,B) = τ(a, b)
where a and b as usual denote the images of A and B in B/L1. The following corollary also follows
from the recent work [39].
Corollary 4.4.3. Joint torsion is a continuous map from (M,d) into C.
Proof. The strategy is to prove this result first on the subset of pairs of invertible operators, then
for index zero operators, and finally in general. Thus, suppose A0 and B0 are invertible. Then
τ(A0, B0) = detA0B0A−10 B−1
0 = det(I + [A0, B0]A−1
0 B−10
).
Suppose (A,B)→ (A0, B0) in M . Since A→ A0 and B → B0 in norm, we may assume that all A
and B are invertible. Hence
[A,B]A−1B−1 → [A0, B0]A−10 B−1
0
in L1. Consequently, τ(A,B)→ τ(A0, B0) since the map
L1 → C, X 7→ det(I +X)
in continuous.
Now suppose that A0 and B0 merely have index zero. Let FA and FB be finite rank operators
such that A0 + FA and B0 + FB are invertible. Suppose (A,B) → (A0, B0) in M . Then we may
assume that all A+FA and B+FB are invertible. Moreover (A+FA, B+FB)→ (A0 +FA, B0 +FB)
in M , so by the above argument,
τ(A+ FA, B + FB)→ τ(A0 + FA, B0 + FB).
56
Hence τ(A,B)→ τ(A0, B0).
Finally suppose indA0 = n and indB0 = m are possibly nonzero. Let Q be a Fredholm
operator with index −n, and let R be a Fredholm operator with index −m. Suppose (A,B) →
(A0, B0) in M . Then we may assume that all A have index n and all B have index m. Define the
following index zero operators on H⊕H⊕H:
A0 = A0 ⊕Q⊕ I and B0 = B0 ⊕ I ⊕R
and A and B similarly. Then (A, B) → (A0, B0) in M . By Lemmas 3.2.4 and 3.3.5, τ(A, B) =
τ(A,B) and τ(A0, B0) = τ(A0, B0). Then the above argument shows that τ(A,B)→ τ(A0, B0).
Recall, however, that joint torsion is defined in terms of the finite dimensional homology
spaces H•(A), H•(B), H•(C), H•(D), and H•(A,B,C,D). In particular, the dimensions of these
spaces are by no means continuous. Thus, the continuity of joint torsion may be seen as analogous
to the continuity of the Fredholm index.
Lemma 4.4.4. If [A,B] ∈ L1, then
τ(eA, eB) = etr[A,B].
Proof. This follows from Theorem 4.3.3 and Example 2.2.12.
It is convenient to state the following variational formula using the logarithmic derivative.
Thus ddz log u should be interpreted as u−1 d
dzu.
Corollary 4.4.5. Suppose A(z) is a differentiable family of operators such that [A(z), B] ∈ L1 for
every z. If, in addition, [A(z), B] is differentiable in L1, then
d
dzlog τ(eA(z), eB) = log τ(e
ddzA(z), eB).
Lemma 4.4.6. Whenever the following joint torsion numbers are defined, we have:
(1) τ(A,B1B2) = τ(A,B1) · τ(A,B2)
57
(2) τ(A, I) = 1
(3) τ(A,B)−1 = τ(B,A)
(4) τ(A, I −A) = 1
(5) τ(A,−A) = 1
(6) τ(A,B) = τ(A∗, B∗)−1
(7) τ(A,B−1) = τ(A,B)−1
(8) τ(A,A) = (−1)indA
Proof. Properties (1)-(6) follow from the corresponding properties of the determinant invariant.
See for instance Lemma 4.2.14 and Theorem 4.2.17 of [50]. Property (7) follows from (1) and (2).
To verify (8), notice that the two torsion factors in the definition of joint torsion are the same.
Thus we are left with (−1)ν(A,A), where ν(A,A) is the sign in the definition of joint torsion. The
result follows since ν(A,A) = indA.
Lemma 4.4.7. Whenever the following joint torsion numbers are defined, we have:
(1) τ(A,A∗) ∈ R.
(2) If A and B are self-adjoint, then |τ(A,B)| = 1.
(3) If B is an idempotent, i.e. B2 = B, then τ(A,B) = 1.
(4) If A is self-adjoint and B is a partial isometry, then τ(A,B) ∈ R.
(5) If A and B are partial isometries, then |τ(A,B)| = 1.
Proof.
(1) By properties (6) and (3) of Lemma 4.4.6,
τ(A,A∗) = τ(A∗, A)−1 = τ(A,A∗).
58
(2) Since A and B are self-adjoint, Lemma 4.4.6(6) implies that
τ(A,B) = τ(A,B)−1.
(3) By Lemma 4.4.6(1), τ(A,B) = τ(A,B)2, and the result follows since joint torsion is nonzero.
(4) First let T be any Fredholm operator which commutes with B modulo L1. Since B is a
Fredholm partial isometry, T also commutes with B∗ modulo L1. Since B∗B is a projection,
(3) implies that
τ(T ∗, B∗) · τ(T ∗, B) = τ(T ∗, B∗B) = 1
so by Lemma 4.4.6(6),
τ(T ∗, B) = τ(T,B). (4.15)
The result follows by setting T = A since A∗ = A.
(5) Applying (4.15) to both A and B yields
τ(A,B) = τ(A∗, B∗)
and the result follows by Lemma 4.4.6(6).
In (4), if A is in fact positive, we will use the behavior of joint torsion under the functional
calculus to show that τ(A,B) > 0 (Proposition 5.1.11).
Lemma 4.4.8. If A and B are commuting Fredholm operators, then for any µ 6= 0,
τ(A,µB) = µindA−dimH0+dimH2τ(A,B)
where H0 = H0(A,B) and H2 = H2(A,B) as usual denote the joint Koszul homology spaces.
Proof. The two long exact sequences EA and EµB in the definition of τ(A,µB) are the same as
those for τ(A,B), except for a factor of µ, given by the exponent on µ above.
Corollary 4.4.9. If A and B are commuting Fredholm operators, then
d
dµlog τ(A,µB) =
indA− dimH0 + dimH2
µ.
59
4.4.3 Joint torsion and Hilbert-Schmidt operators
This section represents a first attempt at understanding torsion invariants in the context of
operator ideals other than the trace class. This work was motivated by Voiculescu’s conjectures on
the existence of lifts of almost normal operators [61]. See Proposition 4.4.10 below.
Smooth non-vanishing functions on the circle may be factored in terms of exponentials and
the function z 7→ z. As we will see, the joint torsion of Toeplitz operators may be calculated as
follows:
(1) τ(Tef , Teg) = exp 12πi
∫D df ∧ dg
(2) τ(Tef , Tz) = exp 12πi
∫S1 f d log z
(3) τ(Tz, Tz) = −1
In (1), f and g are extended to the unit disk D appropriately. Hence (1) recovers the Pincus
principal function [11, 12]. Terms like (3) carry the index data, a fact which seems to hold more
generally. Terms like (2) appear to be new “cross-terms.”
Suppose T is such that [T, T ∗] ∈ L1. As described in Section 3.1, the operator p(T, T ∗) is
well-defined for any p ∈ C[z, z], modulo L1. The exponential of p(T, T ∗) is similarly well-defined,
modulo L1, via the usual power series.
Proposition 4.4.10. Suppose [T, T ∗] ∈ L1, and suppose there exist a unitary U , a normal operator
N , and L ∈ L2 such that T = U∗NU + L. If p, q ∈ C[x, y], then
τ(exp p(T, T ∗), exp q(T, T ∗)) = 1.
The proof of this proposition proceeds by a series of lemmas. If L ∈ L2, then L2 ∈ L1, and
consequently I + L− eL ∈ L1. Hence,
τ(I + L1, I + L2) = τ(eL1 , eL2) = exp tr [L1, L2] = 1
whenever L1, L2 ∈ L2. The third equality follows from Proposition 2.1.6. More generally, one has
the following:
60
Lemma 4.4.11. Suppose A is a Fredholm operator and L1, L2 ∈ L2 such that [A,L2] ∈ L1. Then
τ(A+ L1, I + L2) = τ(A, I + L2).
Proof. If A is invertible, then
τ(A+ L1, I + L2) = τ(A, I + L2) · τ(I +A−1L1, I + L2)
and the result follows by the observation above. In general, let Q be any Fredholm operator with
index opposite that of A, and let F be a finite rank operator such that A ⊕ Q + F is invertible.
The result follows from the invertible case since
τ(A+ L1, I + L2) = τ((A+ L1)⊕Q+ F, (I + L2)⊕ I)
and
τ(A, I + L2) = τ(A⊕Q+ F, (I + L2)⊕ I).
Lemma 4.4.12. Suppose N is a normal Fredholm operator and L ∈ L2 such that [N,L] ∈ L1.
Then
τ(N, I + L) = 1.
Proof. Since N is normal, by [60] we may write N = D+K for some K ∈ L2 and a diagonalizable
operator D. By changing basis, we find that
τ(N, I + L) = τ(D′ +K ′, I + L′)
where L′,K ′ ∈ L2 and D′ is diagonal. By the preceding lemma,
τ(D′ +K ′, I + L′) = τ(D′, I + L′).
By adding a finite rank digaonal operator, we may assume that D′ is invertible and diagonal. Write
D′ = exp d for a diagonal operator d. Then [d, L′] ∈ L1 and
τ(D′, I + L′) = τ(exp d, expL′) = exp tr [d, L′].
Moreover, tr [d, L′] = 0 since d is normal and L′ ∈ L2, and the result follows.
61
More generally, one has the following:
Lemma 4.4.13. Suppose N1 and N2 are normal Fredholm operators and L1, L2 ∈ L2 are such that
[N1, N2], [N1, L2], and [L1, N2] ∈ L1. Then
τ(N1 + L1, N2 + L2) = τ(N1, N2).
Proof. By adding the orthogonal projections onto the kernels of N1 and N2, necessarily finite
dimensional, we may assume N1 and N2 are invertible normal operators. Hence
τ(N1 + L1, N2 + L2) = τ(N1, N2) · τ(I +N−11 L1, N2)
· τ(N1, I +N−12 L2) · τ(I +N−1
1 L1, I +N−12 L2)
The second, third, and fourth factors are all 1 by the preceding lemmas.
Proof of Proposition 4.4.10. We must show that tr[p(T, T ∗), q(T, T ∗)] = 0 since
τ(exp p(T, T ∗), exp q(T, T ∗)) = exp tr[p(T, T ∗), q(T, T ∗)].
First we note that p(T, T ∗) = N1 + L1 and q(T, T ∗) = N2 + L2, modulo L1, for some L1, L2 ∈ L2
and normal operators N1 and N2. If [N1, N2], [N1, L2], [L1, N2] ∈ L1, then the result follows from
the previous lemma. More generally, we find that both UN1U∗ and UN2U
∗ can be expressed as
sums and products of N and N∗. Moreover, [N1, L2] + [L1, N2] ∈ L1. Since N is the sum of a
diagonalizable operator and a Hilbert-Schmidt operator, one has tr([N1, L2]+ [L1, N2]) = 0 and the
result follows.
Chapter 5
Explicit formulas for joint torsion
In this chapter we investigate the transformation of joint torsion under the functional calculus.
Combined with the results of the previous chapter, this will allow us to obtain explicit formulas for
the joint torsion of Toeplitz operators. We also investigate the relationship between joint torsion
and Tate tame symbols. The results in this chapter have appeared in the author’s work [46].
5.1 Transformation rules for joint torsion
5.1.1 Commutators
Let A and B be bounded operators on a fixed Hilbert space. Connes shows in [19] that if
[A,B] ∈ Lp, and either
(1) f is holomorphic on a neighborhood of σ(A), or
(2) A is self-adjoint and f is C∞ on σ(A),
then [f(A), B] ∈ Lp. In this section, we calculate the trace of such a commutator.
Lemma 5.1.1. If [A,B] ∈ L1 and f is an entire function, then
tr[f(A), B] = tr(f ′(A)[A,B]
).
Proof. Write f(z) =∑ckz
k. Then [f(A), B] =∑ck[A
k, B]. Using the identity
[Ak, B] =k∑l=1
Al−1[A,B]Ak−l (5.1)
63
we find that
tr[Ak, B] = tr(k Ak−1[A,B]).
Hence
tr[f(A), B] = tr∑
kckAk−1[A,B]
= tr(f ′(A)[A,B]
)Let f be holomorphic on a neighborhood of the spectrum σ(A) of an operator A. By an
admissible contour Γ for defining f(A), we mean a collection of Jordan curves in the neighborhood
that enclose σ(A) on the left. Thus
f(A) =1
2πi
∫Γ(λ−A)−1f(λ) dλ. (5.2)
Proposition 5.1.2. Suppose [A,B] ∈ L1. If either
(1) f is holomorphic on a neighborhood of σ(A), or
(2) A is self-adjoint and f is C∞ on σ(A)
then
tr[f(A), B] = tr(f ′(A)[A,B]
).
Proof. Recall that in both cases [f(A), B] ∈ L1 by [19], and we adapt arguments therein.
(1) Let Γ be an admissible contour for defining f(A). Since
[(λ−A)−1, B] = (λ−A)−1[A,B](λ−A)−1
we find that λ 7→ [(λ − A)−1, B] is a continuous map from the resolvent set of A into L1,
and
[f(A), B] =1
2πi
∫Γ(λ−A)−1[A,B](λ−A)−1f(λ) dλ.
Moreover,
tr((λ−A)−1[A,B](λ−A)−1
)= tr
((λ−A)−2[A,B]
).
64
Hence
tr[f(A), B] = tr
(1
2πi
∫Γ(λ−A)−2f(λ) dλ [A,B]
)= tr
(f ′(A)[A,B]
)(2) We may assume that f has compact support, so that f = g, the Fourier transform of a
Schwartz class function g. Hence
[f(A), B] =1√2π
∫[e−itA, B]g(t) dt.
By the preceding lemma,
tr[e−itA, B] = tr(−ite−itA[A,B]
)and again by continuity,
tr[f(A), B] = tr1√2π
∫−ite−itAg(t) dt[A,B]
= tr(f ′(A)[A,B]
)Corollary 5.1.3. With the same hypotheses as above,
τ(ef(A), eB) = τ(eA, ef′(A)B).
Proof. Since A and f ′(A) commute, we have f ′(A)[A,B] = [A, f ′(A)B]. The result then follows by
Lemma 4.4.4.
5.1.2 Perturbations
Analogues of Lemma 5.1.1 and Proposition 5.1.2 hold for suitable functions applied to Lp-
perturbations. We will need the following estimate for the exponential function:
Proposition 5.1.4. If A and A′ are self-adjoint with A−A′ ∈ Lp, then eitA − eitA′ ∈ Lp with
‖eitA − eitA′‖p ≤ C (|t|+ 1)
65
where
C = max0≤t≤1
‖eitA − eitA′‖p.
Proof. Using the identity
rn − sn =n∑k=1
sk−1(r − s)rn−k (5.3)
we find that
‖eitnA − eitnA′‖p ≤ n‖eitA − eitA′‖p.
The result then follows by scaling.
Proposition 5.1.5. Let K ∈ Lp. If either
(1) f is holomorphic on a neighborhood of σ(K), or
(2) K is self-adjoint and f is C∞ on σ(K),
then f(K)− f(0)I ∈ Lp.
Note that in (2), σ(K) consists of 0 and real eigenvalues possibly accumulating to 0 by the
spectral theorem for compact self-adjoint operators.
Proof.
(1) Let Γ be an admissible contour for defining f(K). Then
f(K)− f(0)I =
∫Γ
[(λ−K)−1 − (λI)−1
]f(λ) dλ
= K
∫Γ(λ2 − λK)−1f(λ) dλ
The latter integral converges in norm, and the result follows.
(2) We may assume that f has compact support, so that f = g for a Schwartz class function
g. Then
f(K)− f(0)I =
∫ (e−itK − I
)g(t) dt.
The integral converges in the Lp-norm by the preceding proposition with A = K and
A′ = 0.
66
Proposition 5.1.6. Let A−A′ ∈ Lp. If either
(1) f is holomorphic on a neighborhood of σ(A)∪σ(A′) and there is a contour that defines both
f(A) and f(A′), or
(2) A and A′ are self-adjoint and f is C∞ on σ(A) ∪ σ(A′),
then f(A)− f(A′) ∈ Lp.
Proof. The proof proceeds as in the previous proposition. For part (1), one uses the identity
(λ−A)−1 − (λ−A′)−1 = (λ−A)−1(A−A′)(λ−A′)−1.
5.1.3 Joint torsion
For a given Fredholm operator A, we begin with a simple characterization of holomorphic
functions f that preserve the Fredholmness of A. We will use the following factorization of holo-
morphic functions:
Definition 5.1.7. Let f be a holomorphic function on a neighborhood of a compact set K. Then
the collection of zeros λ ∈ K | f(λ) = 0 is finite. Define the polynomial
pK(z) =∏
λ∈K | f(λ)=0
(z − λ)ordλ(f)
where ordλ(f) is the order of the zero of f at λ. Then
f = pKqK
for a holomorphic function qK with no zeros in K.
The index formula (5.4) below is a special case of [29, Theorem 10.3.13]. See also [38, Theorem
1.1].
Proposition 5.1.8. Let A be a Fredholm operator and let f be holomorphic on a neighborhood of
σ(A). Then f(A) is Fredholm if and only if f−1(0) is disjoint from the essential spectrum σe(A).
In this case,
ind f(A) =∑
λ∈σ(A) | f(λ)=0
ordλ(f) · ind(A− λ). (5.4)
67
Proof. Let p = pσ(A) and q = qσ(A) from the definition above. Then f(A) = p(A)q(A) and q
is invertible on a neighborhood of σ(A), so q(A) is invertible. The first assertion then follows
by factoring p, and the index formula follows by the additive property of the index: ind(ST ) =
indS + indT .
More generally one has the following sufficient condition for the Borel functional calculus to
preserve Fredholmness:
Proposition 5.1.9. Let A be a normal Fredholm operator, and let f ∈ L∞(σ(A)). If the sets
f−1(0) and f−1(±∞) are finite and disjoint from σe(A), then f(A) is Fredholm.
Proof. The strategy is to excise the sets f−1(0) and f−1(±∞) and use the resulting function to
construct a parametrix for f(A). Suppose f(λ) = 0, +∞, or −∞. Let Un be a nested sequence of
open subsets of σ(A) such that ∩Un = λ. Then χUn converges to χλ pointwise, so Pn = χUn(A)
converges to P = χλ(A) strongly. Now P is either 0 or the projection onto the λ-eigenspace of A,
which is finite dimensional since A−λ is Fredholm. Since Pn is a descending sequence of projections
that converge to a finite rank projection, there is an N for which Pn is finite dimensional for all
n > N .
Let Uλ = Un and χλ = χUλ for some n > N . By taking n large enough, we may assume
that the open sets Uλ are pairwise disjoint, where λ ranges over all the singularities and zeros of
f . Then
g = (1−∑λ
χλ)f +∑λ
χλ
is invertible in L∞(σ(A)), and g(A)−f(A) is a finite rank operator. Hence, g(A)−1 is a parametrix
for f(A) modulo finite rank operators, so f(A) is Fredholm.
Next we obtain a multiplicative analogue of (5.4):
Proposition 5.1.10. Suppose A and B are Fredholm operators with [A,B] ∈ L1. If f is holomor-
phic on a neighborhood of σ(A) and f(A) is Fredholm, then
τ(f(A), B) =∏
λ∈σ(A) | f(λ)=0
τ(A− λ,B)ordλ(f) · τ(q(A), B)
68
with q = qσ(A) as in Definition 5.1.7, so that in particular q(A) is invertible.
Proof. First we note that [f(A), B] ∈ L1 by Proposition 5.1.2(1). Writing f = pq, we have
[p(A), B] ∈ L1, so [q(A), B] ∈ L1 as well. By multiplicativity,
τ(f(A), B) = τ(p(A), B) · τ(q(A), B).
Since p(A) is a product of factors A − λ, we find that τ(p(A), B) further factors as the product
above.
5.1.4 Positivity
In this section we investigate general conditions under which joint torsion is positive. This
is used to clarify the relationship between joint torsion and the polar decomposition, and also to
obtain variational formulas.
Proposition 5.1.11. Suppose A and B are Fredholm operators and [A,B] ∈ L1. If A is positive
and B is a partial isometry, then τ(A,B) > 0.
Proof. Let F = PkerA be the orthogonal projection onto kerA = imA⊥. Then A + F is positive-
definite. By Proposition 5.1.2(2), B commutes with T = (A+ F )1/2 modulo L1. Hence
τ(A+ F,B) = τ(T,B)2.
By Lemma 4.4.7, τ(T,B) ∈ R since T is self-adjoint. Hence
τ(A,B) = τ(A+ F,B) > 0.
Proposition 5.1.12. Suppose A and B are Fredholm operators with [A,B] ∈ L1 and [A,B∗] ∈ L1.
Then with respect to the polar decompositions
A = PAVA, B = PBVB
we have
|τ(A,B)| = τ(PA, VB) · τ(VA, PB)
69
and consequently,
τ(A,B)
|τ(A,B)|= τ(PA, PB) · τ(VA, VB).
Proof. First notice that PA and VA are Fredholm since A is. Similarly, PB and VB are Fredholm.
We must show that the four joint torsion numbers above are well-defined, that is, the appropriate
commutators lie in L1. Our strategy is to show first that [PA, B], [A,PB], [PA, PB] ∈ L1, then
[VA, PB], [PA, VB] ∈ L1, and finally [VA, VB] ∈ L1.
If FA = PkerA, then A∗A+FA is invertible and commutes with B modulo L1. By Proposition
5.1.2(2), [(A∗A + FA)1/2, B] ∈ L1, so [PA, B] ∈ L1 as well, with PA = (A∗A)1/2. By reversing the
roles of A and B, we find that [A,PB] ∈ L1. Moreover, by replacing B by PB, we find that
[PA, PB] ∈ L1.
Next, we calculate modulo L1:
[VA, PB] ≡ [(PA + FA)−1PAVA, PB]
≡ [(PA + FA)−1A,PB]
≡ (PA + FA)−1[A,PB] + [(PA + FA)−1, PB]A
The first term is in L1 since [A,PB] ∈ L1, and the second term is in L1 since [PA, PB] ∈ L1 as well.
Similarly, [PA, VB] ∈ L1 by reversing the roles of A and B.
Again we calculate modulo L1:
[VA, VB] ≡ [(PA + FA)−1A, (PB + FB)−1B]
≡ (PA + FA)−1[A, (PB + FB)−1]B + (PA + FA)−1(PB + FB)−1[A,B]
+ [(PA + FA)−1, (PB + FB)−1]BA+ (PB + FB)−1[(PA + FA)−1, B]A
As before, all four of the above terms are evidently in L1.
By the multiplicative property of joint torsion,
τ(A,B) = τ(PA, VB) · τ(VA, PB) · τ(PA, PB) · τ(VA, VB).
The first two factors are positive by preceding proposition. The third factor has magnitude one by
Lemma 4.4.7(2), as does the last factor by Lemma 4.4.7(5).
70
Proposition 5.1.13. Suppose A and B are Fredholm operators with [A,B] ∈ L1. If A is positive,
and B is a partial isometry, then for all t ≥ 0,
τ(At, B) = τ(A,B)t.
If A is positive-definite, then the formula holds for all t ∈ R.
Proof. First we note that τ(A,B) > 0 by Proposition 5.1.11, [At, B] ∈ L1 by Proposition 5.1.2(2),
and At is Fredholm with parametrix (A + PkerA)−t. The formula holds for positive integers by
repeated application of Lemma 4.4.6(1), and for t = 0 by Lemma 4.4.6(2). The formula also holds
for all positive rational numbers: if p and q are any positive integers, then
τ(Ap/q, B)q = τ(A,B)p.
If F = PkerA, then A+ F is positive and invertible, and τ(At, B) = τ((A+ F )t, B). We will
show that the map t 7→ ((A + F )t, B), t > 0, is a continuous map into the space M in Corollary
4.4.3. Since t 7→ (A+ F )t is continuous in norm, it suffices to show that
limt→0‖[(A+ F )t, B]‖1 = 0.
Since [log(A+ F ), B] ∈ L1 by Proposition 5.1.2(2), this follows from the estimate
‖[(A+ F )t, B]‖1 ≤ t et‖ log(A+F )‖‖[log(A+ F ), B]‖1.
Joint torsion is continuous on M by Corollary 4.4.3, so the map t 7→ τ(At, B) is continuous.
Thus the result extends from rational t to all t ≥ 0. Finally, if A is positive definite, then A−t is
also positive definite for any t > 0. By the above result for positive t, we find
τ(A−t, B)t = τ(A,B).
A similar result holds when A and B are positive. In this case, |τ(A,B)| = 1 by Lemma
4.4.7. Suppose A and B are positive Fredholm operators with [A,B] ∈ L1. If FA = PkerA and
FB = PkerB, then
φ(A,B) = −i tr [log(A+ FA), log(B + FB)] ∈ R
71
is well-defined by Proposition 5.1.2. Since τ(A,B) = τ(A+ FA, B + FB), Lemma 4.4.4 gives
τ(A,B) = eiφ(A,B)
and φ(A,B) enjoys the additive versions of the properties in Lemma 4.4.6. Moreover, we have:
Proposition 5.1.14. If A and B are positive Fredholm operators with [A,B] ∈ L1, then for all
t > 0,
τ(At, B) = eitφ(A,B) = τ(A,B)t.
If A is positive-definite, then the formula holds for all t ∈ R.
Proof. This follows by noticing that τ(At, B) = τ((A + FA)t, B + FB), then using the fact that
A+ FA and B + FB have logarithms.
Corollary 5.1.15. Suppose A and B are Fredholm operators with [A,B] ∈ L1. If A is positive and
B is either positive or a partial isometry, then
d
dtlog τ(At, B) = log τ(A,B).
5.2 Fredholm modules
Let (A,H, F ) be a 2p-summable Fredholm module, i.e. [φ, F ] ∈ L2p for any φ ∈ A. Let
P = 12(F + I) be the projection onto the +1-eigenspace of F , so in particular, [φ, P ] ∈ L2p for any
φ ∈ A.
Definition 5.2.1. For φ ∈ A, form the abstract Toeplitz operator Tφ = PφP ∈ B(PH).
The main goal of this section is to show that f(Tφ) − Tf(φ) ∈ Lp for suitable functions f .
First let us record a corresponding result for the continuous functional calculus modulo compact
operators:
Proposition 5.2.2. Let T ∈ B be normal and let π : B → B/K be the quotient map onto the Calkin
algebra. If f ∈ C(σ(T )), then π(f(T )) = f(π(T )).
72
Proof. The polynomial functional calculus commutes with the quotient map, so the result follows
from the Stone-Weierstrass Theorem by approximating f by polynomials.
Next we show that the entire functional calculus commutes with the symbol map modulo L2p.
This complements the results of [26] and Proposition 5.2.7 below. Assume that A is closed under
the entire functional calculus. Otherwise, we may replace A by the algebra generated by f(a), for
all a ∈ A and entire functions f . The resulting algebra still has the property that [φ, P ] ∈ L2p.
Indeed, we have already seen that if [a, P ] ∈ L2p and f is holomorphic on a neighborhood of σ(a),
then [f(a), P ] ∈ L2p.
Lemma 5.2.3. For any φ ∈ A and integer k > 1, T kφ − Tφk ∈ L2p, with
‖T kφ − Tφk‖2p ≤k(k − 1)
2‖φ‖k−1‖[φ, P ]‖2p.
Proof. Each term in the identity
(Pφ)kP − PφkP =k−1∑l=1
(Pφ)k−l[φl, P ]P
contains a commutator, so (Pφ)kP − PφkP ∈ L2p. Using the identity (5.1), we estimate
‖[φl, P ]‖2p ≤ l‖φ‖l−1‖[φ, P ]‖2p.
Hence
‖(Pφ)kP − PφkP‖2p ≤k−1∑l=1
l‖φ‖k−1‖[φ, P ]‖2p
and the result follows.
Definition 5.2.4. For an entire function f(z) =∑ckz
k, let f(z) =∑|ck|zk.
Proposition 5.2.5. For any φ ∈ A and any entire function f , Tf(φ) − f(Tφ) ∈ L2p with
‖Tf(φ) − f(Tφ)‖2p ≤‖[φ, P ]‖2p
2‖φ‖f ′′(‖φ‖)
73
Proof. Write f(z) =∑ckz
k. The first two terms in the expansion
Tf(φ) − f(Tφ) =∞∑k=0
ck
(PφkP − (Pφ)kP
)vanish, and by Lemma 5.2.3 we estimate
‖Tf(φ) − f(Tφ)‖2p ≤∞∑k=2
|ck|k(k − 1)
2‖φ‖k−1‖[φ, P ]‖2p
≤ ‖[φ, P ]‖2p2‖φ‖
∞∑k=2
k(k − 1)|ck|‖φ‖k−2
≤ ‖[φ, P ]‖2p2‖φ‖
f ′′(‖φ‖)
In fact, the entire functional calculus commutes with the symbol map modulo Lp. First we
prove the following analogue of Lemma 5.2.3:
Lemma 5.2.6. For any φ ∈ A and integer k > 1, T kφ − Tφk ∈ Lp, with
‖T kφ − Tφk‖p ≤k(k − 1)
2‖φ‖k−2‖[φ, P ]‖22p.
Proof. First one verifies that
(Pφ)kP − PφkP =k−1∑l=1
P [P, φl][P, φ](Pφ)k−l−1P
using the identity P [P,ψ][P, χ]P = Pψ(P − I)χP . Each term of the sum contains a product of
commutators, so it is in Lp. Moreover, by (5.1),
‖[P, φl]‖2p ≤ l‖φ‖l−1‖[φ, P ]‖2p.
Hence
‖(Pφ)kP − PφkP‖p ≤k−1∑l=1
l‖φ‖k−2‖[φ, P ]‖22p
and the result follows.
Proposition 5.2.7. For any φ ∈ A and any entire function f , Tf(φ) − f(Tφ) ∈ Lp with
‖Tf(φ) − f(Tφ)‖p ≤1
2‖[φ, P ]‖22pf ′′(‖φ‖).
74
Proof. Write f(z) =∑ckz
k. The first two terms in the expansion
Tf(φ) − f(Tφ) =∞∑k=0
ck
(PφkP − (Pφ)kP
)vanish, and by Lemma 5.2.6 we estimate
‖Tf(φ) − f(Tφ)‖p ≤∞∑k=2
|ck|k(k − 1)
2‖[φ, P ]‖22p‖φ‖k−2
=1
2‖[φ, P ]‖22p
∞∑k=2
k(k − 1)|ck|‖φ‖k−2
=1
2‖[φ, P ]‖22pf ′′(‖φ‖)
We will need a sharper estimate for the exponential function:
Proposition 5.2.8. If φ is self-adjoint, then eTitφ − Teitφ ∈ Lp for any t ∈ R, and
‖eTitφ − Teitφ‖p ≤ (|t|+ 1)2(c1 + c2)
where
c1 = max0≤t≤1
‖eTitφ − Teitφ‖p and c2 = max0≤t≤1
‖[P, eitφ]‖22p
Proof. Setting r = eTitφ and s = Teitφ in identity (5.3), we obtain
‖eTintφ − (Teitφ)n‖p ≤ ‖eTitφ − Teitφ‖pn∑k=1
‖Teitφ‖k−1‖eTitφ‖n−k.
Since ‖eTitφ‖ = 1 and ‖Teitφ‖ ≤ 1, we find
‖eTintφ − (Teitφ)n‖p ≤ n‖eTitφ − Teitφ‖p. (5.5)
By Lemma 5.2.6,
‖(Tf )n − Tfn‖p ≤n(n− 1)
2‖[P, f ]‖22p‖f‖n−2.
Setting f = eitφ, we have ‖f‖ = 1, so
‖(Teitφ)n − Teintφ‖p ≤n(n− 1)
2‖[P, eitφ]‖22p. (5.6)
Combining (5.5) and (5.6), we find
‖eTintφ − Teintφ‖p ≤ n2(‖eTitφ − Teitφ‖p + ‖[P, eitφ]‖22p)
and the result follows by scaling.
75
We are now able to obtain an analogue of Proposition 5.1.6 for summable Fredholm modules.
Below we regard φ as an operator on H and Tφ as an operator on PH, and we view σ(φ), σ(Tφ),
f(φ), and f(Tφ) accordingly.
Theorem 5.2.9. If either
(1) f is holomorphic on a neighborhood of σ(φ) ∪ σ(Tφ) and there is a contour Γ that defines
both f(φ) and f(Tφ), or
(2) φ is self-adjoint and f is C∞ on σ(φ) ∪ σ(Tφ),
then f(Tφ)− Tf(φ) ∈ Lp.
Proof.
(1) Notice that (λ− PφP )−1 − P (λ− φ)−1P can be written as
P [(λ− φ)−1, P ][φ, P ]P (λ− PφP )−1.
Since [P, φ] ∈ L2p, the assignment
λ 7→ (λ− PφP )−1 − P (λ− φ)−1P
is a continuous map into Lp. Hence
f(Tφ)− Tf(φ) =1
2πi
∫Γ
((λ− PφP )−1 − P (λ− φ)−1P
)f(λ) dλ
converges in Lp.
(2) As in Proposition 5.1.5, we may assume that f has compact support, so that f = g for a
Schwartz class function g. Then
f(Tφ)− Tf(φ) =1√2π
∫ (eT−itφ − Te−itφ
)g(t) dt
and the result follows by Proposition 5.2.8.
76
5.3 Toeplitz operators and tame symbols
In this section we apply our techniques to Toeplitz operators and obtain formulas for joint
torsion in terms of Tate tame symbols. Let P : L2(S1) → H2(S1) be the orthogonal projection
onto the Hardy space H2(S1). Any function φ ∈ L∞(S1) defines a bounded operator on L2(S1) by
multiplication by φ. Then one has the Toeplitz operator Tφ = PφP .
5.3.1 H∞ symbols
Proposition 5.3.1. Suppose φ ∈ C(S1) ∩H∞(S1) is invertible in H∞(S1).
(1) If |λ| > 1, then τ(Tφ, Tz − λ) = 1.
(2) If |λ| < 1, then τ(Tφ, Tz−λ) = φ(λ), with φ extended holomorphically to the interior of the
unit disk.
Proof. First notice that Tφ is invertible with inverse T1/φ. If |λ| > 1, then z − λ is invertible in
H∞(S1) as well. The operators Tφ and Tz − λ commute, so in this case τ(Tφ, Tz − λ) = 1.
Now suppose |λ| < 1. By Lemma 4.4.6(6), it is enough to show that τ(Tz − λ, Tφ) = φ(λ).
In this case, coker(Tz − λ) = 0 and
ker(Tz − λ) = span
(1
1− λz=
∞∑k=0
(λz)k
).
The operator Tφ acts as multiplication by φ(λ) on the one dimensional subspace ker(Tz − λ). In
particular,
detTφ|ker(Tz−λ) = φ(λ).
This is the joint torsion by Lemma 3.2.4 since Tφ is invertible and commutes with Tz − λ.
Proposition 5.3.2. Let λ, µ ∈ C.
(1) If |λ1| > 1 and |λ2| > 1, then τ(Tz − λ1, Tz − λ2) = 1.
(2) If |λ1| < 1 and |λ2| > 1, then τ(Tz − λ1, Tz − λ2) = (λ1 − λ2)−1.
77
(3) If |λ1| > 1 and |λ2| < 1, then τ(Tz − λ1, Tz − λ2) = λ2 − λ1.
(4) If |λ1| < 1 and |λ2| < 1, then τ(Tz − λ1, Tz − λ2) = −1.
Proof. In case (1), both Tz − λ1 and Tz − λ2 are invertible in H∞(S1) and commute with each
other, so τ(Tz − λ1, Tz − λ2) = 1.
Cases (2) and (3) follow from the preceding proposition.
For (4), we use the multiplicative property of joint torsion:
τ(Tz − λ1, Tz − λ2) = τ(Tz, Tz) · τ(Tz, Tz(z−λ2))−1
· τ(Tz(z−λ1), Tz)−1 · τ(Tz(z−λ1), Tz(z−λ2))
The first term is −1, and the last term is 1 since Tz(z−λ1) and Tz(z−λ2) are invertible and commute
with each other. The middle two terms are both 1 by the above proposition. Hence, τ(Tz−λ1, Tz−
λ2) = −1.
Let us recall the notion of tame symbol mentioned in Section 3.1. If f and g are meromorphic
at λ ∈ C, then the quotient
fordλ(g)
gordλ(f)
is regular at λ. Here, ordλ denotes the order of the zero or pole at λ.
Definition 5.3.3. The tame symbol cλ(f, g) of f and g at λ is defined as
cλ(f, g) = (−1)ordλ(f)·ordλ(g) fordλ(g)
gordλ(f)(λ).
Definition 5.3.4. If a ∈ C is nonzero, the Blaschke factor Ba is
Ba(z) =|a|a
a− z1− az
.
Let B0(z) = z and B∞(z) = z. A product of Blaschke factors is known as a Blaschke product.
Notice that for z ∈ S1, we have
Ba(z) = Ba(z) = B1/a(z). (5.7)
The preceding propositions may be rephrased in terms of tame symbols, and in fact we have:
78
Proposition 5.3.5. Suppose f and g are products of
(1) invertible functions in C(S1) ∩H∞(S1),
(2) polynomials, and
(3) Blaschke factors Ba with |a| < 1.
If f and g are non-vanishing on S1, then
τ(Tf , Tg) =∏|λ|<1
cλ(f, g)
Proof. A straightforward calculation with f(z) = z − λ1 and g(z) = z − λ2 verifies that
∏|λi|<1
cλi(z − λ1, z − λ2)
agrees with (1)-(4) in Proposition 5.3.2. The result then holds for polynomials since both joint
torsion and the tame symbol are multiplicative. By Proposition 5.3.1, we find that the result
holds for factors of type (1) and (2). If |a| < 1, then Ba is the product of a polynomial and
(1− az)−1 ∈ H∞(S1). Hence factors of type (3) are products of types (1) and (2).
We will need the following Beurling-Szego factorization into inner and outer functions. See
for instance [18].
Theorem 5.3.6. If f ∈ H∞(S1) is continuous and non-vanishing on S1, then there exists an outer
function φ that is invertible in H∞(S1) such that
f = φ ·∏
Ba
where the above product is taken over finitely many zeros a with |a| < 1.
The following result was first obtained in [15, Proposition 1]. See also [47], and see [40] for a
generalization to the multivariable setting.
79
Theorem 5.3.7. If f, g ∈ H∞(S1) are continuous and non-vanishing on S1, then τ(Tf , Tg) is the
product of tame symbols:
τ(Tf , Tg) =∏|a|<1
ca(f, g).
Proof. As in the preceding theorem, write
f = φf ·∏
Ba, g = φg ·∏
Bb.
By Proposition 5.3.5, the joint torsion numbers
τ(φf , φg), τ(φf , Bb), τ(Ba, φg), τ(Ba, Bb)
agree with the corresponding tame symbols. The result then follows since both joint torsion and
the tame symbol are bimultiplicative.
5.3.2 L∞ symbols
In this section we extend the above result to the noncommutative setting.
Proposition 5.3.8. Suppose φ ∈ C(S1) ∩H∞(S1) is invertible in H∞(S1).
(1) If |λ| > 1, then τ(Tφ, Tz − λ) = φ(1/λ)φ(0) .
(2) If |λ| < 1, then τ(Tφ, Tz − λ) = 1φ(0) .
Proof. If λ = 0, then
τ(Tφ, Tz) · τ(Tφ, Tz) = τ(Tφ, I) = 1.
By Proposition 5.3.1, the second factor is φ(0), so the result follows in this case.
If λ 6= 0, we may write
− 1
λ(z − λ)z = z − 1
λ
so that
τ(Tφ, T−1/λ) · τ(Tφ, Tz−λ) · τ(Tφ, Tz) = τ(Tφ, Tz−1/λ)
80
The first factor is 1 since φ is invertible and the third factor is φ(0). Hence
τ(Tφ, Tz−λ) =τ(Tφ, Tz−1/λ)
φ(0)
and result follows by the Proposition 5.3.1.
Proposition 5.3.9. Let λ, µ ∈ C.
(1) If |λ1| > 1 and |λ2| > 1, then τ(Tz − λ1, Tz − λ2) = 1− (λ1λ2)−1.
(2) If |λ1| < 1 and |λ2| > 1, then τ(Tz − λ1, Tz − λ2) = −λ−12 .
(3) If |λ1| > 1 and |λ2| < 1, then τ(Tz − λ1, Tz − λ2) = −λ−11 .
(4) If |λ1| < 1 and |λ2| < 1, then τ(Tz − λ1, Tz − λ2) = (λ1λ2 − 1)−1.
Proof. This result follows from Proposition 5.3.2, as the preceding proposition follows from Propo-
sition 5.3.1.
Notice that z − λ extends meromorphically to the interior of the unit disk as
1
z− λ
with a simple pole at 0 and a simple zero at 1λ . A straightforward verification shows that the
previous two propositions express the joint torsion as a product of tame symbols. Since a Blaschke
factor is the ratio of two linear factors, we have the following noncommutative generalization of
Proposition 5.3.5:
Proposition 5.3.10. Suppose f and g are products of
(1) invertible functions in C(S1) ∩H∞(S1),
(2) trigonometric polynomials in z and z, and
(3) Blaschke factors Ba with a ∈ C ∪ ∞.
81
If f and g are non-vanishing on S1, then
τ(Tf , Tg) =∏|λ|<1
cλ(f, g).
Here f and g have been extended meromorphically to the interior of the unit disk.
Now suppose f ∈ L∞(S1) such that Tf is Fredholm. Then f is continuous and non-vanishing
on S1, say with winding number n. The function z−nf(z) has winding number zero, so there is a
continuous function f such that
ef(z) = z−nf(z).
Let f+ = P f , f− = (I − P )f , where P : L2(S1) → H2(S1) is the orthogonal projection as usual.
Then
f(z) = z−nef−ef+ . (5.8)
Thus we may write f = f−f+ with f+, f− ∈ H∞ continuous and non-vanishing. By Theorem 5.3.6,
we may write
f+ = f1 ·∏
Ba, f− = f2 ·∏
Bb
where f1, f2 ∈ H∞ are invertible in H∞, the zeros a satisfy |a| < 1, and the zeros b satisfy |b| > 1.
Letting f0 be the product of Blaschke factors above, we have the factorization
f = f0f1f2.
If f is smooth, then so is z−nf , and we can take f to be smooth as well. Consequently f+
and f− are smooth, for example because the projection P can be expressed in terms of the Hilbert
transform, which preserves regularity. Hence the related functions f±, fi, i = 0, 1, 2, are smooth as
well. Define gi, i = 0, 1, 2, similarly. As in Proposition 5.1.10, we see that joint torsion factors as a
discrete part (tame symbols) and a continuous part (a determinant):
Theorem 5.3.11. If f, g ∈ C∞(S1) are non-vanishing on S1, then
τ(Tf , Tg) =∏|a|<1
ca(f0f1, g0g1) · ca(g0, f2)
ca(f0, g2)·τ(Tf1 , Tg2)
τ(Tg1 , Tf2).
82
Here fi, gi are as above, and
τ(Tf1 , Tg2) = exp
(1
2πi
∫log f1 d(log g2)
).
for continuous choices of logarithms of f1 and g2, and similarly for τ(Tg1 , Tf2).
Proof. By the multiplicative property of joint torsion, we find that
τ(Tf , Tg) = τ(Tf0f1 , Tg0g1) · τ(Tf0f1 , Tg2) · τ(Tf2 , Tg0g1) · τ(Tf2 , Tg2).
The first factor is the product of tame symbols by Theorem 5.3.7. The fourth factor is 1 since Tf2
and Tg2 are invertible commuting operators. Next we calculate the second factor; the third factor
is dealt with similarly. Again using multiplicativity, the second factor is
τ(Tf0 , Tg2) · τ(Tf1 , Tg2).
For the first factor above, notice that f0 is still a Blaschke product and g2 ∈ H∞ is invertible
in H∞. Hence τ(Tf0 , Tg2) = τ(Tf0, Tg2)−1 by Lemma 4.4.6(6), and the latter is ca(f0, g2)−1 by
Proposition 5.3.10. In the second factor both Tf1 and Tg2 are invertible. Hence their joint torsion
is the multiplicative commutator
det(Tf1Tg2Tf−1
1Tg−1
2
)which is calculated as the claimed integral by Lemma 4.4.4 and Theorem 2.3.6.
5.3.3 An integral formula
Now we apply and refine the results of Section 5.2 in the case of Toeplitz operators. Let
L2 = L2(S1) and H2 = H2(S1). As a consequence of Proposition 2.3.4, we obtain the following:
Theorem 5.3.12. Let φ ∈ L∞(S1) ∩W12,2(S1). If either
(1) f is holomorphic on a neighborhood of φ(S1), or
(2) φ is real-valued and f is C∞ on φ(S1),
then f(Tφ)− Tfφ ∈ L1(H2).
83
Proof. Let Γ be an admissible contour for defining f(Tφ) ∈ B(H2), as in (5.2). By the above
discussion Γ can also be used to define f(φ) ∈ B(L2), that is,
f(φ) =1
2πi
∫Γ(λ− φ)−1f(λ) dλ
The result then follows by Theorem 5.2.9 with p = 1.
We conclude with an illustration of the above results by deriving an integral formula for the
joint torsion of Toeplitz operators [15, Theorem 7]. See also [30].
Theorem 5.3.13. If f, g ∈ C∞(S1) are non-vanishing functions, then
τ(Tf , Tg) = exp1
2πi
(∫S1
log f d(log g)− log g(p)
∫S1
d(log f)
).
The integrals are taken counterclockwise starting at any point p = eiα ∈ S1. If h(eiθ) =
|h(eiθ)|eiφ(θ) for a continuous function φ : [α, α+ 2π]→ R, then we take log h(eiθ) = log |h|+ iφ(θ).
Any other choice of log h will differ by a multiple of 2πi and hence will leave the quantity in the
theorem unaffected.
Proof. Let n and m be the winding numbers of f and g, respectively. Define f , f+, and f− as in
(5.8), and similarly for g. By Theorem 5.3.12, Tef− eTf ∈ L1, so
τ(Tf , Tg) = τ(Tz, Tz)mn · τ(Tz, Teg)
n · τ(Tef, Tz)
m · τ(eTf , eTg).
The first factor is (−1)mn by Proposition 5.3.2. By applying Proposition 5.3.1 with λ = 0 and both
φ = eg+ and φ = eg− , we find that the second term is e−ng+(0). Similarly, the third term is emf+(0).
By Lemma 4.4.4 and Theorem 2.3.6, the fourth term is
exp
(1
2πi
∫f dg
).
Hence
τ(Tf , Tg) = exp
(πimn+mf+(0)− ng+(0) +
1
2πi
∫f dg
). (5.9)
84
Now we calculate the last term in the exponential:∫f dg =
∫log(e−inθf) d(log(e−imθg)
=
∫−inθ dg +
∫log f d(log(e−imθg))
Integration by parts gives ∫−inθ dg = −inθg|α+2π
α +
∫ing dθ.
The first term is −2πing(p) since g has winding number zero. By writing g in terms of the
orthonormal basis elements eikθ, we see that the second term is 2πing+(0). Next we calculate∫log f d(log(e−imθg)) =
∫f · −imdθ +
∫inθ · −imdθ +
∫log f d(log g).
As before the first term is −2πimf+(0), and the second term is 2mnπ2 + 2πmnα. Combining this
with (5.9) gives
τ(Tf , Tg) = exp
(−ng(p)− imnα+
1
2πi
∫log f d(log g)
).
The result follows since the first term is
−n(−imα+ log g(p)) = imnα− 1
2πilog g(p)
∫d(log f).
Bibliography
[1] J. Anderson and L. N. Vaserstein. Commutators in ideals of trace class operators. IndianaUniv. Math. J., 354(2):345–372, 1986.
[2] William B. Arveson. p-summable commutators in dimension d. J. Operator Theory, 54(1):101–117, 2005.
[3] M. F. Atiyah and R. Bott. A Lefschetz fixed point formula for elliptic complexes. Ann. ofMath., 86:374–407, 1967.
[4] A. A. Beılinson. Higher regulators and values of L functions. J. Soviet Math., 30:2036–2070,1985.
[5] Jean-Michel Bismut and Daniel S. Freed. The analysis of elliptic families. I. Metrics andconnections on determinant bundles. Comm. Math. Phys., 106(1):159–176, 1986.
[6] Jean-Michel Bismut, Henri Gillet, and Christophe Soule. Analytic torsion and holomorphicdeterminant bundles. i. bott-chern forms and analytic torsion. Comm. Math. Phys. 115,115(1):49–78, 1988.
[7] L. G. Brown. The determinant invariant for operators with compact self-commutators. In Proc.Conf. on Operator Theory, volume 345 of Lecture Notes in Math., pages 210–228. Springer-Verlag, Berlin, Heidelberg, and New York, 1973.
[8] L. G. Brown. Operator algebras and algebraic k-theory. Bull. Amer. Math. Soc., 81:1119–1121,1975.
[9] J. Bruning and M. Lesch. Hilbert complexes. J. Funct. Anal., 108(1):88–132, 1992.
[10] Ulrich Bunke. A regulator for smooth manifolds and an index theorem. arXiv:1407. 1379,pages 1–53, 2014.
[11] Richard Carey and Joel Pincus. An exponential function for determining functions. IndianaUniv. Math. J., 23:1155–1165, 1974.
[12] Richard Carey and Joel Pincus. Commutators, symbols and determining functions. J.Functional Analysis, 19:50–80, 1975.
[13] Richard Carey and Joel Pincus. Reciprocity for Fredholm operators. Integral Equations andOperator Theory, 9:469–501, 1986.
86
[14] Richard Carey and Joel Pincus. Joint torsion. Preprint, pages 1–174, 1995-1996.
[15] Richard Carey and Joel Pincus. Joint torsion of Toeplitz operators with H∞ symbols. IntegralEquations and Operator Theory, 33:273–304, 1999.
[16] Richard Carey and Joel Pincus. Perturbation Vectors. Integral Equations and OperatorTheory, 35:271–365, 1999.
[17] L. A. Coburn. Toeplitz operators on odd spheres. In Proceedings of the Conference OperatorTheory (Dalhousie Univ. Halifax, N.S., 1973), volume 345 of Lecture Notes in Math., pages7–12. Springer, Berlin, 1973.
[18] Peter Colwell. Blaschke products. Bounded analytic functions. University of Michigan Press,Ann Arbor, MI, 1985.
[19] Alain Connes. Non-commutative differential geometry. IHES Publ. Math., 62:257–360, 1985.
[20] Alain Connes and Max Karoubi. Caractere multiplicatif d’un module de Fredholm. K-Theory,2(3):431–463, 1988.
[21] Alain Connes and Henri Moscovici. Cyclic cohomology, the Novikov conjecture and hyperbolicgroups. Topology, 29(3):345–388, 1990.
[22] Raul E. Curto. Fredholm and invertible n-tuples of operators. The deformation problem.Trans. Amer. Math. Soc., 266:129–159, 1981.
[23] P. Deligne. Le symbol modere. Inst. Hautes Etudes Sci. Publ. Math., 73:147–181, 1991.
[24] Ronald G. Douglas. Banach algebra techniques in operator theory. Springer-Verlag, 2nd ed.edition, 1998.
[25] Ronald G. Douglas. A new kind of index theorem. In Analysis, geometry and topology ofelliptic operators, pages 369–382. World Sci. Publ., Hackensack, NJ, 2006.
[26] T. Ehrhardt. A generalization of Pincus’ formula and Toeplitz operator determinants. Arch.Math. (Basel), 80(3):302–309, 2003.
[27] Jorg Eschmeier. Quasicomplexes and Lefschetz numbers. Acta. Math. Sci. (Szeged), 79(3-4):611–621, 2013.
[28] Jorg Eschmeier and Mihai Putinar. Spectral theory and sheaf theory. iii. J. Reine Agnew.Math., 354:150–163, 1984.
[29] Jorg Eschmeier and Mihai Putinar. Spectral decompositions and analytic sheaves, volume 10of London Mathematical Society Monographs. New Series. The Clarendon Press, OxfordUniversity Press, 1996.
[30] H. Esnault and E. Viehweg. Deligne-Beılinson cohomology. In Beılinson’s conjectures on specialvalues of L-functions, volume 4 of Perspect. Math., pages 43–91. Academic Press, Boston, MA,1988.
[31] Bjorn Gustafsson and Vladimir G. Tkachev. The resultant on compact Riemann surfaces.Comm. Math. Phys., 286(1):313–358, 2009.
87
[32] J. W. Helton and R. Howe. Integral operators: commutators, traces, index and homology.In Proc. Conf. on Operator Theory, volume 345 of Lecture Notes in Math., pages 141–209.Springer-Verlag, 1973.
[33] J. William Helton and Roger E. Howe. Traces of commutators of integral operators. ActaMath., 135(3-4):271–305, 1975.
[34] Nigel Higson and John Row. Analytic K-Homology. Oxford University Press, 2000.
[35] Jens Kaad. A calculation of the multiplicative character. J. Noncommut. Geom., 5(3):351–385,2011.
[36] Jens Kaad. Comparison of secondary invariants of algebraic k-theory. J. K-Theory, 8(1):169–182, 2011.
[37] Jens Kaad. Joint torsion of several commuting operators. Adv. Math., 229:442–486, 2012.
[38] Jens Kaad and Ryszard Nest. A transformation rule for the index of commuting operators.arXiv:1208.1862, pages 1–28, 2012.
[39] Jens Kaad and Ryszard Nest. Canonical holomorphic sections of determinant line bundles.arXiv:1403.7937, pages 1–45, 2014.
[40] Jens Kaad and Ryszard Nest. Tate tame symbol and the joint torsion of commuting operators.arXiv:1408.3851, pages 1–26, 2014.
[41] F. F. Knudsen and D. Mumford. The projectivity of the moduli of stable curves. i. Math.Scand., 39(1):19–55, 1976.
[42] Maxim Kontsevich and Simeon Vishik. Geometry of determinants of elliptic operators. InFunctional analysis on the eve of the 21st century (New Brunswick, NJ, 1993), volume 131 ofProgr. Math., pages 173–197. Birkhauser Boston, Boston, MA, 1995.
[43] Matthias Lesch. Pseudodifferential operators and regularized traces. In Motives, QuantumField Theory, and Pseudodifferential Operators, volume 12 of Clay Mathematics Proceedings,pages 37–72. Amer. Math. Soc., 2010.
[44] Jean-Lois Loday. Cyclic homology. Springer-Verlag, 1992.
[45] Joseph Migler. Functional calculus and joint torsion of pairs of almost commuting operators.arXiv:1409.6289, 2014.
[46] Joseph Migler. Joint torsion equals the determinant invariant. To appear in the Journal ofOperator Theory, arXiv:1403.4882, 2015.
[47] Efton Park. The tame symbol and determinants of Toeplitz operators. arXiv:0905.4510, pages1–10, 2009.
[48] Daniel Quillen. Determinants of Cauchy-Riemann operators over a Riemann surface.Funktsional. Anal. i Prilozhen., 19(1):37–41, 1985.
[49] D. B. Ray and I. M. Singer. Analytic torsion for complex manifolds. Ann. of Math., 98:154–177,1973.
88
[50] Jonathan Rosenberg. Algebraic K-theory and its applications. Springer-Verlag, 1994.
[51] Barry Simon. Trace ideals and their applications, 2nd ed. American Mathematical Society,2005.
[52] R. G. Swan. Vector bundles and projective modules. Trans. Amer. Math. Soc., 105:264–277,1962.
[53] G. Szego. Ein Grenzwertsatz uber die Toeplitzschen Determinanten einer reelen positivenfunction. Math. Ann., 76:490–503, 1915.
[54] G. Szego. On certain Hermitian forms associated with the Fourier series of a positive function.Festschrift Marcel Riesz, pages 228–238, 1952.
[55] N. Tarkhanov. The Euler characteristic of a Fredholm quasicomplex. Funktsional. Anal. iPrilozhen., 41(4):87–93, 2007.
[56] N. Tarkhanov and D. Wallenta. The Lefschetz number of sequences of trace class curvature.Mathematical Sciences, 6(44), 2012.
[57] N. N. Tarkhanov. Euler characteristic of Fredholm quasicomplexes. Functional Analysis andIts Applications, 41(4):318–322, 2007.
[58] John Tate. Symbols in arithmetic. In Actes du Congres International des Mathematiciens(Nice, 1970), Tome 1, pages 201–211. Gauthier-Villars, Paris, 1971.
[59] Joseph L. Taylor. A joint spectrum for several commuting operators. J. Functional Analysis,6:172–191, 1970.
[60] Dan Voiculescu. Some results on norm-ideal perturbations of Hilbert space operators. J.Operator Theory, 2:3–37, 1979.
[61] Dan Voiculescu. Almost normal operators mod Hilbert-Schmidt and the K-theory of thealgebras EΛ(Ω). arXiv:1112.4930, 2011.
[62] Antony Wassermann. Analysis of operators. www.dpmms.cam.ac.uk/%7Eajw, 2006.
[63] Mariusz Wodzicki. Algebraic K-theory and functional analysis. In First European Congress ofMathematics, Vol. II (Paris, 1992), volume 120 of Progr. Math., pages 485–496. Birkhauser,Basel, 1994.
Appendix A
The existence of perturbations
Let R be a ring and let J be a two-sided ideal. Suppose a1, . . . , an ∈ R/J are commuting
elements. We begin by describing some obstructions to lifting the ai to commuting Ai ∈ R. The
strategy is to replace R and R/J by simple models – either a free ring or one with the minimal
number of relations – and look at homomorphisms out of them obtained by lifting commuting
elements. Later we specialize the case when R = B, the algebra of bounded operators on a Hilbert
space, and J = L1, the ideal of trace class operators.
Let q : R → R/J be the quotient map. First suppose R is a unital ring. Let ϕ :
Z[x1, . . . , xn] → R/J be the unital homomorphism that maps xi to ai. Let F be a covariant
functor on rings. If there exist commuting lifts Ai ∈ R of the ai, then we can define a homorphism
ϕ : Z[x1, . . . , xn]→ R as above, and the following diagram commutes:
F(R)
Fq
F(Z[x1, . . . , xn])
Fϕ66
Fϕ// F(R/J)
If F(Z[x1, . . . , xn]) 6= 0, pick any nonzero x ∈ F(Z[x1, . . . , xn]). Then Fϕ(x) ∈ imFq, thus
providing an obstruction to lifting the ai to commuting Ai.
Likewise, if G is a contravariant functor on rings, then the following diagram commutes:
G(R)Gϕ
ww
G(Z[x1, . . . , xn]) G(R/J)
Gq
OO
Gϕoo
90
In general, if R is unital or otherwise, let
Xn = 〈x1, . . . , xn |xixj = xjxi〉
be the free commutative ring on n generators. Then one may replace Z[x1, . . . , xn] above everywhere
with Xn.
Next consider the “AB = CD” problem raised in Section 3.3. Thus let a1, a2 ∈ R/J be two
commuting elements and consider the ring
Y2 = 〈x1, x2, y1, y2 |x1x2 = y2y1〉.
Let ψ : Y2 → R/J be the homomorphism defined by xi, yi 7→ ai. This map factors through X2. If
there exist lifts Ai, Bi ∈ R of ai modulo J such that A1A2 = B2B1, then let ψ : Y2 → R be the
homomorphism defined by xi 7→ Ai, yi 7→ Bi. One may then proceed as above.
If one of the ai is invertible, it turns out that there is no obstruction to finding lifts Ai, Bi ∈ R
of ai modulo J such that A1A2 = B2B1. In fact, we have the following slightly stronger result,
which was obtained in [27] in the case when R = B and J = Lp for some p.
Proposition A.0.14. Let R be a unital ring and let J be a two-sided ideal in R. Let a1, a2 ∈ R/J
be commuting elements.
(1) If a1 is invertible, then for any lifts A1, B1 ∈ R of a1, there exist lifts A2, B2 of a2 such
that A1A2 = B2B1.
(2) If a2 is invertible, then for any lifts A2, B2 ∈ R of a2, there exist lifts A1, B1 of a1 such
that A1A2 = B2B1.
Proof. For the proof of (1), let C2 be a lift of a2 and let Q1 be any lift of a−11 . Then we may take
A2 = C2Q1B1, B2 = A1C2Q1
since
A1A2 = A1C2Q1B1
= B2B1
91
The second claim is verified similarly.
In light of the assumptions in [16], there is also the case when A1, A2 are specified and
required to appear in the answer. Of course, Theorem 4.3.3 and the above proposition show that
these questions are no longer necessary for the study of joint torsion of two almost commuting
Fredholm operators.
So far the treatment has been purely algebraic. Now in the case when R = B and J is any
operator ideal, we have the following consequence of the Open Mapping Theorem:
Proposition A.0.15. Let a1, a2 ∈ B/J be commuting invertible elements. Suppose Ai ∈ B are lifts
of ai such that the operator −A2
A1
: H → H2
has closed range. Then there exist lifts Bi of ai such that
A1B2 = A2B1.
Let a1, a2 ∈ B/J be commuting invertible elements. We conclude with some partial positive
results and negative results to the following question: Given lifts Ai ∈ B of ai, when do there exist
lifts Bi such that A1A2 = B2B1?
Proposition A.0.16. Let R be a unital ring and let J be a two-sided ideal. Let a1, a2 ∈ R/J be
commuting elements with lifts A1, A2 ∈ R. If either
(1) a1 has a left invertible lift, or
(2) a2 has a right invertible lift,
then there exist lifts Bi ∈ R of ai such that A1A2 = B2B1.
Proof. In the case of (1), let F1 ∈ J be such that A1 + F1 has a left inverse (A1 + F1)L. Then
A1A2 =(A1A2(A1 + F1)L
)(A1 + F1)
92
so we may take B1 = A1 + F1 and B2 = A1A2(A1 + F1)L. The proof the second claim follows
similarly.
Corollary A.0.17. Let J ∈ B be a two-sided ideal. Let a1, a2 ∈ B/J be commuting elements with
lifts A1, A2 ∈ B. If either
(1) A1 is Fredholm with indA1 ≤ 0, or
(2) A2 is Fredholm with indA2 ≥ 0,
then there exist lifts Bi ∈ B of ai such that A1A2 = B2B1.
Proof. Every two-sided ideal J contains the finite rank operators. Moreover, a Fredholm operator
with non-positive (resp. non-negative) index can be perturbed by a finite rank operator to an
injective (resp. surjective) operator. The Open Mapping Theorem then guarantees the existence of
a left (resp. right) inverse.
Example A.0.18. The above conditions are sufficient but not necessary. Indeed, let H = l2(Z+),
let J be any two-sided ideal, and let S be the unilateral shift with index −1. Let A1 = SS∗S∗,
A2 = SSS∗, B1 = S∗, and B2 = S. Then indA1 > 0, indA2 < 0, and
A1A2 = SS∗S∗SSS∗
= SS∗
= B2B1
Example A.0.19. This example shows that the conclusion of the above corollary is not true in
general. Let A1 be a bounded surjective operator with nontrivial kernel, for example a unilateral
shift, and let A2 be a bounded right inverse. Suppose there exist perturbations Bi of Ai modulo any
ideal such that
A1A2 = B2B1.
93
Since A1A2 = I, this implies that B1 is injective, and hence has non-positive index. This contradicts
the fact that A1, and hence also B1, have positive index. Thus the conclusion of the corollary fails
to hold in this case.
We do not yet have a complete characterization of pairs (A1, A2) that satisfy the conclusion
of the above corollary, but at least we have the following:
Corollary A.0.20. Let J ∈ B be a two-sided ideal. Let a1, a2 ∈ B/J be commuting invertible
elements with lifts A1, A2 ∈ B. Then there exist lifts Bi ∈ B of ai such that either
A1A2 = B2B1 or A2A1 = B1B2.
![Conformal designs based on vertex operator algebras · Extremal vertex operator algebras have been introduced in [19] and are vertex operator algebras V with only one irreducible](https://img.pdfslide.us/doc/110x75/5f8abfe07b0d5b260a39e9ab/conformal-designs-based-on-vertex-operator-algebras-extremal-vertex-operator-algebras.jpg)
