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Banach function algebras and BSEnorms
H. G. Dales, Lancaster
Joint work with Ali Ulger, Istanbul
Graduate course during 23rd Banach algebra
conference, Oulu, Finland
July 2017
1
Some references
H. G. Dales and A. Ulger, Approximate identi-
ties in Banach function algebras, Studia Math-
ematica, 226 (2015), 155–187.
H. G. Dales and A. Ulger, Banach function
algebras and BSE norms, in preparation.
E. Kaniuth and A. Ulger, The Bochner–Schoen-
berg–Eberlein property for commutative Ba-
nach algebras, especially Fourier and Fourier–
Stieltjes algebras, Trans. American Math. Soc.,
362 (2010), 4331–4356.
S.-E. Takahasi and O. Hatori, Commutative
Banach algebras that satisfy a Bochner–Schoen-
berg–Eberlein-type theorem, Proc. American
Math. Soc., 37 (1992), 47–52.
2
Banach spaces
Let E be a normed space. The closed unit ballis E[1] = x ∈ E : ‖x‖ ≤ 1.
The dual space of E is E′. This is the spaceof continuous = bounded linear functionals onE, and its norm is given by
‖λ‖ = sup |〈x, λ〉| = |λ(x)| : x ∈ E[1],so that (E′, ‖ · ‖) is a Banach space.
The weak-∗ topology on E′ is σ(E′, E). Thus(E′[1], σ(E′, E)) is compact.
The bidual of E is E′′ = (E′)′, and we regardE as a closed subspace of E′′; the canonicalembedding is κE : E → E′′, where
〈κE(x), λ〉 = 〈x, λ〉 (x ∈ E, λ ∈ E′).
For a closed subspace F of E, the annihilatorof F is
F⊥ = λ ∈ E′ : λ | F = 0 .
3
Algebras
All algebras are linear and associative and taken
over the complex field, C. The identity of
a unital algebra A is eA; the unitisation of a
(non-unital) algebra A is A].
For S, T ⊂ A, set
S · T = ab : a ∈ S, b ∈ T, ST = lin S · T ;
set A[2] = A · A and A2 = lin A[2].
An ideal in A is a linear subspace I such that
AI ⊂ I and IA ⊂ I.
4
The radical
We set A• = A \ 0; an element a of A is
quasi-nilpotent if zeA − a is invertible in A]
for each z ∈ C•, and the set of quasi-nilpotent
elements is denoted by Q(A).
The (Jacobson) radical of an algebra A is de-
noted by radA; it is the intersection of the
maximal modular left ideals; it is an ideal in A.
The algebra A is semi-simple if radA = 0and radical if radA = A, so that A is radical
if and only if A = Q(A).
5
Characters on algebras
A character = multiplicative linear functional
on an algebra A is a linear functional ϕ : A→ Csuch that ϕ(ab) = ϕ(a)ϕ(b) (a, b ∈ A) and also
ϕ 6= 0.
The character space of an algebra, the col-
lection of characters on A, is denoted by ΦA.
The centre of A is
Z(A) = a ∈ A : ab = ba (b ∈ A);
A is commutative if Z(A) = A.
6
Banach algebras
An algebra A with a norm ‖ · ‖ is a Banachalgebra (BA) if (A, ‖ · ‖) is a Banach space and
‖ab‖ ≤ ‖a‖ ‖b‖ (a, b ∈ A).
When A is unital, we also require that ‖eA‖ = 1.
Standard non-commutative example: A = B(E),the algebra of all bounded linear operators on aBanach space E, with operator norm ‖ · ‖op. Here
(ST )(x) = S(Tx) (x ∈ E)
for S, T ∈ A.
Each character ϕ on a BA is continuous, with‖ϕ‖ ≤ 1, and so ΦA ⊂ A′[1]; ΦA is a locally
compact subspace of (A′, σ(A′, A)), and it iscompact when A is unital.
In a BA, each maximal (modular) ideal is closedand so rad A is closed. Further
Q(A) = a ∈ A : limn→∞ ‖a
n‖1/n = 0.
7
Continuous functions
Let K be a locally compact space. Then Cb(K)is the algebra of all bounded, continuous func-tions on K, with the pointwise operations;
C0(K) consists of the continuous functions thatvanish at infinity;
C00(K) consists of the continuous functionswith compact support.
We define
|f |K = sup |f(x)| : x ∈ K (f ∈ Cb(K)) ,
so that | · |K is the uniform norm on K and(Cb(K), | · |K) is a commutative, semisimpleBanach algebra;
C0(K) is a closed ideal in Cb(K);
C00(K) is an ideal in Cb(K).
The topology of pointwise convergence on Cb(K)is called τp.
8
Function algebras
A function algebra on K is a subalgebra A of
Cb(K) that separates strongly the points of K,
in the sense that, for each x, y ∈ K with x 6= y,
there exists f ∈ A with f(x) 6= f(y), and, for
each x ∈ K, there exists f ∈ A with f(x) 6= 0.
Banach function algebras
A Banach function algebra (= BFA) on K
is a function algebra A on K with a norm ‖ · ‖such that (A, ‖ · ‖) is a Banach algebra.
The BFA A is natural if all characters on A
have the form εx : f 7→ f(x) for some x ∈ K;
equivalently, all maximal modular ideals are of
the form
Mx = ker εx = f ∈ A : f(x) = 0 .
9
Gel’fand theory
Let A be a BA. Define a(ϕ) = ϕ(a) (ϕ ∈ ΦA).
Then a ∈ C0(ΦA), and the Gel’fand transform
G : a 7→ a, (A, ‖ · ‖)→ (C0(ΦA), | · |ΦA),
is a continuous linear operator that is an alge-
bra homomorphism.
In the case where A is a CBA = commutative
Banach algebra, ker G = rad A = Q(A), and
so G is injective iff A is semi-simple.
Thus natural BFAs correspond to semi-simple
CBAs on their character space.
In the case where A is a commutative C∗-algebra, Gel’fand theory shows that A is iso-
metrically and algebraically ∗-isomorphic
to C0(ΦA).
10
More on BFAs
Henceforth K will be a non-empty, locally com-
pact (Hausdorff) space, and usually A will be
a natural BFA on K.
The closure of A ∩ C00(K) in A is called A0.
The BFA A is Tauberian if A = A0.
The ideal Jx in A consists of the functions in
A ∩ C00(K) that are 0 on a neighbourhood of
x, so that Jx ⊂Mx; A is strongly regular if Jxis dense in Mx for each x ∈ X.
11
Locally compact groups
Let G be a locally compact group with left
Haar measure mG. Then the group algebra
is (L1(G), ? , ‖ · ‖1) and the measure algebra
is (M(G), ? , ‖ · ‖), so that L1(G) is a closed
ideal in M(G). Both are semi-simple Banach
algebras. As a Banach space, M(G) = C0(G)′,and the product µ ? ν of µ, ν ∈ M(G) is given
by:
〈f, µ ? ν〉 =∫G
∫Gf(st) dµ(s) dν(t) (f ∈ C0(G)) .
The product of f, g ∈ L1(G) is given by
(f ? g)(t) =∫Gf(s)g(s−1t) dmG(s) (t ∈ G).
There is always one character on L1(G), namely
f 7→∫G f dmG; its kernel is the augmentation
ideal L10(G).
12
Dual Banach algebras
A Banach algebra A is a dual Banach algebra
if there is a closed submodule F of A′ such that
F ′ ∼ A, and then F is the predual of A. In this
case, we can write
A′′ = A⊕ F⊥
as a Banach space.
Key example: M(G) is a dual Banach algebra,
with predual C0(G).
13
Locally compact abelian groups
Let G be a locally compact abelian (LCA) group.
A character on G is a group homomorphism
from G onto the circle group T. The set Γ = G
of all continuous characters on G is an abelian
group with respect to pointwise multiplication
given by:
(γ1 + γ2)(s) = γ1(s)γ2(s) (s ∈ G, γ1, γ2 ∈ Γ) .
The topology on Γ is that of uniform cov-
ergence on compact subsets of G; with this
topology, Γ is also a LCA group, called the
dual group to G.
It is standard that the dual group of a compact
group is discrete and that the dual group of a
discrete group is compact.
For example, Z = T, T = Z, and R = R.
14
Pontryagin duality theorem
For each s ∈ G, the map
γ 7→ γ(s), Γ→ T,
is a continuous character on Γ, and the famous
Pontryagin duality theorem asserts that each
continuous character on Γ has this form and
that the topology of uniform convergence on
compact subsets of Γ coincides with the origi-
nal topology on Γ, so that Γ = G.
HenceG = G.
15
Fourier transform
Let G be a LCA group. The Fourier trans-
form of f ∈ L1(G) is f = Ff , so that
(Ff)(γ) = f(γ) =∫Gf(s)〈−s, γ〉dmG(s) (γ ∈ Γ) ,
and
A(Γ) =f : f ∈ L1(G)
,
is a natural, Tauberian BFA on Γ.
The Fourier–Stieltjes transform of µ ∈M(G)
is µ = Fµ, so that
(Fµ)(γ) = µ(γ) =∫G〈−s, γ〉dµ(s) (γ ∈ Γ) ,
and
B(Γ) = µ : µ ∈M(G) ,
is a Banach function algebra on Γ.
Of course, F : (M(G), ? ) → (B(Γ), · ) is a lin-
ear contraction that is an algebra isomorphism.
16
The group C∗-algebra
Here Γ is a locally compact group.
Let π be a representation of (L1(Γ), ? ), so that
π : L1(Γ)→ B(Hπ)
is a contractive ∗-homomorphism for some Hilbert
space Hπ. For f ∈ L1(Γ), define
|||f ||| = sup ‖π(f)‖ : π is a representation of L1(Γ) ,
so that |||f ||| ≤ ‖f‖1. Then ||| · ||| is a norm on
L1(Γ) such that
|||f∗ ? f ||| = |||f |||2 (f ∈ L1(Γ)) ,
and the completion of (L1(Γ), ||| · |||) is a C∗-algebra, called C∗(Γ), the group C∗-algebra
of Γ.
17
Fourier and Fourier–Stieltjes algebras
For a function f on a group Γ, we set
f(s) = f(s−1) (s ∈ Γ) .
Let Γ be a locally compact group.
The Fourier algebra on Γ is
A(Γ) = f ? g : f, g ∈ L2(Γ) .
Let Γ be a locally compact group. A functionf : Γ → C is positive-definite if it is continu-ous and if, for each n ∈ N, t1, . . . , tn ∈ G, andα1, . . . , αn ∈ C, we have
n∑i,j=1
αiαjf(t−1i tj) ≥ 0 .
The space of positive-definite functions on Γis denoted by P (Γ).
The Fourier–Stieltjes algebra on Γ, calledB(Γ), is the linear span of the positive-definitefunctions.
18
Properties of A(Γ) and B(Γ)
First, in the case where Γ is abelian, these twoalgebras agree with those previously defined.
Their theory originates in the seminal work ofEymard of 60 years ago.
The norm on B(Γ) comes from identifying itwith the dual of C∗(Γ), the group C∗-algebraof Γ.
For details of all this, see Lecture 1 of JorgeGalindo.
Theorem Let Γ be a locally compact group.Then A(Γ) is a natural, strongly regular, self-adjoint BFA on Γ, and B(Γ) is a self-adjointBFA on Γ. Further, A(Γ) is the closed ideal inB(Γ) that is the closure of B(Γ)∩C00(Γ). 2
Usually, A(Γ) ( B(Γ).
19
Facts about A(Γ) and B(Γ)
These facts will not be used, and terms are
not defined.
Facts A(Γ) is complemented in B(Γ); A(Γ) is
weakly sequentially complete; the dual space
A(Γ)′ is V N(Γ), the group von Neumann alge-
bra of Γ; A(Γ) is an ideal in its bidual iff Γ is
discrete. 2
Facts B(Γ) is a dual BFA, with predual C∗(Γ);
A(Γ) is a dual BFA iff A(Γ) = B(Γ) iff Γ is
compact [iff B(Γ) has the Schur property]. 2
20
Banach sequence algebras
Let S be a non-empty set, usually N. We write
c0(S) and `∞(S) for the Banach spaces of null
and bounded functions on S, respectively; the
algebra of all functions on S of finite support
is c00(S).
A Banach sequence algebra (= BSA) on S
is a BFA A on S such that
c00(S) ⊂ A ⊂ `∞(S) .
Thus A is Tauberian iff c00(S) is dense in A.
For example, ` p = ` p(N) and A(Z) with point-
wise product are Tauberian BSAs.
21
Biduals of Banach algebras
Let A be a Banach algebra. Then there are two
products 2 and 3 on A′′, the first and second
Arens products, that extend the given prod-
uct on A. Roughly:
Take M,N ∈ A′′, say M = limα aα and
N = limβ bβ, where (aα) and (bβ) are nets in A
(weak-∗ limits). Then
M2N = limα
limβaαbβ, M3N = lim
βlimαaαbβ .
The basic theorem of Arens is that κA : A→ A′′
is an isometric algebra monomorphism of A
into both (A′′,2) and (A′′,3).
We shall usually write just A′′ for (A′′,2).
22
Arens regularity
A Banach algebra A is Arens regular = AR if2 and 3 coincide on A′′. A commutative Ba-nach algebra is AR iff (A′′,2) is commutative.
Fact A′′ is a dual Banach algebra iff A is AR. 2
Let A be a C∗-algebra. Then A is AR and(A′′,2) is also a C∗-algebra, called the en-veloping von Neumann algebra.
In particular, (C0(K)′′,2) is a commutative C∗-algebra, and so has the form C(K) for a com-pact space K, called the hyper-Stonean en-velope of K. For K = N, we have K = βN,the Stone–Cech compactification of N.
Advertisement: this is discussed at length –with several ‘constructions’ and characteriza-tions of K – in
H. G. Dales, F. K. Dashiell, Jr., A. T.-M. Lau,and D. Strauss, Banach spaces of continuousfunctions as dual spaces, Springer, 2016
23
Strong Arens irregularity
Let A be a Banach algebra. Then the left andright topological centres are
Z(`)t (A′′)=
M ∈ A′′ : M2N = M3N (N ∈ A′′)
and
Z(r)t (A′′)=
M ∈ A′′ : N2M = N3M (N ∈ A′′)
,
respectively. Thus the algebra A is Arens reg-ular if and only if
Z(`)t (A′′) = Z
(r)t (A′′) = A′′ ;
A is strongly Arens irregular = SAI ifZ
(`)t (A′′) = Z
(r)t (A′′) = A.
In the case where A is commutative,
Z(`)t (A′′) = Z
(r)t (A′′) = Z(A′′).
Example Each group algebra L1(G) is SAI(Lau and Losert). Indeed, each measure al-gebra M(G) is SAI (Neufang et al). 2
24
Ideals in biduals
Let A be an algebra. For a ∈ A, we define La
and Ra by
La(b) = ab , Ra(b) = ba (b ∈ A) .
They are multipliers in an appropriate sense.
Let A be a BFA. Then A is an ideal in its
bidual if A is a closed ideal in A′′. This hap-
pens iff each La and Ra is a weakly compact
operator.
Fact Let A be a Tauberian BSA. Then Lf is
compact for each f ∈ A, and so A is an ideal
in its bidual. 2
There are non-Tauberian BSAs on N that are
ideals in their biduals, and there is a BSA on
N that is AR, but not an ideal in its bidual.
25
Tensor products
Let E and F be Banach spaces. Then (E ⊗F, ‖ · ‖π)
is their projective tensor product. Each ele-
ment z of E ⊗F can be expressed in the form
z =∞∑i=1
xi ⊗ yi ,
where xi ∈ E, yi ∈ F and∑∞i=1 ‖xi‖ ‖yi‖ < ∞,
and then ‖z‖π is the infimum of these sums
over all such representations.
The basic property of E ⊗F is the following:
for Banach spaces E, F , and G and each bounded
bilinear operator S : E × F → G, there is a
unique bounded linear operator TS : E ⊗F → G
such that TS(x ⊗ y) = S(x, y) (x ∈ E, y ∈ F )
and such that ‖TS‖ = ‖S‖.
26
Duals of tensor products
We have (E ⊗F )′ ∼= B(E,F ′), where the iso-
metric isomorphism
T : λ 7→ Tλ , (E ⊗F )′ → B(E,F ′) ,
satisfies the condition that
〈y, Tλx〉 = 〈x⊗y, λ〉 (x ∈ E, y ∈ F, λ ∈ (E ⊗F )′) .
This duality prescribes a weak-∗ topology on
B(E,F ′).
We use the following result of Cabello Sanchez
and Garcia:
Theorem Suppose that E′′ has the bounded
approximation property (BAP). Then the nat-
ural embedding of E ⊗F into (E ⊗F )′′ extends
to an isomorphic embedding of E′′ ⊗F ′′ onto a
closed subspace of (E ⊗F )′′. 2
27
Tensor products of BFAs
Let A and B be algebras, and set A = A⊗B.
Then there is a unique product on A with res-
pect to which A is an algebra and such that
(a1 ⊗ b1)(a2 ⊗ b2) = a1a2 ⊗ b1b2for a1, a2 ∈ A and b1, b2 ∈ B.
Fact Let A and B be natural BFAs on K and
L, respectively, and suppose that A has the ap-
proximation property. Then A ⊗B is a natural
BFA on K × L. If A has BAP, then A′′ ⊗B′′ is
a closed subalgebra of (A ⊗B)′′. 2
General question Suppose that A and B are
BFAs that are AR. Is A ⊗B AR?
A criterion involving biregularity and many ex-
amples (both ways) were given by Ali Ulger,
TAMS, 1988. See later.
28
Uniform algebras
A BFA A is a uniform algebra if it is closedin (Cb(K), | · |K), and so the norm is equivalentto the uniform norm.
For example, C0(K) is a natural uniform alge-bra on K. A natural uniform algebra A on K
is trivial if A = C0(K).
The disc algebra consists of all f analytic onD = z ∈ C : |z| < 1 and continuous on D.
A point x in K is a strong boundary pointfor A if, for each neighbourhood U of x, thereexists f ∈ A such that f(x) = |f |X = 1 and|f(y)| < 1 (y ∈ K \ U).
For x, y ∈ ΦA, say x ∼ y if ‖εx − εy‖ < 2. Thisis an equivalence relation that divides ΦA intoequivalence classes, called Gleason parts. Astrong boundary point is a singleton part, butnot conversely.
29
Approximate identities
Let A be a CBA. A net (eα) in A is an approx-imate identity for A if
limαaeα = a (a ∈ A) ;
an approximate identity (eα) is bounded ifsup α ‖eα‖ <∞, and then sup α ‖eα‖ is the bound;an approximate identity is contractive if it hasbound 1.
We refer to a BAI and a CAI, respectively, inthese two cases.
A natural BFA A on K is contractive if Mx
has a CAI for EACH x ∈ K.
Obvious example Take A = C0(K). Then Ais contractive. Are there any more contractiveBFAs? See later.
Group algebras have a CAI, but the augmen-tation ideal L1
0(G) has a BAI (of bound 2 - seelater), not a CAI, and so L1(G) is not contrac-tive.
30
Pointwise approximate identities
We shall consider (natural) BFAs on a locallycompact space K.
Let A be a natural BFA on K. A net (eα) in A
is a pointwise approximate identity (PAI) if
limαeα(x) = 1 (x ∈ K) ;
the PAI is bounded, with bound m > 0, ifsup α ‖eα‖ ≤ m, and then (eα) is a boundedpointwise approximate identity (BPAI); abounded pointwise approximate identity of bound1 is a contractive pointwise approximateidentity (CPAI).
Clearly a BAI is a BPAI and a CAI is a CPAI.
The algebra A is pointwise contractive if Mx
has a CPAI for each x ∈ K.
Also clearly a contractive BFA is pointwise con-tractive. But we shall give examples to showthat the converse is not true.
31
Contractive uniform algebras
Theorem Let A be a uniform algebra on a
compact space K, and take x ∈ K. Then the
following conditions on x are equivalent:
(a) εx ∈ exKA, where
KA = λ ∈ A′ : ‖λ‖ = 〈1K, λ〉 = 1 ;
(b) x is a strong boundary point;
(c) Mx has a BAI;
(d) Mx has a CAI.
Proof Most of this is standard.
(c) ⇒ (d) M ′′x is a maximal ideal in A′′, a closed
subalgebra of C(K)′′ = C(K). A BAI in Mx
gives an identity in M ′′x , hence an idempotent
in C(K). The latter have norm 1. So there is
a CAI in Mx. 2
32
Cole algebras
Definition Let A be a natural uniform algebra
on a compact space K. Then A is a Cole al-
gebra if every point of K is a strong boundary
point.
Theorem A uniform algebra is contractive if
and only if it is a Cole algebra. 2
There are non-trivial Cole algebras (but they
took some time to find). One is R(X) for a
certain compact set X in C2.
Theorem A natural uniform algebra on X is
pointwise contractive if and only if each set
x is a singleton Gleason part. 2
Standard examples now give separable uniform
algebra that are pointwise contractive, but not
contractive.33
The BSE norm
Definition Let A be a natural Banach functionalgebra on a locally compact space K. ThenL(A) is the linear span of εx : x ∈ K as asubset of A′, and
‖f‖BSE = sup |〈f, λ〉| : λ ∈ L(A)[1] (f ∈ A) .
Clearly K ⊂ L(A)[1] ⊂ A′[1], and so
|f |K ≤ ‖f‖BSE ≤ ‖f‖ (f ∈ A) .
In fact, ‖ · ‖BSE is an algebra norm on A - seelater.
Definition A BFA A has a BSE norm if thereis a constant C > 0 such that
‖f‖ ≤ C ‖f‖BSE (f ∈ A) .
Clearly each uniform algebra has a BSE norm.
A closed subalgebra of a BFA with BSE normalso has a BSE norm.
34
BSE algebras
Let A be a natural BFA on locally compact K.Then
M(A) = f ∈ Cb(K) : fA ⊂ A ,the multiplier algebra of A. It is a unital BFAon K with respect to the operator norm ‖ · ‖op.
For example, the multiplier algebra of L1(G) isM(G) (Wendel). This applies to all G: eachtwo-sided multiplier on (L1(G), ? ) has the formf 7→ f ? µ for some µ ∈M(G).
Let A be a natural Banach function algebra onK. Then
‖f‖BSE = sup |〈f, λ〉| : λ ∈ L(A)[1] (f ∈ Cb(K)) ,
and
CBSE(A) = f ∈ Cb(K) : ‖f‖BSE <∞ .The algebra A is a BSE algebra wheneverM(A) = CBSE(A). (It does not necessarilyhave a BSE norm.) For unital algebras, thecondition is that A = CBSE(A).
35
Basic theorem on CBSE(A)
The following is in TH in 1992.
Theorem Let A be a natural Banach func-tion algebra on K. Then (CBSE(A), ‖ · ‖BSE)is a Banach function algebra on K. Further,CBSE(A) is the set of functions f ∈ C b(K) forwhich there is a bounded net (fν) in A withlimν fν = f in (C b(K), τp); for f ∈ CBSE(A),the infimum of the bounds of such nets is equalto ‖f‖BSE.
Proof Certainly CBSE(A) is a linear subspaceof C b(K), and ‖ · ‖BSE is a norm on CBSE(A).It is a little exercise to check that(CBSE(A), ‖ · ‖BSE) is a Banach space.
Now take f1, f2 ∈ CBSE(A). We show that
‖f1f2‖BSE ≤ ‖f1‖BSE ‖f2‖BSE ;
we shall suppose that ‖f1‖BSE , ‖f2‖BSE = 1.
36
Proof continued
Take λ =∑ni=1αiεxi ∈ L(A)[1], and fix ε > 0.
First, set µ1 =∑ni=1αif1(xi)εxi, so that µ1 ∈ A′.
Then there exists g1 ∈ A[1] with |〈g1, µ1〉| >‖µ1‖ − ε. Next, set µ2 =
∑ni=1αig1(xi)εxi, so
that µ2 ∈ A′. Then there exists g2 ∈ A[1] with
|〈g2, µ2〉| > ‖µ2‖ − ε. We see that
〈f1, µ2〉 = 〈g1, µ1〉 and 〈g2, µ2〉 = 〈g1g2, λ〉 ,
and hence that |〈g2, µ2〉| ≤ ‖g1g2‖ ‖λ‖ ≤ 1. We
now have
|〈f1f2, λ〉| =
∣∣∣∣∣∣n∑i=1
αif1(xi)f2(xi)
∣∣∣∣∣∣= |〈f2, µ1〉| ≤ ‖µ1‖ < |〈g1, µ1〉|+ ε
= |〈f1, µ2〉|+ ε ≤ ‖µ2‖+ ε
< |〈g2, µ2〉|+ 2ε ≤ 1 + 2ε .
This holds for each λ ∈ L(A)[1] and each ε > 0,
and so ‖f1f2‖BSE ≤ 1, as required.
37
Proof concluded
Take f ∈ C b(K) to be such that there is a
bounded net (fν) in A[m] for some m > 0
such that limν fν = f in (C b(K), τp). For each
λ ∈ L(A)[1], we have
|〈f, λ〉| = limν|〈fν, λ〉| ≤ m,
and hence f ∈ CBSE(A)[m].
Conversely, suppose that f ∈ CBSE(A)[m], where
m > 0. For each non-empty, finite subset F
of K and each ε > 0, it follows from Helly’s
theorem that there exists fF,ε ∈ A such that
fF,ε(x) = f(x) (x ∈ F ) and∥∥∥fF,ε∥∥∥ ≤ m + ε.
Then the net (fF,ε) converges to f in
(C b(K), τp). 2
38
Sample general theorems – 1
Theorem Let A be a natural BFA. Then A is
a BSE algebra if and only if A has a BPAI and
the set
f ∈M(A) : ‖f‖BSE ≤ 1
is closed in (C b(ΦA), τp). 2
Theorem Let A be a natural BSA. Then CBSE(A)
is isometrically isomorphic to the Banach al-
gebra A′′/L(A)⊥, and CBSE(A) is a dual BFA,
with predual L(A). 2
Theorem Let A be a natural BFA. Then A
has a BSE norm iff the subalgebra A+ L(A)⊥
is closed in A′′. 2
39
Sample general theorems – 2
Theorem Let A be a dual BFA with predual
F . Suppose that the space ΦA∩F[1] is dense in
ΦA. Then A = CBSE(A) and ‖f‖ = ‖f‖BSE (f ∈ A).
Then A has a BSE norm.
Proof Take f ∈ CBSE(A), with ‖f‖BSE = m,
say. Then there is a bounded net (fν) in A[m]with limν fν = f in (C b(K), τp). Let g be an
accumulation point of this net in (A, σ(A,F )).
Then g(ϕ) = f(ϕ) (ϕ ∈ ΦA ∩ F[1]), and so
g = f . Thus f ∈ A[m] with ‖f‖ = ‖f‖BSE,
showing that A = CBSE(A). 2
Corollary Let G be a compact group. Then
M(G) has a BSE norm. 2
Theorem A BSE algebra has a BSE norm iff
it is closed in (M(A), ‖ · ‖op). 2
40
Sample general theorems – 3
The `1-norm on L(A) is given by∥∥∥∥∥∥n∑i=1
αiεxi
∥∥∥∥∥∥1
=n∑i=1
|αi| .
Theorem Let A be a BFA on K. Then
CBSE(A) = Cb(K) iff the usual norm on L(A)
is equivalent to the `1-norm. 2
41
Ideals in biduals
Theorem Let (A, ‖ · ‖) be a natural BFA on K
such that A is an ideal in its bidual. Then A isan ideal in CBSE(A) and
| · |K ≤ ‖ · ‖op ≤ ‖ · ‖BSE ≤ ‖ · ‖
on A. 2
Theorem (KU) Let A be a BFA that is an idealin its bidual. Then the following are equivalent:
(a) A is a BSE algebra;
(b) A has a BPAI;
(c) A has a BAI. 2
Theorem Let A be a dual BFA that is an idealin its bidual. Then A = CBSE(A) is AR, A hasa BSE norm, and A′′ = A⊕ L(A)⊥. 2
Theorem (*) Let A be a BFA that is an idealin its bidual, is AR, and has a BAI. Then A′′ isa BFA and has BSE norm. 2
42
Easy examples of BSAs
Here all algebras have coordinatewise products.
Example 1 Look at c0. Here c′′o = `∞, so c0 isan ideal in its bidual and is AR; it is not a dualalgebra. It has a BSE norm, and it is a BSEalgebra becauseM(c0) = `∞ = CBSE(c0). 2
Example 2 Look at `1, a Tauberian BSA, sothat `1 is an ideal in its bidual; it is a dual BSAwith predual c0. Here (`1)′ = `∞ = C(βN) and(`1)′′ = M(βN). Further, `1 = CBSE(`1) isAR, and M(βN) = `1nM(N∗), with the product
(α, µ)2 (β, ν) = (αβ,0) (α, β ∈ `1, µ, ν ∈M(N∗)) .
No BPAI, so not a BSE algebra; since L(`1)[1]is weak-∗ dense in (`1)′[1], the BSA `1 has aBSE norm. 2
Example 3 Look at ` p, where 1 < p <∞. Thisis a Tauberian BSA and is a reflexive Banachspace, and so ` p is an ideal in its bidual and adual algebra. It has a BSE norm, but it is nota BSE algebra. 2
43
General results
BSE algebras and BSE norms were introducedin 1990 by Takahasi and Hatori (TH) as anabstraction of a classical theorem of harmonicanalysis, the Bochner–Schoenberg–Eberleintheorem; see later.
Quite a few papers have discussed specific ex-amples. Our work seeks to give an underlyinggeneral theory, and applications to more exam-ples.
General questions Does every dual BFA havea BSE norm? Does every (even Tauberian)BSA have a BSE norm?
In both cases, we can give positive answerswith the help of modest extra hypotheses; wehave no counter-examples.
We can resolve these questions for many, butnot all, specific examples that we have lookedat – see below.
44
Contractive results
Theorem A contractive BFA with a BSE norm
is a Cole algebra. 2
Theorem Let A be a pointwise contractive
BFA with a BSE norm. Then the norms | · |Kand ‖ · ‖BSE on A are equivalent, and A is a
uniform algebra for which each singleton in ΦA
is a one-point Gleason part. Further, A is a
BSE algebra if and only if A = C(K). 2
Thus, to find (pointwise) contractive BFAs that
are not equivalent to uniform algebras, we must
look for those that do not have a BSE norm;
see later.
45
Queries for uniform algebras
Caution It is not true that every natural uni-
form algebra on a compact K is a BSE alge-
bra - a Cole algebra on a compact K is a BSE
algebra iff it is C(K), and so we can take a
non-trivial Cole algebra as a counter-example.
The disc algebra is a BSE algebra.
Query What is CBSE(A) for a uniform alge-
bra A? How do we characterize the uniform
algebras that are BSE algebras?
Query For example, what is CBSE(R(K)) for
compact K ⊂ C? Look at a Swiss cheese K.
46
Banach sequence algebras, bis
BSAs are more complicated than you mightsuspect. Does each natural BSA on N have aBSE norm? We have a general theorem thatat least covers the following example.
Example For α = (αk) ∈ CN, set
pn(α) =1
n
n∑k=1
k∣∣∣αk+1 − αk
∣∣∣ ,p(α) = sup pn(α) : n ∈ N .
Define A to be α ∈ c0 : p(α) <∞, so that Ais a self-adjoint BSA on N for the norm
‖α‖ = |α|N + p(α) (α ∈ A) .
Then A is a natural; A2 = A20 = A0 ( A; A is
not Tauberian; A is non-separable; A is not anideal in its bidual. The algebra A is not Arensregular.
This example does have a BSE norm, and it isa BSE algebra. 2
47
Tensor products of BSAs – 1
Guess Suppose that A and B are BFAs thatare BSE algebras/have BSE norms. Then A ⊗Bhas the corresponding property.
Let A and B be natural BSAs on S and T andsuppose that A has AP as a Banach space.Then A ⊗B is a natural BSA on S × T .
Example 1 Take p and q with 1 < p, q < ∞,and set A = ` p ⊗ ` q, so that A′ = B(` p, ` q
′).
Then A is a Tauberian BSA on N × N, and soan ideal in its bidual; it is the dual of K(` p, ` q
′);
it is AR.
It is reflexive iff pq > p+ q (Pitt) (this fails forp = q = 2).
Here A = CBSE(A), so A has a BSE norm, butA is not a BSE algebra. 2
48
Tensor products of BSAs – 2
Example 2 Let A = c0 ⊗ c0, so A is a Taube-
rian BSA on N×N, hence an ideal in its bidual.
It is AR, has a BSE norm, and it is a BSE
algebra. Here M(A) = CBSE(A) = A′′.
By Theorem (*), A′′ has a BSE norm.
Of course c′′0 = `∞ = C(βN); by an earlier
result, C(βN) ⊗C(βN) is a closed subalgebra
of A′′, and so also has a BSE norm.
We do not know if either C(βN) ⊗C(βN) or A′′
is a BSE algebra.
Neufang has shown that A′′ is not AR - see
his lecture. What about C(βN) ⊗C(βN)?
49
Varopoulos algebra
Let K and L be compact spaces, and set
V (K,L) = C(K) ⊗C(L) ,
the projective tensor product of C(K) and C(L);
this algebra is the Varopoulos algebra.
It is a natural, self-adjoint BFA on K×L, dense
in C(K ×L). The dual space is identified with
B(C(K),M(L)).
To show that V = V (K,L) has a BSE norm,
we must show that L(V )[1] is weak-∗ dense in
V ′[1] = B(C(K),M(L))[1]:
given T ∈ B(C(K),M(L))[1], ε > 0, n ∈ N,
f1, . . . , fn ∈ C(K), and g1, . . . , gn ∈ C(L), we
must find S ∈ L(V )[1] such that
|〈g, (T − S)f〉| < ε
whenever f ∈ f1, . . . , fn and g ∈ g1, . . . , gn.50
Varopoulos algebra, continued
We can do this by choosing suitable partitions
of unity in C(K) and C(L). Thus:
Theorem For compact K and L, V (K,L) has
a BSE norm. 2
Question Is V = V (K,L) a BSE algebra?
For this, we would have to show that CBSE(V ) = V .
At least we know that CBSE(V ) ( C(K × L),
using a result in the book of Helemskii.
51
Tensor products of uniform algebras
Let A and B be natural uniform algebras on
K and L, respectively. It is natural to ask if
A ⊗B always has a BSE norm. Clearly this
would follow immediately from the above if we
knew that A ⊗B were a closed subalgebra of
V (K,L). However this is not easily seen: it
is not immediate because a proper uniform al-
gebra A on a compact space K is never com-
plemented in C(K). The result is true in the
special case where A and B are the disc alge-
bra, as shown by Bourgain
Theorem Let A := A(D) to be the disc al-
gebra. Then A ⊗A is a closed subalgebra of
V (D,D), and so A ⊗A has a BSE norm. 2
Query What happens for different uniform al-
gebras? Is A(D) ⊗A(D) a BSE algebra?
52
Group and measure algebras
Let G be an infinite, LCA group with dual Γ.
Theorem (i) L1(G) is a BSE algebra, and
M(L1(G)) = CBSE(L1(G)) = M(G) .
(ii) ‖µ‖ = ‖µ‖BSE (µ ∈ M(G)), and so M(G)
and L1(G) each have a BSE norm.
(iii) M(G) is a BSE algebra iff G is discrete.
Proof (i) Classical Bochner–Schoenberg–Eberlein
theorem.
(ii) Uses almost periodic functions on G.
(iii) It is easy to find functions in CBSE(M(G))
that are not in M(G) when G is not discrete. 2
53
Compact abelian groups
Take G to be a compact, abelian group. For
1 ≤ p ≤ ∞, (Lp(G), ? ) is a semi-simple CBA.
For 1 < p <∞, F(Lp(G), ? ) is a Tauberian BSA
on Γ; it is reflexive; and hence AR and an ideal
in its bidual and a dual BFA; it does not have
a BPAI.
Further, F(L∞(G), ? ) is a natural BSA on Γ,
but it is not Tauberian. It is a dual BSA with
predual A(Γ); it is AR; it is an ideal in its bid-
ual.
Theorem For 1 < p ≤ ∞, F(Lp(G)) has a BSE
norm, but it is not a BSE algebra. 2
54
Beurling algebras on Z
A weight on Z is a function ω : Z → [1,∞)
such that ω(0) = 1 and
ω(m+ n) ≤ ω(m)ω(n) (m,n ∈ Z) .
Then `1(Z, ω) is the space of functions
f =∑f(n)δn such that
‖f‖ω =∑|f(n)|ω(n) <∞ .
This is a commutative Banach algebra for con-
volution. Via the Fourier transform, `1(Z, ω) is
a BFA on the circle or an annulus in C.
The algebra `1(Z, ω) is a dual BFA, with pre-
dual c0(Z,1/ω); it is not an ideal in its bidual.
Examples show that `1(Z, ω) may be AR, that
it may be that ω is unbounded and it is SAI; it
may be neither AR nor SAI (D-Lau).
55
Beurling algebras as BSE algebras
Theorem Beurling algebras Aω are BSE alge-
bras with a BSE norm for most, may be all,
weights.
Proof This works when ΦAω ∩ c0(Z,1/ω)[1] is
dense in ΦAω. 2
Trouble for weights ω with lim supn→∞ ω(n) =∞and lim infn→∞ ω(n) = 1; they exist.
56
Figa-Talamanca–Herz algebras
Let Γ be a locally compact group.and take p
with 1 < p < ∞. The Figa-Talamanca–Herz
(FTH) algebra is Ap(Γ). Formally, Ap(Γ) is
the collection of sums
f =∞∑n=1
gn ? hn
where gn ∈ Lp(Γ) and hn ∈ Lp′(Γ) for each
n ∈ N and∑∞n=1 ‖gn‖p ‖hn‖p′ < ∞, and ‖f‖ is
the infimum of such sums.
Thus Ap(Γ) is a self-adjoint, Tauberian, nat-
ural, strongly regular Banach function algebra
on Γ.
[See papers of Herz, a book and lectures of
Derighetti.]
57
BAIs and BPAIs in FTH algebras
Theorem (mainly Leptin) Let Γ be a locally
compact group, and take p > 1. Then the
following are equivalent:
(a) Γ is amenable;
(b) Ap(Γ) has a BAI;
(c) Ap(Γ) has a BPAI;
(d) Ap(Γ) has a CAI. 2
58
Arens regularity of Fourier algebras
Theorem (Lau–Wong) Let Γ be a LC group,
and suppose that A(Γ) is AR. Then Γ is dis-
crete, and every amenable subgroup is finite.
May be Γ must be finite. 2
Suppose that Γ is discrete. If Γ is amenable,
then A(Γ) is SAI (Lau–Losert, 1988), but not
if Γ contains F2 (Losert, 2016).
For the case where Γ is not discrete, and es-
pecially when Γ is compact, see the lectures of
Jorge Galindo in Oulu.
59
BSE properties
Theorem (essentially Eymard) Let Γ be a LC
group. Then
‖f‖ = ‖f‖BSE (f ∈ B(Γ)) ,
and so A(Γ) and B(Γ) have a BSE norm.
Proof Since B(Γ) = C∗(Γ)′, we can use Ka-
plansky’s density theorem for C∗-algebras. 2
Theorem (KU) A(Γ) is a BSE algebra iff Γ is
amenable. 2
60
B(Γ) as a BSE algebra
Let Γ be a LC group.
In the case where Γ is compact, A(Γ) = B(Γ),
and so B(Γ) is a BSE algebra.
In the case where Γ is not compact, there is,
as shown in KU, surprising diversity: there are
amenable groups for which B(Γ) is and is not
a BSE algebra, and there are non-amenable
groups for which B(Γ) is and is not a BSE
algebra.
61
Tensor products of Fourier algebras
Let Γ1 and Γ2 be locally compact groups. Sup-
pose that
A(Γ1) ⊗A(Γ2) = A(Γ1 × Γ2) . (∗)
Then A(Γ1) ⊗A(Γ2) has a BSE norm, and it
is a BSE algebra if and only if both Γ1 and
Γ2 are amenable. But (*) is not always true
(Losert).
Guess A(Γ1) ⊗A(Γ2) always has a BSE norm,
and is a BSE algebra if and only if both Γ1 and
Γ2 are amenable.
62
BAIs and BPAIs in maximal ideals ofFourier algebras
Let Γ be an infinite, amenable locally compactgroup, and let M be a maximal modular idealof A(Γ).
It is standard that M has a BAI of bound 2.By a theorem of Delaporte and Derighetti,the number 2 is the minimum bound for sucha BAI. We now consider pointwise versions ofthis.
Theorem Let Γ be an infinite locally compactgroup such that Γd is amenable. Then theminimum bound of a BPAI in M is also 2. Inparticular, A(Γ) is not pointwise contractive.2
Query What happens if Γ is amenable, butΓd is not? (Eg., Γ = SO(3).) The minimumbound is > 1.
Query What happens for Ap(Γ) when p > 1and p 6= 2?
63
FTH algebras Ap(Γ)
Here Γ is a LC group and 1 < p <∞.
Theorem (Forrest) Ap(Γ) is an ideal in its bi-
dual iff Γ is discrete. 2
Theorem (Forrest) Suppose that Ap(Γ) is AR.
Then Γ is discrete and every abelian subgroup
is finite. May be Γ must be finite. 2
Apparently nothing is known of when Ap(Γ) is
SAI.
There are varying definitions of Bp(Γ). The
first was by Herz. Cowling said it wasM(Ap(Γ));
Runde gave a definition involving representa-
tion theory; we prefer Runde’s definition be-
cause it gives the previous Bp(Γ) when p = 2.
The definitions all agree when Γ is amenable.
64
BSE properties of FTH algebras
This is harder than for the case p = 2 because
we have no help from C∗-algebra theory. Take
p with 1 < p <∞.
Theorem Let Γ be a locally compact group.
Then Ap(Γ) is a BSE algebra if and only if Γ
is amenable. In this case,
Bp(Γ) = CBSE(Ap(Γ)) =M(Ap(Γ)) ,
and Ap(Γ) and Bp(Γ) have BSE norms.
Proof Uses interplay with Bp(Γd) and results
of Herz and of Derighetti. 2
Query Does Ap(Γ) have a BSE norm for each
Γ? This is true for p = 2.
65
Segal algebras
Definition Let (A, ‖ · ‖A) be a natural Banachfunction algebra on a locally compact spaceK. A Banach function algebra (B, ‖ · ‖B) is anabstract Segal algebra (with respect to A) ifB is an ideal in A and there is a net in B thatis an approximate identity for both (A, ‖ · ‖A)and (B, ‖ · ‖B).
Classical Segal algebras are abstract Segalalgebras with respect to L1(G).
Let S be a Segal algebra with respect to L1(G).Then F(S) is a natural, Tauberian BFA on Γ;it is an ideal in its bidual iff G is compact.
The norm is equivalent to ‖ · ‖1 iff S = L1(G).
Always M(G) ⊂M(S) (but not necessarily equal).
Theorem A Segal algebra S is a BSE algebraiff S has BPAI, and then
M(G) =M(S) = CBSE(S) . 2
66
BSE norms for Segal algebras
Let S be a Segal algebra on a LC group G.
Suppose that S has a CPAI. Then we can iden-
tify the BSE norm.
Indeed, for f ∈ S, we have∣∣∣f ∣∣∣Γ≤ ‖f‖BSE,S = ‖f‖op,S = ‖f‖1 ≤ ‖f‖S ,
and so S has a BSE norm iff S = L1(G).
67
An example of a Segal algebra
Example Let G be a non-discrete LCA group
with dual group Γ. Take p ≥ 1, define
Sp(G) = f ∈ L1(G) : f ∈ Lp(Γ) ,
and set
‖f‖Sp = max‖f‖1 ,
∥∥∥f ∥∥∥p
(f ∈ Sp(G)) .
Then (Sp(G), ? , ‖ · ‖Sp) is a Segal algebra with
respect to L1(G) and a natural, Tauberian BFA
on Γ. Since Sp(G)2 ( Sp(G), Sp(G) does not
have a BAI. However, by a result of Inoue and
Takahari, Sp(R) has a CPAI.
Thus Sp(R) is a BSE algebra without a BSE
norm. 2
68
Final example
Example We give a BFA A on the circle T, butwe identify C(T) with a subalgebra of C[−1,1].We fix α with 1 < α < 2.
Take f ∈ C(T). For t ∈ [−1,1], the shift of fby t is defined by
(Stf)(s) = f(s− t) (s ∈ [−1,1]) .
Define
Ωf(t) = ‖f − Stf‖1 =∫ 1
−1|f(s)− f(s− t)| ds
and
I(f) =∫ 1
−1
Ωf(t)
|t|αdt .
Then A = f ∈ C(T) : I(f) <∞ and
‖f‖ = |f |T + I(f) (f ∈ A) .
We see that (A, ‖ · ‖) is a natural, unital BFAon T; it is homogeneous.
69
Final example continued
Let en be the trigonometric polynomial given
by en(s) = exp(iπns) (s ∈ [−1,1]). Then
en ∈ A, and so A is uniformly dense in C(T).
But ‖en‖ ∼ nα−1, and so (A, ‖ · ‖) is not equiv-
alent to a uniform algebra.
We claim that A is contractive. We show that
M := f ∈ A : f(0) = 0 has a CAI.
For this, define
∆n(s) = max 1− n |s| ,0 (s ∈ [−1,1], n ∈ N) .
Then we can see that I(∆n) ∼ 1/n2−α, and so
‖1−∆n‖ ≤ 1+O(1/n2−α) = 1+o(1). Further,
a calculation shows that (1−∆n : n ∈ N) is an
approximate identity for M .
We conclude that ((1−∆n)/ ‖1−∆n‖ : n ∈ N)
is a CAI in M , and so A is contractive. 2
70
Conclusions
We have a contractive BFA not equivalent
to a uniform algebra.
Here the BSE norm is equal to the uniform
norm, and CBSE(A) = C(I), whereas M(A) = A,
so A is not a BSE algebra.
Thus our example is neither a BSE algebra nor
has a BSE norm.
71