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THE HAAGERUP APPROXIMATION PROPERTY AND

RELATIVE AMENABILITY

JON P. BANNON

Abstract. In this paper we provide a description of the Haagerup approx-imation property in terms of Connes’s theory of correspondences. We showthat if N ⊂ M is an amenable inclusion of finite von Neumann algebras andN has the Haagerup approximation property, then M also has the Haagerupapproximation property. This answers in the affirmative a question of SorinPopa.

1. Introduction

An action α of a locally compact group G on an affine Hilbert space H is metri-cally proper if for every bounded subset B of H, the set g ∈ G | α(g)B ∩B 6= ∅is relatively compact in H. If a locally compact group admits a metrically properisometric action on some affine Hilbert space, that group is said to be a-T-menable,or to have the Haagerup property. The class of groups having the Haagerup prop-erty is quite large. It includes all compact groups, all locally compact amenablegroups and all locally compact groups that act properly on trees (e.g. Fn). Study-ing the class of Haagerup groups has been a fertile endeavor. For example, theBaum-Connes conjecture is solved for this class [HigKa].

Along with the above definition due to Gromov in 1992, there are many equiv-alent characterizations of the Haagerup property. A locally compact group G hasthe Haagerup property if it admits a strongly continuous unitary representation,weakly containing the trivial representation, with matrix coefficients vanishing atinfinity on G. Equivalently, G has the Haagerup property if there is a sequence ofnormalized continuous positive definite functions on G vanishing at infinity on Gand converging to 1 uniformly on compact subsets. Also, G has the Haagerup prop-erty if there is a continuous positive real valued function on G that is conditionallynegative definite. An extensive treatment of the Haagerup property for groups canbe found in the book [CCJJV].

The Haagerup property is a strong negation to Kazhdan’s property T, in thateach of the equivalent definitions above stands opposite to a definition of propertyT cf. [CCJJV]. A. Connes and V. Jones introduced a notion of property T for vonNeumann algebras using Connes’s theory of correspondences [CoJo]. A group hasKazhdan’s property T if and only if the associated group von Neumann algebra hasthe Connes-Jones property T. Since property T has a natural analogue in the theoryof von Neumann algebras, we may expect that the same should be possible for theHaagerup property. Such an analogue indeed exists, as M. Choda proved in [Cho]

Date: 24 September 2007.Key words and phrases. Operator Algebras, Functional Analysis, von Neumann algebras.

1

2 JON P. BANNON

that a group has the Haagerup property if and only if its associated group von Neu-mann algebra has the Haagerup approximation property first introduced in [Haa1]:A finite von Neumann algebra M with trace τ has the Haagerup approximationproperty if there exists a net (ϕα)α∈Λ of normal completely positive maps from Mto M that for every α ∈ Λ satisfy (1) τ ϕα(x

∗x) ≤ τ(x∗x) for all x ∈ M, (2) ϕα

induces a compact bounded operator on L2(M) and (3) limα ||ϕα(x)− x||2 = 0 forall x ∈ M. In this paper we provide a description of the Haagerup approximationproperty in the language of correspondences.

In recent breakthrough work, Sorin Popa has combined relative versions of prop-erty T and the Haagerup property to create “deformation malleability” techniquesthat he, Adrian Ioana and Jesse Peterson have used to solve old open questionsabout type II1 factors [Po2], [IPP]. In particular, the authors construct factorswith trivial fundamental group and factors with no outer automorphism. In thefirst of these papers Popa proves that if a type II1 factor possesses a Cartan sub-algebra relative to which the factor has property T and the Haagerup propertythen, up to inner automorphisms of the factor, this Cartan subalgebra is unique.It follows that any invariant of the measurable equivalence relation associated tothe Cartan subalgebra is actually an invariant of the factor. In Section 3.5.2 ofthe first paper [Po2] above, Popa points out connections between the Haagerupapproximation property and the amenability of inclusions of finite von Neumannalgebras. An inclusion N ⊆ M of finite von Neumann algebras is amenable if thereexists an M-hypertrace on the result 〈M, eN 〉 of the basic construction, i.e. a stateρ on 〈M, eN 〉 with ρ(xy) = ρ(yx) for all x ∈ M and y ∈ 〈M, eN 〉. Popa asks thefollowing question: If N ⊆ M is an inclusion of finite von Neumann algebras andN has the Haagerup approximation property, then does M also have the Haagerupapproximation property? This question is motivated by the analogous known re-sult for groups: If G is a closed subgroup of G0, and G is co-Følner in G0 in thesense of Eymard, then whenever G has the Haagerup property so does G0. A proofof this result can be found in [CCJJV]. The condition that G is co-Følner in G0

in the sense of Eymard is equivalent to the amenability of the functorial inclusionL(G) ⊆ L(G0) of group von Neumann algebras [Po2].

The affirmative answer to Popa’s question for group von Neumann algebras isfound in the work of Anantharaman-Delaroche [AD2]. She proves that the com-pact approximation property is equivalent to the Haagerup approximation propertyin the group von Neumann algebra case. In a bit more detail, a separable finitevon Neumann algebra M has the compact approximation property if there ex-ists a net (φα)α∈Λ of normal completely positive maps from M to M such thatfor all x ∈ M we have limα φα(x) = x ultraweakly and for all ξ ∈ L2(M) andα ∈ Λ, the map x 7→ φα(x)ξ is the restriction to M of a compact operator onL2(M). Anantharaman-Delaroche proved that if N ⊆ M is an amenable inclusionand N has the compact approximation property, then M must have the compactapproximation property. Using properties (2) and (3) in the above definition of theHaagerup approximation property she proves that the Haagerup approximationproperty implies the compact approximation property. It follows that if N ⊆ Mis an amenable inclusion and N has the Haagerup approximation property, thenM has the compact approximation property. Popa’s question is then answered byappealing to the above result of Choda to establish that for group von Neumann

HAAGERUP APPROXIMATION PROPERTY AND RELATIVE AMENABILITY 3

algebras the compact approximation property implies the Haagerup approximationproperty.

In this paper, we answer Popa’s question affirmatively for all finite von Neumannalgebras. We point out that if it is the case that the Haagerup approximationproperty is equivalent to the compact approximation property for all finite vonNeumann algebras, then our result would follow from Anantharaman-Delaroche’stheorem.

The author wishes to express his deepest thanks to Paul Jolissaint and MingchuGao for carefully reading an early version of this paper.

1.1. Background on Correspondences. We aim to express the Haagerup ap-proximation property in the language of correspondences, so we will give a briefreview of some basic facts from the theory of correspondences in this section. Amore extensive treatment of the theory of correspondences can be found in [Co],[CoJo] , [Po1], [AD1], [AD2].

Let M and N be W ∗-algebras. We first give the standard definition of corre-spondence between M and N as binormal bimodule. Let N o denote the oppositealgebra of N , which is identical in every regard with N except that the productis reversed: x ·

opy = yx for all x, y in N . A representation of N o is an anti-

representation of N . A correspondence between M and N is a binormal M-Nbimodule, i.e. a triple (H, πM, πNo), where H is a Hilbert space with commutingunital normal representations πM of M and πNo of N o on B(H).

Correspondences can equivalently described as certain self-dual Hilbert modules.This shift in point of view is essential for our purposes. We recall below some factsabout self-dual Hilbert modules that will be needed in the sequel. For a morethorough treatment of Hilbert module theory, see [Pas], [Rie1], [Rie2].

We present next the definition of right Hilbert N -module as a right N -modulewith N -valued inner product. We follow the standard practice of naming suchmodules with capital roman letters X,Y, Z in what follows. By an algebraic rightN -module we mean a right N -module in the algebraist’s sense, i.e. only the ringstructure of N o is considered. A right pre-Hilbert N -module is an algebraic rightN -module X equipped with a conjugate- bilinear map1 〈·, ·〉N : X → N satisfyingfor all ξ, η ∈ X and y ∈ N : (1) 〈ξ, ξ〉N ∈ N+ and equals zero only if ξ = 0, (2)〈ξ, η〉N = 〈η, ξ〉∗N and (3) 〈ξ, ηy〉N = 〈ξ, η〉N y. Such a map 〈·, ·〉N is called an

N -valued inner product. We define a norm on X by ||ξ||N = ||〈ξ, ξ〉N ||1/2, where||〈ξ, ξ〉N || is the C∗-algebra norm of 〈ξ, ξ〉N in N . If X is complete with respectto this norm and the ultraweak closure of the linear span of the range 〈X,X〉N of〈·, ·〉N is all of N , we say that X is a right Hilbert N -module.

Notice that in the above definition that if we begin with an algebraic right N -module and then take the completion with respect to the prescribed norm, theresulting right Hilbert N -module need not be a Hilbert space with respect to thatnorm. A simple example of a right Hilbert N -module is the W ∗-algebra N itselfwith obvious right action and N -valued inner product given by 〈n1, n2〉N = n∗

1n2

for all n1, n2 ∈ N . Note that even if N = M2(C), the parallelogram law fails forthe norm || · ||N , so the right Hilbert module is not a Hilbert space.

1We follow the standard practice and regard the map as conjugate linear in the first variable.With Hilbert spaces, however, we maintain conjugate linearity in the second variable.

4 JON P. BANNON

Suppose that X and Y are right Hilbert N -modules. Let BN (X,Y ) denote theset of all bounded N -linear maps from X into Y . We say that X is self-dual iffor every ϕ ∈ BN (X,N ) there exists ξϕ ∈ X such that ϕ(ξ) = 〈ξϕ, ξ〉N for allξ ∈ X. By the double commutant theorem, the W ∗-algebra N viewed as a rightHilbert N -module as above is self-dual. Unlike in Hilbert space, the completenessof right Hilbert N -modules doesn’t guarantee that every right Hilbert N -moduleis self-dual (cf. [Pas]).

An M-N correspondence X is a pair (X, π), with X a self-dual right HilbertN -module and π a faithful unital ∗-homomorphism π from M into BN (X) =BN (X,X), such that for every ξ ∈ X , the map φX,ξ : m 7→ 〈ξ, π(m)ξ〉N = 〈ξ,mξ〉Nfrom M to N is ultraweakly continuous. These are also called normal M-rigged Nmodules in Definition 5.1 of [Rie2]. The completely positive maps φX,ξ are calledthe coefficients of X . Hence we may informally rephrase the definition as: An M-Ncorrespondence is a self-dual right Hilbert N -module along with a left action of Mthat intertwines the right action of N such that every coefficient of X is ultraweaklycontinuous. The W ∗-algebra N viewed as a right Hilbert N -module with obviousleft N -action is a simple example of an N -N correspondence. This correspondenceis called the trivial, or identity, correspondence. In order to economically indicatethat a correspondence X is an M-N correspondence we may write X as MXN .

We reiterate that a correspondence between M and N is necessarily a Hilbertspace and an M-N correspondence is not. We nevertheless can relate the twoconcepts in a very natural way. Let L2(N ) be the Hilbert space of a standard formof N with right N -action written in terms of the modular conjugation operatorJ and the standard action by ξn = Jn∗Jξ for ξ ∈ L2(N ) and n ∈ N . To everyM-N correspondence X we associate the correspondence H(X) between M and Nobtained by inducing the standard representation of N up to M via X in the wayprescribed by Rieffel (cf. Theorem 5.1 of [Rie1]). Very roughly,H(X) is constructedby separation and completion of the algebraic bimodule tensor product X⊗

NL2(N )

with respect to the conjugate-bilinear form defined on simple tensors by

〈x1 ⊗ ξ1, x2 ⊗ ξ2〉 = 〈〈x2, x1〉N ξ1, ξ2〉.

The right and left actions of M and N on L2(N ) are given on simple tensors bym(x ⊗ ξ)n = mx ⊗ ξn for m ∈ M and n ∈ N . With these actions H(X) is abinormal M-N bimodule. Let R be a W ∗-algebra. Suppose that H is a correspon-dence between M and N and K is a correspondence between R and N . Denoteby BN (H,K) the set of all bounded N -linear maps from H into K. To every cor-respondence H between M and N we associate an M-N correspondence X(H) asfollows: Regard the standard Hilbert space L2(N ) as an N -N correspondence withthe standard left action and the right action given above. The M-N correspon-dence X(H) is the algebraic right N -module BN (H, L2(N )) with right N -actiongiven by the composition of operators and N -valued inner product given by

〈ϕ, φ〉N = ϕ∗φ

for all ϕ, φ ∈ BN (H, L2(N )), along with the left M action given by composi-tion of operators. Note that ϕ∗φ ∈ N above since the Hilbert space adjointϕ∗ is in BN (L2(N ),H) and BN (L2(N ), L2(N )) = N by the double commutanttheorem. The maps X 7→ H(X) and H 7→X(H) above are inverses up to uni-tary equivalence (cf. Theorem 6.2 of [Rie2]). To illustrate with a simple exam-ple: if we begin with the correspondence H = L2(N ) between N and itself, then

HAAGERUP APPROXIMATION PROPERTY AND RELATIVE AMENABILITY 5

X(L2(N )) = BN (L2(N ), L2(N )) = N and conversely if we begin with the trivialN -N correspondence X = N , then H(N ) = N ⊗

NL2(N ) = L2(N ). We identify

the Hilbert module picture and the bimodule picture via the above procedure, andthereby need not distinguish “correspondence between M and N” and “M-N cor-respondence” in what follows.

We describe the direct sum of correspondences. Let Xii∈S be a family of M-N correspondences, and let 〈·, ·〉N ,i denote the N -valued inner product of Xi. LetF be set of all finite subsets of S directed by inclusion. For S-tuples x = (xi)i∈S

and y = (yi)i∈S with xi, yi ∈ Xi and S ∈ F define 〈x, y〉SN =∑

i∈S〈xi, yi〉N ,i.

Define⊕

i∈SXi to be the set of all S-tuples x = (xi)i∈S satisfying sup||〈x, x〉SN ||: S ∈ F < ∞. Under coordinatewise operations,

⊕i∈SXi is a right Hilbert N -

module that is self-dual if and only if eachXi is self-dual (cf. [Pas]). With pointwiseleft M-actions,

⊕i∈SXi is an M-N correspondence.

To any normal completely positive map ϕ : M → N , we can associate an M-N correspondence Xϕ via the Stinespring construction. In the Hilbert modulepicture, Xϕ is the result of gifting the algebraic tensor product M⊙N with rightN action and intertwining left M action given by m0(m⊗ n)n0 = m0m⊗ nn0 andthe N -valued inner product

〈m⊗ n,m1 ⊗ n1〉N = n∗ϕ(m∗m1)n1

withm0,m,m1 ∈ M and n0, n, n1 ∈ N , and then separating and taking the comple-

tion with respect to the ||〈·, ·〉N ||1/2 norm. As a simple example, we note that thecorrespondence X = N is equivalent to the N -N correspondence obtained whenone lets ϕ be the identity map on N . Additionally, every M-N correspondenceX is a direct sum of cyclic correspondences of the form Xϕ (cf. Lemma 1.2.20 of[Po1]).

Let Corr(M,N ) denote the set of all (unitary equivalence classes of) M-Ncorrespondences. We define a topology on Corr(M,N ) as follows: Let X0 ∈Corr(M,N ). Given an ultraweak neighborhood W of zero in N , E a finite sub-set of M, and S = ξ1, ..., ξp a finite subset of X0, the basic neighborhoodV (X0;W,E, S) is the set of (equivalence classes of) correspondences X such thatthere are η1, ..., ηp ∈ X such that

〈ξi,mξj〉N − 〈ηi,mηj〉N ∈W

for all m ∈ E and i, j ∈ 1, 2, ..., p. In the case where X0 has a cyclic vector ξ0,then X0 has a neighborhood basis of sets of the form V (X0;W,E, ξ0), and in thiscase, X0 is in the closure of X with X ∈ Corr(M,N ) if and only if there is anet (ξi)i∈Λ in X such that limi〈ξi,mξi〉N = 〈ξ0,mξ0〉N ultraweakly for all m ∈ M.The trivial correspondence NNN is in the closure of X, with X ∈ Corr(N ,N ),if and only if there is a net (φα)α∈Λ of coefficients of X such that limα φα(n) = n

ultraweakly for all n ∈ N .We shall use the concept of weak containment of correspondences. If X and

Y are both M-N correspondences then we say that X is weakly contained in Y

(written X ≺ Y ) if X belongs to the closure in Corr(M,N ) of the set of finitedirect sums of copies of Y. In particular, the trivial correspondence N is weaklycontained in an N -N correspondence X if and only if there is a net (ϕα)α∈I ofcompletely positive maps from N to N each of which is a finite sum of coefficientsof X , such that limα ϕα(n) = n ultraweakly for all n ∈ N . (cf. [AD1], [AD2])

6 JON P. BANNON

By Theorem 2.1.3 of [Po1] and a straightforward modification of Proposition1.12 of [AD1], the topology on Corr(M,N ) given above is equivalent to the onegiven analogously by everywhere replacing “ultraweak” by “ultrastrong”. This isparallel to what happens with the topology on the space of representations of agroup. We note that as a result of this, closure and weak containment of the trivialcorrespondence can be characterized by pointwise ultrastrong convergence insteadof pointwise ultraweak convergence.

We now define the tensor product, or composition, correspondence. Supposethat M,N and R are W ∗-algebras, X is an M-N correspondence and Y is an N -R correspondence. Endow the algebraic bimodule tensor product X ⊗

NY with the

left and right actions of M and R given on simple tensors by m(ξ⊗ η)n = mξ⊗ ηnfor m ∈ M and n ∈ N , and define the R-valued inner product of simple tensorsas 〈ξ ⊗ η, ξ1 ⊗ η1〉R = 〈η, 〈ξ, ξ1〉N η1〉R. The M-R correspondence obtained as theself dual completion of X⊗

NY is written as MX⊗

NYR and is called the composition

correspondence ofX and Y . The tensor product of correspondences is “associative”,in that if M,N ,R,Q are W ∗-algebras, X is an M-N correspondence, Y is an N -R correspondence and Z is an R-Q correspondence, the M-Q correspondences(MX ⊗

NYR)⊗

RZQ and MX ⊗

N(NY ⊗

RZQ) are isomorphic.

Suppose that N ⊂ M is an inclusion of von Neumann algebras and that X is anN -N correspondence. We define the correspondence induced by X from N up toM to be IndMN X = MMN ⊗

NX ⊗

NNMM. Here MMN = X(ML2(M)N ), where

ML2(M)N is the correspondence obtained by restricting the right action of M

to N in the correspondence ML2(M)M, and NMM = X(NL2(M)M), where the

adjoint correspondence NL2(M)M is obtained by imposing the left N and right

M actions given by nξm = m∗ξn∗ on the conjugate Hilbert space L2(M). Weshall freely use the following well-established facts: If there is a faithful normalconditional expectation E from M onto N then in the correspondence MMN isequivalent to XE, which is equivalent to the correspondence obtained from M byrestricting the right action to N and with the N -valued inner product given by〈m,m1〉N = E(m∗m1) for m,m1 ∈ M. The correspondence NMM is equivalentto Xι, where ι : N → M is the inclusion map, and that this is equivalent to thecorrespondence obtained by restricting the left action of MMM to N and with theinherited natural M-valued inner product. For more on these facts, see [AD1] and[AD2].

Corollary 1.8 in [AD2] gives us that induction is continuous with respect to weakcontainment: if N ⊂ M is an inclusion of von Neumann algebras and X1, X2 areN -N correspondences such that X1 ≺ X2, then Ind

MN X1 ≺ IndMN X2.

Suppose N ⊆ M is an inclusion of finite von Neumann algebras. By Theorem3.2.3 in [Po1], there is a state ρ on 〈M, eN 〉 with ρ(xy) = ρ(yx) for all x ∈ M andy ∈ 〈M, eN 〉 if and only if MMM ≺ IndMN (NNN ).

2. Main Results

Let M be a finite von Neumann algebra equipped with a given faithful normaltracial state τ , and denote by M+ the positive part of M. For p ∈ 1, 2 andx ∈ M the element |x|p ∈ M+ with |x| = (x∗x)1/2. Assigning ||x||p = τ(|x|p)1/p

for each x ∈ M defines a norm on M. We denote by Lp(M) the completion of

HAAGERUP APPROXIMATION PROPERTY AND RELATIVE AMENABILITY 7

M with respect to || · ||p. The Hilbert space L2(M) agrees with the standardrepresentation space obtained from the GNS construction applied to M and τ. LetΩ denote a joint generating trace vector for M and its commutant M′ with respectto this standard action of M on L2(M). When we regard an element x of M as avector in L2(M), we shall write it as xΩ.

We let J = J∗ = J2 denote the modular conjugation operator on L2(M) ex-tending the map xΩ 7→ x∗Ω on MΩ. Recall that JMJ = M′. Given x ∈ M, whenit becomes necessary to regard the element Jx∗J of M′ as a vector in L2(M), weshall write it as Jx∗JΩ.

Recall that a closed, densely defined linear operator T on L2(M) is affiliatedwith M if it commutes with every operator in M′. Every ξ ∈ Lp(M) may beregarded as an operator affiliated with L2(M) with domain M′Ω by identifying itwith the operator Lξ, defined by Lξx

′Ω = x′ξ. Conversely, let U(M) denote the∗-algebra of operators affiliated with M. Every T ∈ U(M) is integrable, in thesense that for every projection pn = χ[0,n](|T |) satisfies τ(I − pn) < ∞. Thereforeif we denote by U(M)+ the set of all positive self-adjoint operators in U(M), thetrace τ extends to a positive tracial functional on U(M)+ and we may identifyLp(M) with the set of all T ∈ U(M) with τ(|T |p) < ∞. We define Lp(M)+ =U(M)+ ∩Lp(M). By the Borel functional calculus, M+ is dense in Lp(M)+. Formore, see [Se], [PiXu].

If ϕ : M → M is a linear map, we let ϕΩ : MΩ → MΩ be the map satisfyingϕΩ(xΩ) = yΩ if and only if ϕ(x) = y for all x, y ∈ M. In the event where ϕΩ

has a continuous extension to L2(M) we shall denote this extension also by ϕΩ,risking no confusion. It is evident that if ϕ and ψ are linear maps on M, then(ϕ ψ)Ω = ϕΩ ψΩ. The inner product on L2(M) extending 〈xΩ, yΩ〉 = τ(y∗x)for all x, y ∈ M will be written as 〈ξ, η〉 for ξ, η ∈ L2(M). We use the notationassociated to the theory of correspondences found in the introduction above.

Definition 1. A linear map ϕ : M → M is positive if ϕ(x) ≥ 0 whenever x ≥ 0,or equivalently, if ϕ(x∗x) ≥ 0 for all x ∈ M.

Definition 2. Let n be a positive integer. A linear map ϕ : M → M is n-positiveif the maps ϕk :Mk(M) → Mk(M) defined by ϕk((xij)ij) = (ϕ(xij))ij are positivefor all k ∈ 1, 2, ..., n. If the ϕ is n− positive for all n ∈ N, we say that ϕ is acompletely positive map on M.

Recall that for any k ∈ N we can associate to the given faithful normal tracialstate τ on M the faithful normal tracial state on Mk(M) given by

1

kT r(τk((nij)ij)) =

1

k

∑ki=1τ(nii).

Definition 3. We say that a completely positive map ϕ on M is subtracial ifτ ϕ ≤ τ . We say ϕ is completely subtracial if ϕk is subtracial with respect to thetrace 1

kTr τk for all k ∈ N.

Proposition 1. If ϕ is a subtracial completely positive map on M, then ϕ iscompletely subtracial.

8 JON P. BANNON

Proof. Suppose that ϕ is subtracial and k ∈ N. We obtain

1

kT r(τk(ϕk((x

∗x)ij))) =1

kT r((

∑lτ(ϕ(x

∗lixlj)))ij)

=1

k

∑i

∑lτ(ϕ(x

∗lixli)) ≤

1

k

∑i

∑lτ(x

∗lixli)

=1

kT r(τk((x

∗x)ij)).

It follows that ϕk is subtracial for all positive integers k, hence ϕ is completelysubtracial.

A trivial modification of the above proof gives us the following.

Corollary 1. If ϕ is a completely positive map on M and there exists c > 0 withτ ϕ ≤ cτ , then 1

kTr τk ϕk ≤ c 1kTr τk for all positive integers k.

Definition 4. We say that M has the Haagerup approximation property if thereexists a net (ϕα)α∈Λ of normal completely positive maps from M to M satisfying:

(1) τ ϕα ≤ τ for every α ∈ Λ;(2) Each ϕΩ

α extends by continuity to a compact operator on L2(M);(3) limα||ϕ

Ωα(xΩ) − xΩ||2,τ = 0 for all x ∈ M.

Definition 5. Suppose X is an M-M correspondence. We say that X is a C0-correspondence if X ∼=

⊕α∈SXϕα

with each ϕα : M → M a normal completelypositive map satisfying:

(1) τ ϕα ≤ τ ;(2) ϕΩ

α extends by continuity to a compact operator on L2(M).

Although it appears that the Haagerup approximation depends on the choice oftrace on M, P. Jolissaint has shown that this is not so.

Proposition 2. [Joli] Let τ and τ ′ be two normal, faithful, finite normalized traceson M. If M has the Haagerup approximation property with respect to τ , then Mhas the Haagerup approximation property with respect to τ ′.

The above proposition justifies our choosing τ once and for all at the outset.

Definition 6. Suppose X is an M-M correspondence. We say that X is aHaagerup correspondence if MMM ≺ X and X is a C0-correspondence.

Given a completely positive map ϕ on M, we denote by ϕΩ the adjoint of ϕΩ in

B(L2(M)), i.e. the unique bounded operator satisfying 〈ϕΩ(ξ), η〉 = 〈ξ, ϕΩ(η)〉 forall ξ, η ∈ L2(M).

By Theorem 14 of [Se], for every normal linear functional ρ on M there existsH ∈ L1(M)+ such that ρ(x) = τ(Hx) for all x ∈ M. In fact, by Lemma 14.1 of[Se], provided there exists c > 0 such that τ ρ ≤ cτ this H ∈ M and is unique. We

denote H , the Radon-Nikodym derivative of τ ρ with respect to τ , by dρdτ . We now

prove that if the Radon-Nikodym derivative dτϕdτ is bounded then ϕΩ(M) ⊆ M.

We use the following result, found in [KRII].

Lemma 1. (7.3.4 of [KRII]) If ρ is a state of M and A ∈ M such that B → ρ(BA)is hermitian, then |ρ(AH)| ≤ ||A||ρ(H) for each positive H in M.

We shall need Sakai’s quadratic Radon-Nikodym Theorem [Sak]. A proof of thisresult is also found in [KRII].

HAAGERUP APPROXIMATION PROPERTY AND RELATIVE AMENABILITY 9

Theorem 1. (Sakai-Radon-Nikodym) If ρ and ρ0 are normal positive linear func-tionals on M, and ρ0 ≤ ρ, then there is a positive operator H0 in the unit ball ofM such that ρ0(A) = ρ(H0AH0) for all A ∈ M.

Note that by the uniqueness of the Radon-Nikodym derivative proved in [Se],

when ρ = τ we have dρ0

dτ = H20 , where H0 is as in Theorem 1.

Notation 1. Given y ∈ M, let Ad(y) be the map x 7→ y∗xy on M.

Proposition 3. Suppose that τ ϕ ≤ cτ for some c > 0 and y is in M. Thenτ Ad(y) ϕ ≤ c||y||2τ .

Proof. Suppose that c > 0 and that τ ϕ ≤ cτ . By Theorem 1, there exists H0

in the unit ball of M such that τ(ϕ(x)) = cτ(H20x) = τ(cH2

0x) for all x ∈ M.

Therefore by the uniqueness of dτϕdτ , we have that cH2

0 = dτϕdτ . Therefore ||dτϕdτ ||

= c||H20 || ≤ c, and therefore dτϕ

dτ is in ball of radius c in M. We see that the mapB → τ Ad(y)(B) = τ(Byy∗) is hermitian, so by Lemma 1, τ Ad(y)(x) = τ(y∗xy)= τ(yy∗x) ≤ ||yy∗||τ(x) = ||y||2τ(x) for all positive x in M. It follows that

τ Ad(y)(x) = τ(yy∗ϕ(x)) ≤ ||y||2τ(ϕ(x)) ≤ c||y||2τ(x)

for all positive x in M.

Corollary 2. Suppose c > 0 and y is in M. If dτϕdτ is in the || · ||-ball of radius c

in M, then dτAd(y)ϕdτ is in || · ||-ball of radius c||y||2 in M, for all y ∈ M.

Proof. Suppose y is in M and that τ Ad(y)ϕ ≤ c||y||2τ . It follows from Theorem1 that there exists H0 in the unit ball of M such that

τ Ad(y) ϕ(x) = τ(yy∗ϕ(x)) = c||y||2τ(H0x) = τ(c||y||2H0x)

for all x ∈ M. Therefore, by uniqueness of the Radon Nikodym derivative, we have

dτ Ad(y) ϕ

dτ= c||y||2H0.

It follows that dτAd(y)ϕdτ is in the ball of radius c||y||2 in M.

Theorem 2. Suppose ϕ is a completely positive map on M and c > 0 is such that

τ ϕ ≤ cτ . Then ϕΩ(M) ⊆ M.

Proof. The map ϕΩ is a densely defined linear operator on L2(M). We shall prove

that ϕΩ(MΩ) ⊆ MΩ. Given ϕΩ(yΩ) ∈ L2(M), consider the densely defined

operator Ty = LcϕΩ(yΩ)on L2(M) affiliated with M given by Tyx

′Ω = x′ϕΩ(yΩ)

for all x′Ω ∈ M′Ω. We note that if y = y∗ then since ϕ is ∗-preserving and

10 JON P. BANNON

JMJ = M′

〈TyJx∗JΩ, Jz∗JΩ〉 = 〈Jx∗JϕΩ(yΩ), Jz∗JΩ〉

= 〈ϕΩ(yΩ), JxJJz∗JΩ〉

= 〈ϕΩ(y), Jxz∗JΩ〉

= 〈ϕΩ(y), zx∗Ω〉

= 〈yΩ, ϕΩ(zx∗Ω)〉

= 〈Ω, yϕΩ(zx∗Ω)〉

= 〈Ω, LϕΩ(zx∗Ω)yΩ〉

= 〈Ω, Lϕ(zx∗)yΩ〉

= 〈L∗ϕ(zx∗)Ω, yΩ〉

= 〈Lϕ(zx∗)∗Ω, yΩ〉

= 〈Lϕ(xz∗)Ω, yΩ〉

= 〈ϕ(xz∗)Ω, yΩ〉

= 〈ϕΩ(xz∗Ω), yΩ〉

= 〈xz∗Ω, ϕΩ(yΩ)〉

= 〈Jzx∗JΩ, ϕΩ(yΩ)〉

= 〈JzJJx∗JΩ, ϕΩ(yΩ)〉

= 〈Jx∗JΩ, Jz∗JϕΩ(yΩ)〉

= 〈Jx∗JΩ, TyJz∗JΩ〉

for all x, z ∈ M, so Ty is self-adjoint.If y∗y ∈ M+ then

〈Ty∗yJx∗JΩ, Jx∗JΩ〉 = 〈Jx∗JϕΩ(y∗yΩ), Jx∗JΩ〉

= 〈ϕΩ(y∗yΩ), JxJJx∗JΩ〉

= 〈ϕΩ(y∗yΩ), xx∗Ω〉

= 〈y∗yΩ, ϕΩ(xx∗Ω)〉

= 〈y∗yΩ, ϕ(xx∗)Ω〉

= τ(ϕ(xx∗)∗y∗y)

= τ(ϕ(xx∗)y∗y)

= τ(yϕ(xx∗)y∗)

≥ 0

for all x ∈ M. It follows that the operator Ty∗y is a positive self-adjoint oper-ator affiliated with M. If Ty∗y is also bounded, then Ty∗y ∈ M, and therefore

ϕΩ(y∗yΩ) = LcϕΩ(y∗yΩ)Ω ∈ MΩ. We now proceed to show that every Ty∗y is

bounded.Every closed densely defined positive self-adjoint operator A possesses a unique

positive square root which commutes with every bounded operator that commutes

HAAGERUP APPROXIMATION PROPERTY AND RELATIVE AMENABILITY 11

with A (cf. for example [Wo]). Since Ty∗y is a positive self-adjoint operator affiliated

with M, its square root T1/2y∗y is a positive self-adjoint operator affiliated with M

and

||T1/2y∗yJx

∗JΩ||22 = 〈T1/2y∗yJx

∗JΩ, T1/2y∗yJx

∗JΩ〉

= 〈Ty∗yJx∗JΩ, Jx∗JΩ〉

= 〈Jx∗JϕΩ(y∗yΩ), Jx∗JΩ〉

= 〈ϕΩ(y∗yΩ), Jxx∗JΩ〉

= 〈ϕΩ(y∗yΩ), xx∗Ω〉

= 〈y∗yΩ, ϕΩ(xx∗Ω)〉

= 〈y∗yΩ, ϕ(xx∗)Ω〉

= τ(ϕ(xx∗)y∗y)

= τ(yϕ(xx∗)y∗)

= τ Ad(y∗) ϕ(xx∗)

≤ c||y∗||2τ(xx∗)

= c||y||2||Jx∗JΩ||22.

for every x ∈ M and therefore that T1/2y∗y could have norm at most c1/2||y||, and T

1/2y∗y

extends to a bounded operator on L2(M). Therefore (T1/2y∗y )

2 = Ty∗y extends to a

bounded operator on L2(M), and ||Ty∗y|| ≤ ||T1/2y∗y ||

2 ≤ c||y||2. We have establishedthat Ty∗y is a bounded operator for all y ∈ M. Since every T in M is a finite linear

combination of positive elements in M, ϕΩ is linear, and Laξ+η = aLξ + Lη for alla ∈ C, ξ, η ∈ L2(M), we have that Ty is bounded for every y ∈ M.

Note that if ϕΩ(yΩ) = z1Ω = z2Ω for z1, z2 ∈ M, then 〈z1Ω, wΩ〉 = 〈z2Ω, wΩ〉for all z ∈ M, hence 〈z1Ω, (z1 − z2)Ω〉 = 〈z2Ω, (z1 − z2)Ω〉, and hence ||(z1 −z2)Ω||

22,τ = 0 and z1 = z2. We define the map ϕ on M such that ϕ(y) = z if and

only if ϕΩ(yΩ) = zΩ. Notice then, that ϕΩ(yΩ) = ϕ(y)Ω for all y ∈ M.

Proposition 4. If ϕ : M → M is a completely positive map and c > 0 is suchthat τ ϕ ≤ cτ . Then ϕ : M → M is a positive map.

12 JON P. BANNON

Proof. By Theorem 2, the hypothesis guarantees that the map ϕ exists. The posi-tivity of ϕ yields that ϕ is also positive, since

0 ≤ 〈ϕ(x∗x)yΩ, yΩ〉 = 〈y∗ϕ(x∗x)yΩ,Ω〉

= τ(y∗ϕ(x∗x)y)

= τ(yy∗ϕ(x∗x))

= 〈yy∗ϕ(x∗x)Ω,Ω〉

= 〈ϕ(x∗x)Ω, yy∗Ω〉

= 〈ϕΩ(x∗xΩ), yy∗Ω〉

= 〈x∗xΩ, ϕΩ(yy∗Ω)〉

= 〈x∗xΩ, ϕ(yy∗)Ω〉

= 〈Ω, x∗xϕ(yy∗)Ω〉

= τ(x∗xϕ(yy∗))

= τ(xϕ(yy∗)x∗)

= 〈xϕ(yy∗)x∗Ω,Ω〉

= 〈ϕ(yy∗)x∗Ω, x∗Ω〉

for all x, y ∈ M.

We now note that by the above positivity of ϕ:

τ(y∗ϕ(x)) = 〈y∗ϕ(x)Ω,Ω〉

= 〈ϕ(x)Ω, yΩ〉 = 〈ϕΩ(xΩ), yΩ〉

= 〈xΩ, ϕΩ(yΩ)〉 = 〈xΩ, ϕ(y)Ω〉

= 〈ϕ(y)∗xΩ,Ω〉 = τ(ϕ(y)∗x)

= τ(ϕ(y∗)x)

for all x, y ∈ M.In the following theorem, let Ωn be the joint cyclic trace vector for Mn(M) and

Mn(M)′ acting on L2(Mn(M), 1nTr τn).

Theorem 3. If ϕ : M → M is a completely positive map and c > 0 is such thatτ ϕ ≤ cτ . Then ϕ : M → M is a completely positive map.

Proof. Suppose that ϕ : M → M is a completely positive map and c > 0 is suchthat τ ϕ ≤ cτ . The maps ϕn are positive for all n, since ϕ is completely positive.Equivalently, 〈ϕn(xx

∗)yΩn, yΩn〉 ≥ 0 for all n ∈ N and x, y ∈ Mn(M). By Corol-lary 1 we have 1

nTr τn ϕn ≤ c 1nTr τn for all n. It follows from Theorem 2 that

HAAGERUP APPROXIMATION PROPERTY AND RELATIVE AMENABILITY 13

the map (ϕn) is defined on Mn(M), and

〈(ϕn)(yy∗)xΩn, xΩn〉 = 〈x∗ (ϕn)(yy

∗)xΩn,Ωn〉

=1

nTr(τn(x

∗ (ϕn)(yy∗)x))

=1

nTr(τn(xx

∗ (ϕn)(yy∗)))

= 〈(ϕn)(yy∗)Ω, xx∗Ω〉

= 〈(ϕΩnn )(yy∗Ωn), xx

∗Ωn〉

= 〈yy∗Ωn, ϕΩnn (xx∗Ωn)〉

= 〈yy∗Ωn, ϕn(xx∗)Ωn〉

= 〈Ωn, yy∗ϕn(xx

∗)Ωn〉

=1

nTr(τn(yy∗ϕn(xx∗)))

=1

nTr(τn(y∗ϕn(xx∗)y))

= 〈y∗ϕn(xx∗)yΩn,Ωn〉

= 〈ϕn(xx∗)yΩn, yΩn〉 ≥ 0

for all x, y ∈ Mn(M). To prove our claim it suffices to show that (ϕ)n = (ϕn) forall positive integers n. Let n be a given positive integer. We need to show that forall x, y ∈Mn(M) that

〈ϕn(x)Ωn, yΩn〉 = 〈xΩn, ϕn(y)Ωn〉.

Let x = (xij)ij and y = (yij)ij , then ϕn((yij)ij) = (ϕ(yij))ij . The positivity of ϕestablished in Proposition 4 guarantees that ϕ(yji)

∗ = ϕ(y∗ji) and hence

〈xΩn, ϕn(y)Ωn〉 = 〈ϕn(y)∗xΩn,Ωn〉

=1

nTr(τn((ϕn(y)

∗x))

=1

nTr(τn((ϕ(yij))

∗ijx))

=1

nTr(τn((ϕ(y

∗ji))ijx))

=1

nTr(τn(

∑kϕ(y

∗ki)xkj)ij)

=1

nTr((τ(

∑kϕ(y

∗ki)xkj))ij)

=1

nTr((

∑kτ(ϕ(y

∗ki)xkj))ij)

=1

n

∑ni=1

∑nk=1τ(ϕ(y

∗ki)xki))ii)

=1

n

∑ni=1

∑nk=1τ(y

∗kiϕ(xki)))ii)

=1

nTr((

∑kτ(y

∗kiϕ(xkj)))ij)

14 JON P. BANNON

=1

nTr((τ(

∑ky

∗kiϕ(xkj)))ij)

=1

nTr(τn(

∑ky

∗kiϕ(xkj))ij)

=1

nTr(τn(y

∗(ϕ(xij))ij))

=1

nTr(τn(y

∗ϕn(x)))

= 〈y∗ϕn(x)Ωn,Ωn〉

= 〈ϕn(x)Ωn, yΩn〉

it follows that (ϕn) = ϕn for every positive integer n, and hence ϕ is completelypositive.

Definition 7. The completely bounded (or cb) norm of a linear map ϕ on M is

||ϕ||cb = supn∈N

||ϕΩnn ||

where ||ϕΩnn || is +∞ if ϕΩn

n is an unbounded operator on L2(Mn(M), 1nTrτn) and

the operator norm of ϕΩnn otherwise.

Theorem 4. If ϕ is a nonzero completely positive map on M such that there is ac > 0 is such that τ ϕ ≤ cτ then the following are equivalent:

(a) τ ϕ ≤ τ ;(b) ϕ(I) ≤ I;(c) ||ϕ||cb ≤ 1;(d) ϕ(I) ≤ I;(e) ||ϕ||cb ≤ 1;(f) ||ϕ||cb = 1;(g) ||ϕ||cb = 1;(h) τ ϕ ≤ τ.

Proof. (a⇔b) Suppose τ(ϕ(xx∗)) ≤ τ(xx∗) for all x ∈ M. Then

〈ϕ(I)xΩ, xΩ〉 = τ(x∗ϕ(I)x) = τ(ϕ(I)xx∗)

= τ(ϕ(xx∗)) ≤ τ(xx∗) = 〈xΩ, xΩ〉

for all x ∈ M, hence ϕ(I) ≤ I. Conversely, ϕ(I) ≤ I implies τ(ϕ(xx∗)) =τ(x∗ϕ(I)x) ≤ τ(xx∗) for all x ∈ M.

(d⇔h) Obtained from proof of (a⇔b) mutatis mutandis.(b⇔c) If ϕ(I) ≤ I, then ||ϕ(I)|| ≤ 1. By Proposition 3.6 of [Paul], ||ϕ(I)|| =

||ϕ||cb, so ||ϕ||cb ≤ 1. Conversely, if ||ϕ||cb ≤ 1, then ||ϕ(I)|| ≤ 1. Since M is a C∗-algebra and ϕ(I) is self-adjoint, ||ϕ(I)||I−ϕ(I) ≥ 0, so I−ϕ(I) ≥ ||ϕ(I)||I−ϕ(I) ≥0.

(d⇔e) Obtained from (b⇔c) mutatis mutandis.

(c⇔e and f⇔g) It follows from the proof of Theorem 3 that ϕk = (ϕk) for all

k ∈ N. Therefore ||ϕk|| = ||(ϕk)|| = ||ϕk||. It follows that ||ϕ||cb = ||ϕ||cb.(a and e ⇒ f) Suppose τ ϕ ≤ τ , and k ∈ N. By the Cauchy-Schwartz inequality

for 2-positive maps, we have that for all x ∈Mk(M)

ϕk(x)∗ϕk(x) ≤ ||ϕk(I)||ϕk(x

∗x).

HAAGERUP APPROXIMATION PROPERTY AND RELATIVE AMENABILITY 15

Therefore by Proposition 1, each ϕk is subtracial and hence

||ϕk(x)||22,τ = ||ϕk(I)||τ(ϕk(x

∗x))

≤ ||ϕk(I)||||x||22,τ .

By Proposition 3.6 of [Paul] ||ϕk(I)|| = ||ϕk||cb, and hence

||ϕk||1/2cb ≤ ||ϕk||.

Therefore 1 ≤ ||ϕk||||ϕk||cb

||ϕk|| ≤ ||ϕk||. Since (a⇔e), we have that ||ϕ||cb ≤ 1. We

conclude that ||ϕk|| = 1 for all k ∈ N, it follows that ||ϕ||cb = 1.(g⇒c) This is obvious.

Recall that a net (xα)α∈Λ inM converges to x ∈ M in the ultrastrong topology ifand only if for all ε > 0 and all sequences (ξm)∞m=1 in L2(M) with

∑∞m=1||ξm||22,τ <

∞ there is an L ∈ Λ such that α ≥ L implies that∑∞

m=1||(xα − x)ξm||22,τ < ε.Consider a net (xα)α∈Λ in the norm unit ball ofM. If xα → x ultrastrongly, then

clearly xα → x in the strong operator topology. If xα → x in the strong operatortopology, then given a sequence (ξm)∞m=1 in L2(M) with

∑∞m=1||ξm||22,τ <∞, and

ε > 0 choose a large enough finite set F of N so that∑

m∈N\F||ξm||22,τ < ε4 and

choose L ∈ Λ such that α ≥ L implies∑

m∈F||(xα − x)ξm||22,τ <

ε2 . Under these

circumstances, if α ≥ L then∑∞

m=1||(xα − x)ξm||22,τ =∑

m∈F||(xα − x)ξm||22,τ +

∑m∈N\F||(xα − x)ξm||22,τ

≤∑

m∈F||(xα − x)ξm||22,τ +

∑m∈N\F||ξm||22,τ < ε.

It follows that on any bounded subset ofM that xα → x in the ultrastrong topologyif and only if xα → x in the strong operator topology. Similarly, the ultraweaktopology agrees with the weak operator topology on bounded subsets of M.

Notice that if the net (xα)α∈Λ in the norm unit ball of M converges to xα in thestrong operator topology then ||(xα−x)Ω||2,τ → 0. Conversely, if ||(xα−x)Ω||2,τ →0, then given ξ ∈ L2(M) and ε > 0 choose x ∈ M such that ||Jx∗JΩ− ξ||2,τ <

ε4

and L ∈ Λ such that α ≥ L implies ||(xα − x)Ω||2,τ <ε

2||x|| . Then whenever α ≥ L

||(xα − x)ξ||2,τ ≤ ||(xα − x)Jx∗JΩ||2,τ + ||(xα − x)||||Jx∗JΩ− ξ||2,τ

≤ ||x||||(xα − x)Ω||2,τ + 2||Jx∗JΩ− ξ||2,τ < ε.

It follows that on any given bounded subset of M, ||(xα − x)Ω||2,τ → 0 if and onlyif xα → x in the strong operator topology.

Theorem 5. If M has the Haagerup approximation property then there exists anM-M Haagerup correspondence.

Proof. Suppose thatM has the Haagerup approximation property and that (ϕα)α∈Λ

is a net of normal completely positive maps satisfying (1)-(3) of Definition 4 above.Consider the M-M correspondence X =

⊕α∈Λ

Xϕα. This correspondence is C0 by

construction. Since each ϕα is subtracial, we have by Theorem 4 that ||ϕα||cb ≤ 1for all α. Therefore the set ϕα(x) − x|α ∈ Λ lies entirely in the || · || ballof radius 2||xΩ||2,τ for any given x ∈ M. Since the ultrastrong topology agreeswith the strong operator topology on bounded subsets of M and on bounded setsstrong operator convergence agrees with 2-norm convergence, ||ϕΩ

α(xΩ)−xΩ||2,τ =||(ϕα(x) − x)Ω||2,τ → 0 for x ∈ M implies that ϕα(x) → x ultrastrongly. Since

16 JON P. BANNON

each ϕα is a coefficient of X , we have exhibited a net of coefficients convergingpointwise ultrastrongly to the identity map on M, and therefore MMM ≺ X . Itfollows that X is a Haagerup M-M correspondence.

In the following theorem, recall that ‖·‖Mk(M) and || · ||M denote the operator

norms in Mk(M) and M, respectively.

Lemma 2. A completely positive map ϕ on M is subtracial if and only if for all∑ki=1 xi ⊗ yi = ξ ∈ M⊙M,

τ φXϕ,ξ(x∗x) ≤ ‖diag(x1, ..., xk)‖

2Mk(M) ||

k∑

i=1

yi||2Mτ(x∗x)

for all x ∈ M.

Proof. First suppose that for every∑k

i=1 xi⊗yi = ξ ∈ M⊙M that τ φXϕ,ξ(x∗x) ≤

‖diag(x1, ..., xk)‖Mk(M) ||∑k

i=1 yi||Mτ(x∗x) for all x ∈ M. Then in particular, for

I⊗I ∈ M⊙M, we have that τφXϕ,ξ(x∗x) = τ(ϕ(x∗x))≤ ||I||M1(M)||I||Mτ(x∗x) =

τ(x∗x).Conversely, suppose that ϕ is subtracial. By Proposition 1, ϕ is completely

subtracial as well. Therefore, by Corollary 2, we have that for any positive integerk and y ∈Mk(M) that

1

kT r τk Ad(y) ϕk ≤ ||y||2

1

kT r τk ϕk.

Suppose that∑k

i=1 xi ⊗ yi = ξ ∈ M⊙M and x ∈ M and consider

A =

x∗ 0 · · · 0x∗ 0 · · · 0... 0 · · · 0x∗ 0 · · · 0

x x · · · x

0 0 0 0...

......

...0 0 0 0

B =

x1 0 · · · 0

0 x2 0...

... 0. . . 0

0 · · · 0 xk

C =

y1 0 · · · 0

0 y2 0...

... 0. . . 0

0 · · · 0 yk

D =

I 0 · · · 0

I 0 0...

... 0. . . 0

I · · · 0 0

HAAGERUP APPROXIMATION PROPERTY AND RELATIVE AMENABILITY 17

in Mk(M). Now by Corollary 2 and the complete subtraciality of ϕ,

τ φXϕ,ξ(x∗x) =

1

kT r τk((CD)∗ϕk(B

∗AB)CD)

≤ ||CD||21

kT r τk(ϕk(B

∗AB))

≤ ||CD||21

kT r τk(B

∗AB)

≤ ||CD||2||B||21

kT r τk(A)

= ||CD||2||B||2τ(x∗x).

We note that ||CD||2Mk(M) = ||∑k

i=1 yi||2M, and the proof is finished.

We shall need the following ”smooth approximation” lemma.

Lemma 3. Let X be a Haagerup M-M correspondence. If (ϕ0α)α is a bounded

net of completely positive maps on M , each of which is a finite sum of coefficientsof X, so that ϕ0

α converges pointwise in the weak operator topology to the identitymap on M, then there exists a bounded net (ϕα)α of completely positive maps on

M, each of which is a finite sum of coefficients of X with dτϕα

dτ ∈ M so that ϕα

converges pointwise in the weak operator topology to the identity map on M.

Proof. Let (ϕ0α)α∈Λ be a bounded net of completely positive maps from M to

M, each of which is a finite sum of coefficients of X , so that ϕ0α(x) → x in the

weak operator topology for all x ∈ M. Suppose that ϕ0α = 1

n(α)

∑n(α)i=1 φX,ξα

iwith

ξαi ∈ X . We assume for simplicity that X is cyclic and there exists a normalcompletely positive map Φ from M to M so that X = XΦ, and therefore we mayregard M⊙M as a dense subset of X . Suppose α ∈ Λ, i ∈ 1, 2, ..., n(α), x ∈ Mand ε > 0 are given. We can find Tα ∈ M⊙M such that ||Tα||M ≤ ||ξαi ||M + ε

and ||ξαi − Tα||M < −||ξαi ||M +√||ξαi ||

2M + ε. We then have

φX,ξαi(x)− φX,Tα(x) = 〈ξαi − Tα, x(ξαi − Tα)〉M + 〈ξαi − Tα, xTα〉M

+ 〈Tα, x(ξαi − Tα)〉M

for all x ∈ M. Therefore, by the triangle inequality and the Cauchy-Schwartzinequality for Hilbert modules (see, for example [Pas]):

||φX,ξαi(x) − φX,Tα(x)|| ≤ ||〈ξαi − Tα, x(ξαi − Tα)〉M||+ ||〈ξαi − Tα, xTα〉M||

+ ||〈Tα, x(ξαi − Tα)〉M||

≤ ||ξαi − Tα||M||x(ξαi − Tα)||M + ||ξαi − Tα||M||xTα||M

+ ||Tα||M||x(ξαi − Tα)||M.

Since X is self-dual and the left action of x is a bounded right M-module map, byCorollary 3.5 of [Pas] we have that every x ∈ M has an adjoint x∗ in the algebraB(X) of || · ||M-bounded linear maps on X , i.e. 〈ξ, xη〉M = 〈x∗ξ, η〉M for all ξ, ηin X . We denote by || · ||X the operator norm in B(X). Therefore, by Proposition2.6 of [Pas], we obtain that ||x(ξαi −Tα

i )||M ≤ ||x||X ||(ξαi −Tαi )||M. It follows that

18 JON P. BANNON

for all x ∈ M

||φX,ξαi(x)− φX,Tα

i(x)|| ≤ (||ξαi − Tα

i ||M + 2||Tαi ||M)||ξαi − Tα

i ||M||x||X

≤ (||ξαi − Tαi ||M + 2(||ξαi ||M + ε))||ξαi − Tα

i ||M||x||X

< (−||ξαι ||M +√||ξαι ||

2M + ε+ 2||ξαi ||M + 2ε)·

(−||ξαι ||M +√

||ξαι ||2M + ε)||x||X

= (√||ξαι ||

2M + ε+ ||ξαi ||M + 2ε)·

(√

||ξαι ||2M + ε− ||ξαι ||M)||x||X

≤ (√||ξαι ||

2M + ε+ ||ξαi ||M)·

(√

||ξαι ||2M + ε− ||ξαι ||M)||x||X

= (||ξαι ||2M + ε− ||ξαι ||

2M)||x||X = ε||x||X .

With very little extra work, we deduce that given ε > 0 we have for all x ∈ M that

||ϕ0α(x) −

1

n(α)

∑n(α)i=1 φX,Tα

i(x)|| < ε||x||X .

Now let ε > 0 and ω in the predual M∗ be given. For each α in Λ, let ϕα =1

n(α)

∑n(α)i=1 φX,Tα

ibe chosen so that ||ϕ0

α(x) − ϕα(x)|| <ε

2||ω|| . Choose L ∈ Λ so

that if α ≥ L then |ω(ϕ0α(x)−x)| < ε

2 . Now |ω(ϕα(x)− x)| = |ω(ϕα(x)−ϕ0α(x)) +

ω(ϕ0α(x)−x)| ≤ |ω(ϕα(x)−ϕ0

α(x))|+ |ω(ϕ0α(x)−x)| ≤ ||ω||||ϕα(x)−ϕ0

α(x)||2,τ +ε2

≤ ||ω||||ϕα(x)−ϕ0α(x)||+

ε2 = ||ω|| ε

2||ω||+ε2 = ε. As a result, we see that ϕα(x) → x

in the weak operator topology. By Lemma 2 and Lemma 14.1 of [Se], each dτϕα

dτ ∈M.

The following Lemma of Anantharaman-Delaroche and Havet (Lemma 2.2 of[AdH]) plays a crucial role in the proof of Theorem 6.

Lemma 4. Let CP (M) denote the set of completely positive maps from M to M,equipped with the topology of pointwise ultraweak convergence. Let F be a convexsubcone of CP (M) such that for φ ∈ F and b ∈ M, the completely positive mapAd(b) φ belongs to F . Let φ be an element of the closure of F in CP (M). Thenthere exists a net (φι) in F such that φι(I) ≤ φ(I) for all ι, which converges to φ.

Theorem 6. If there exists a Haagerup M-M correspondence, then M has theHaagerup approximation property.

Proof. Suppose X =⊕

α∈ΛXϕαis a given Haagerup M-M correspondence. Let F

be the subcone of all ϕ ∈ CP (M) that are finite sums of coefficients of X . Since

MMM ≺ X , we know there exists a net (ϕ0β) in F , so that ϕ0

β(x) → x ultraweakly

for all x ∈ M. By Lemma 4, we may replace the net (ϕ0β) by a net (ϕ1

ι ) in F with

ϕ1ι (I) ≤ I for all ι that also satisfies ϕ1

ι (x) → x ultraweakly for all x ∈ M. ByLemma 3, there is a bounded net (ϕι) in F converging pointwise ultraweakly to theidentity, with each ϕι having bounded Radon-Nikodym deriviative with respect toτ , and therefore by Theorem 4, each ϕι is subtracial. Thus (ϕι) a net of normal

HAAGERUP APPROXIMATION PROPERTY AND RELATIVE AMENABILITY 19

compact subtracial completely positive maps converging pointwise ultraweakly tothe identity. We have that for every H ∈ L1(M)+ and x ∈ M that

τ(Hϕι(x)) → τ(Hx).

Therefore, for any y ∈ M+

||(ϕι(y)− y)Ω||22,τ

= τ(ϕι(y)ϕι(y))− τ(yϕι(y))

− τ(ϕι(y)y) + τ(yy)

≤ ||ϕι||cbτ(ϕι(yy))− τ(yϕι(y))

− τ(ϕι(y)y) + τ(yy)

≤ τ(yy)− τ(yϕι(y))

− τ(ϕι(y)y) + τ(yy)

= 2(τ(yy)− τ(yϕι(y))).

Where the first inequality follows from the Cauchy-Schwartz inequality for 2-positivemaps, and the second inequality follows from the fact that each ϕι is subtracial and||ϕι||cb ≤ 1. The right hand side converges to zero since τ(yϕι(y)) → τ(yy). For ageneral y ∈ M, we have ||(ϕι(y) − y)Ω||22,τ → 0 since ϕι is linear and y is a linearcombination of positive elements in M. It follows that (ϕι) is a net of normalcompletely positive maps on M satisfying the hypotheses of Definition 4. It followsthat M has the Haagerup approximation property.

Putting together Theorems 5 and 6, we obtain the following:

Theorem 7. M has the Haagerup approximation property if and only if thereexists an M-M Haagerup correspondence.

The following two propositions are certainly well-known. We include proofsfor the reader’s convenience. In the proofs, we will take M and N -valued innerproducts in different correspondences. Which correspondence we are working withwill be clear from the context, but to clarify further we will sometimes theM-valuedinner product on correspondence X by 〈·, ·〉M,X .

Recall that given a cyclic N -N correspondence, ξ ∈ Xϕ is called a cyclic vectorif span(N ξN ) is dense in Xϕ.

Lemma 5. If I ∈ N ⊂ M is an inclusion of finite von Neumann algebras, E isthe normal τ-preserving conditional expectation of M onto N , and X = Xϕ is acyclic N -N correspondence, then IndMN (X) ∼= XϕE.

20 JON P. BANNON

Proof. Given m1,m2 ∈ M and a cyclic vector ξ =∑

iξi ⊗ ηi ∈ N ⊙ N ⊆ Xϕ wehave, appealing to the N -bimodule property of E, that

〈(m1 ⊗ ξ)⊗m2, (m1 ⊗ ξ)⊗m2〉M,IndMN

(X)

= 〈m2, 〈(m1 ⊗ ξ), (m1 ⊗ ξ)〉Nm2〉M

= 〈m2, 〈ξ, 〈m1,m1〉N ξ〉Nm2〉M

= m∗2〈ξ, E(m∗

1m1)ξ〉Nm2

= m∗2

∑i,jη

∗i ϕ(ξ

∗i E(m∗

1m1)ξj)ηjm2

=∑

i,jm∗2η

∗i ϕ(E(ξ∗im

∗1m1ξj))ηjm2

=∑

i,j(ηim2)∗ϕ(E((m1ξi)

∗m1ξj))ηjm2

= 〈∑

im1ξi ⊗ ηim2,∑

im1ξi ⊗ ηim2〉M

= 〈m1ξm2,m1ξm2〉M,XϕE.

Therefore the map defined on simple tensors by (m1 ⊗ ξ)⊗m2 7→ m1ξm2 extendsto an M-linear isometry of IndMN (X) onto XϕE that intertwines the left actions ofM. This isometry, in conjunction with the isometries giving the “associativity” ofthe composition of correspondences, provides us with the desired isomorphism.

Remark 1. Note that the proof above didn’t depend on the choice of cyclic vectorξ. Of course, one can simply use ξ = I ⊗ I.

Lemma 6. If I ∈ N ⊂ M is an inclusion of finite von Neumann algebras andX =

⊕α∈SXα is an N -N correspondence, then IndMN (X) ∼=

⊕α∈SInd

MN (Xα).

Proof. Letm1,m2 ∈ M and (ξαi)Ni=1 ∈ (wα1

, ..., wαk, 0, 0, ..) : k ∈ N, i ∈ 1, 2, ..., k,

wαi∈ Xϕαi

⊆⊕

α∈SXα.

〈[m1 ⊗ (ξαi)Ni=1]⊗m2, [m1 ⊗ (ξαi

)Ni=1]⊗m2〉M,IndMN

(X)

= 〈m2, 〈(m1 ⊗ (ξαi)Ni=1), (m1 ⊗ (ξαi

)Ni=1)〉Nm2〉M

= 〈m2, 〈(ξαi)Ni=1, 〈m1,m1〉N (ξαi

)Ni=1〉Nm2〉M

= 〈m2,∑N

i=1〈ξαi, 〈m1,m1〉N ξαi

〉Nm2〉M

=∑N

i=1〈m2, 〈ξαi, 〈m1,m1〉N ξαi

〉Nm2〉M

= 〈(m1 ⊗ ξαi⊗m2)

Ni=1, (m1 ⊗ ξαi

⊗m2)Ni=1〉M,

⊕α∈SIndM

N(Xα)

.

Therefore the map defined on simple tensors by [m1⊗ (ξαi)Ni=1]⊗m2 7→ (m1⊗ξαi

⊗m2)

Ni=1 extends to an M-linear isometry of IndMN (X) onto

⊕α∈SInd

MN (Xα) that

intertwines the left actions of M. This isometry, in conjunction with the isometriesgiving the “associativity” of the composition of correspondences, provides us withthe desired isomorphism.

Theorem 8. If I ∈ N ⊂ M is an inclusion of finite von Neumann algebras andX is a C0 N -N correspondence, then IndMN X is a C0 M-M correspondence.

Proof. Suppose X =⊕

α∈SXϕαis a C0 N -N correspondence and let E be the τ -

preserving normal conditional expectation of M onto N . By Lemma 5 and Lemma6 we have that IndMN (

⊕α∈SXϕα

) ∼=⊕

α∈SXϕαE . Since each ϕα is subtracial, each

ϕΩα is compact and E is trace-preserving, we have that each ϕα E is subtracial

and each (ϕα E)Ω = ϕΩα EΩ is compact.

HAAGERUP APPROXIMATION PROPERTY AND RELATIVE AMENABILITY 21

Theorem 9. If I ∈ N ⊆ M is an amenable inclusion of finite von Neumannalgebras and N has the Haagerup approximation property then M also has theHaagerup approximation property.

Proof. Let X be a N -N Haagerup correspondence. By Lemma 8, IndMN X is a C0

N -N -correspondence. We also have that

NNN ≺ X .

By the continuity of induction we see that

IndMN (NNN ) ≺ IndMN X .

Since N ⊆ M is an amenable inclusion, we have

MMM ≺ IndMN (NNN ),

so by transitivity of ≺ we obtain

MMM ≺ IndMN X .

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Department of Mathematics, Siena College, Loudonville, NY 12211

E-mail address: jbannon@siena.edu