Banach function algebras and BSE norms

H. G. Dales, Lancaster

Joint work with Ali Ulger,¨ Istanbul

Graduate course during 23rd Banach algebra conference, Oulu, Finland

July 2017

[email protected]

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 ball is E[1] = {x ∈ E : kxk ≤ 1}.

The dual space of E is E0. This is the space of continuous = bounded linear functionals on E, and its norm is given by

kλk = sup {|hx, λi| = |λ(x)| : x ∈ E[1]}, so that (E0, k · k) is a Banach space.

The weak-∗ topology on E0 is σ(E0,E). Thus 0 0 (E[1], σ(E ,E)) is compact.

The bidual of E is E00 = (E0)0, and we regard E as a closed subspace of E00; the canonical 00 embedding is κE : E → E , where 0 hκE(x), λi = hx, λi (x ∈ E, λ ∈ E ).

For a closed subspace F of E, the annihilator of F is F ⊥ = {λ ∈ E0 : 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 rad A; it is the intersection of the maximal modular left ideals; it is an ideal in A. The algebra A is semi-simple if rad A = {0} and radical if rad A = 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 → C such 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 k · k is a Banach algebra (BA) if (A, k · k) is a Banach space and kabk ≤ kak kbk (a, b ∈ A).

When A is unital, we also require that keAk = 1.

Standard non-commutative example: A = B(E), the algebra of all bounded linear operators on a Banach space E, with operator norm k · kop. Here (ST )(x) = S(T x)(x ∈ E) for S, T ∈ A.

Each character ϕ on a BA is continuous, with 0 kϕk ≤ 1, and so ΦA ⊂ A[1];ΦA is a locally compact subspace of (A0, σ(A0,A)), and it is compact when A is unital.

In a BA, each maximal (modular) ideal is closed and so rad A is closed. Further Q(A) = {a ∈ A : lim kank1/n = 0}. n→∞

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 that vanish at infinity;

C00(K) consists of the continuous functions with compact support.

We define b |f|K = sup {|f(x)| : x ∈ K} (f ∈ C (K)) , so that | · |K is the uniform norm on K and b (C (K), | · |K) is a commutative, semisimple Banach algebra;

b C0(K) is a closed ideal in C (K);

b C00(K) is an ideal in C (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 k · k such that (A, k · k) 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 ab(ϕ) = ϕ(a)(ϕ ∈ ΦA). Then ab ∈ C0(ΦA), and the Gel’fand transform G : a 7→ a, (A, k · k) → (C (Φ ), | · | ), b 0 A 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 Jx is 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 1 is (L (G),?, k · k1) and the measure algebra is (M(G),?, k · k), so that L1(G) is a closed ideal in M(G). Both are semi-simple Banach 0 algebras. As a Banach space, M(G) = C0(G) , and the product µ ? ν of µ, ν ∈ M(G) is given by: Z Z hf, µ ? νi = f(st) dµ(s) dν(t)(f ∈ C0(G)) . G G

The product of f, g ∈ L1(G) is given by Z −1 (f ? g)(t) = f(s)g(s t) dmG(s)(t ∈ G). G

There is always one character on L1(G), namely R f 7→ G f dmG; its kernel is the augmentation 1 ideal L0(G). 12 Dual Banach algebras

A Banach algebra A is a dual Banach algebra if there is a closed submodule F of A0 such that F 0 ∼ A, and then F is the predual of A. In this case, we can write

A00 = 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 = Gb 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, Zb = T, Tb = Z, and Rb = 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 Γb = G.

Hence Gbb = G.

15 Fourier transform

Let G be a LCA group. The Fourier trans- form of f ∈ L1(G) is fb = Ff, so that Z (Ff)(γ) = fb(γ) = f(s)h−s, γi dmG(s)(γ ∈ Γ) , G and n 1 o A(Γ) = fb : f ∈ L (G) , is a natural, Tauberian BFA on Γ.

The Fourier–Stieltjes transform of µ ∈ M(G) is µb = Fµ, so that Z (Fµ)(γ) = µb(γ) = h−s, γi dµ(s)(γ ∈ Γ) , G and

B(Γ) = {µb : 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 1 π : L (Γ) → B(Hπ) is a contractive ∗-homomorphism for some Hilbert 1 space Hπ. For f ∈ L (Γ), define

|||f||| = sup {kπ(f)k : π is a representation of L1(Γ)} , so that |||f||| ≤ kfk1. 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

fe(s) = f(s−1)(s ∈ Γ) .

Let Γ be a locally compact group.

The Fourier algebra on Γ is 2 A(Γ) = {f ? ge : f, g ∈ L (Γ)} .

Let Γ be a locally compact group. A function f :Γ → C is positive-definite if it is continu- ous and if, for each n ∈ N, t1, . . . , tn ∈ G, and α1, . . . , αn ∈ C, we have n X −1 αiαjf(ti tj) ≥ 0 . i,j=1 The space of positive-definite functions on Γ is denoted by P (Γ).

The Fourier–Stieltjes algebra on Γ, called B(Γ), is the linear span of the positive-definite functions. 18 Properties of A(Γ) and B(Γ)

First, in the case where Γ is abelian, these two algebras agree with those previously defined.

Their theory originates in the seminal work of Eymard of 60 years ago.

The norm on B(Γ) comes from identifying it with the dual of C∗(Γ), the group C∗-algebra of Γ.

For details of all this, see Lecture 1 of Jorge Galindo.

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-adjoint BFA on Γ. Further, A(Γ) is the closed ideal in B(Γ) that is the closure of B(Γ) ∩ C 00(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)0 is VN(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 ∞ c 0(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 c 00(S).

A Banach sequence algebra (= BSA) on S is a BFA A on S such that

∞ c 00(S) ⊂ A ⊂ ` (S) .

Thus A is Tauberian iff c 00(S) is dense in A. p p For example, ` = ` (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 A00, the first and second Arens products, that extend the given prod- uct on A. Roughly:

00 Take M, N ∈ A , say M = limα aα and N = limβ bβ, where (aα) and (bβ) are nets in A (weak-∗ limits). Then

M 2 N = lim lim aαbβ, M 3 N = lim lim aαbβ . α β β α

00 The basic theorem of Arens is that κA : A → A is an isometric algebra monomorphism of A into both (A00, 2) and (A00, 3).

We shall usually write just A00 for (A00, 2).

22 Arens regularity

A Banach algebra A is Arens regular = AR if 2 and 3 coincide on A00. A commutative Ba- nach algebra is AR iff (A00, 2) is commutative.

Fact A00 is a dual Banach algebra iff A is AR. 2

Let A be a C∗-algebra. Then A is AR and (A00, 2) is also a C∗-algebra, called the en- veloping von Neumann algebra.

00 ∗ In particular, (C0(K) , 2) is a commutative C - algebra, and so has the form C(Kf) for a com- pact space Kf, called the hyper-Stonean en- velope of K. For K = N, we have Kf = βN, the Stone–Cechˇ compactification of N. Advertisement: this is discussed at length – with several ‘constructions’ and characteriza- tions of Kf – in

H. G. Dales, F. K. Dashiell, Jr., A. T.-M. Lau, and D. Strauss, Banach spaces of continuous functions as dual spaces, Springer, 2016 23 Strong Arens irregularity

Let A be a Banach algebra. Then the left and right topological centres are (`) 00 n 00 00 o Zt (A )= M ∈ A :M 2 N = M 3 N (N ∈ A ) and (r) 00 n 00 00 o Zt (A )= M ∈ A :N 2 M = N 3 M (N ∈ A ) , respectively. Thus the algebra A is Arens reg- ular if and only if (`) 00 (r) 00 00 Zt (A ) = Zt (A ) = A ; A is strongly Arens irregular = SAI if (`) 00 (r) 00 Zt (A ) = Zt (A ) = A.

In the case where A is commutative, (`) 00 (r) 00 00 Zt (A ) = Zt (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 A00. 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 ⊗b F, k · kπ) is their projective tensor product. Each ele- ment z of E ⊗b F can be expressed in the form ∞ X z = xi ⊗ yi , i=1 P∞ where xi ∈ E, yi ∈ F and i=1 kxik kyik < ∞, and then kzkπ is the infimum of these sums over all such representations.

The basic property of E ⊗b 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 ⊗b F → G such that TS(x ⊗ y) = S(x, y)(x ∈ E, y ∈ F ) and such that kTSk = kSk.

26 Duals of tensor products

0 ∼ 0 We have (E ⊗b F ) = B(E,F ), where the iso- metric isomorphism

0 0 T : λ 7→ Tλ , (E ⊗b F ) → B(E,F ) , satisfies the condition that

0 hy, Tλxi = hx⊗y, λi (x ∈ E, y ∈ F, λ ∈ (E ⊗b F ) ) . This duality prescribes a weak-∗ topology on B(E,F 0).

We use the following result of Cabello S´anchez and Garcia:

Theorem Suppose that E00 has the bounded approximation property (BAP). Then the nat- 00 ural embedding of E ⊗b F into (E ⊗b F ) extends 00 00 to an isomorphic embedding of E ⊗b F onto a 00 closed subspace of (E ⊗b 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 ⊗ b1b2 for 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 B is a natural 00 00 BFA on K × L. If A has BAP, then A ⊗b B is 00 a closed subalgebra of (A ⊗b B) . 2

General question Suppose that A and B are BFAs that are AR. Is A ⊗b 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 closed b in (C (K), | · |K), and so the norm is equivalent to 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 on D = {z ∈ C : |z| < 1} and continuous on D.

A point x in K is a strong boundary point for A if, for each neighbourhood U of x, there exists f ∈ A such that f(x) = |f|X = 1 and |f(y)| < 1 (y ∈ K \ U).

For x, y ∈ ΦA, say x ∼ y if kεx − εyk < 2. This is an equivalence relation that divides ΦA into equivalence classes, called Gleason parts.A strong boundary point is a singleton part, but not conversely. 29 Approximate identities

Let A be a CBA. A net (eα) in A is an approx- imate identity for A if lim ae = a (a ∈ A); α α an approximate identity (eα) is bounded if sup α keαk < ∞, and then sup α keαk is the bound; an approximate identity is contractive if it has bound 1.

We refer to a BAI and a CAI, respectively, in these 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 A is contractive. Are there any more contractive BFAs? See later.

Group algebras have a CAI, but the augmen- 1 tation ideal L0(G) has a BAI (of bound 2 - see later), not a CAI, and so L1(G) is not contrac- tive. 30 Pointwise approximate identities

We shall consider (natural) BFAs on a locally compact space K.

Let A be a natural BFA on K. A net (eα) in A is a pointwise approximate identity (PAI) if lim e (x) = 1 (x ∈ K); α α the PAI is bounded, with bound m > 0, if sup α keαk ≤ m, and then (eα) is a bounded pointwise approximate identity (BPAI); a bounded pointwise approximate identity of bound 1 is a contractive pointwise approximate identity (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 show that 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 0 KA = {λ ∈ A : kλk = h1K, λi = 1} ;

(b) x is a strong boundary point;

(c) Mx has a BAI;

(d) Mx has a CAI.

Proof Most of this is standard.

00 00 (c) ⇒ (d) Mx is a maximal ideal in A , a closed 00 subalgebra of C(K) = C(Kf). A BAI in Mx 00 gives an identity in Mx , hence an idempotent in C(Kf). 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 2 certain compact set X in C .

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 function algebra on a locally compact space K. Then L(A) is the linear span of {εx : x ∈ K} as a subset of A0, and kfkBSE = sup {|hf, λi| : λ ∈ L(A)[1]} (f ∈ A) .

0 Clearly K ⊂ L(A)[1] ⊂ A[1], and so

|f|K ≤ kfkBSE ≤ kfk (f ∈ A) .

In fact, k · kBSE is an algebra norm on A - see later.

Definition A BFA A has a BSE norm if there is a constant C > 0 such that

kfk ≤ C kfkBSE (f ∈ A) .

Clearly each uniform algebra has a BSE norm.

A closed subalgebra of a BFA with BSE norm also 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 BFA on K with respect to the operator norm k · kop.

For example, the multiplier algebra of L1(G) is M(G) (Wendel). This applies to all G: each two-sided multiplier on (L1(G),? ) has the form f 7→ f ? µ for some µ ∈ M(G).

Let A be a natural Banach function algebra on K. Then b kfkBSE = sup {|hf, λi| : λ ∈ L(A)[1]} (f ∈ C (K)) , and b CBSE(A) = {f ∈ C (K): kfkBSE < ∞} . The algebra A is a BSE algebra whenever M(A) = CBSE(A). (It does not necessarily have a BSE norm.) For unital algebras, the condition 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), k · kBSE) is a Banach function algebra on K. Further, b CBSE(A) is the set of functions f ∈ C (K) for which there is a bounded net (fν) in A with b limν fν = f in (C (K), τp); for f ∈ CBSE(A), the infimum of the bounds of such nets is equal to kfkBSE.

Proof Certainly CBSE(A) is a linear subspace b of C (K), and k · kBSE is a norm on CBSE(A). It is a little exercise to check that (CBSE(A), k · kBSE) is a Banach space.

Now take f1, f2 ∈ CBSE(A). We show that

kf1f2kBSE ≤ kf1kBSE kf2kBSE ; we shall suppose that kf1kBSE , kf2kBSE = 1. 36 Proof continued

Pn Take λ = i=1 αiεxi ∈ L(A)[1], and fix ε > 0.

Pn 0 First, set µ1 = i=1 αif1(xi)εxi, so that µ1 ∈ A . Then there exists g1 ∈ A[1] with |hg1, µ1i| > Pn kµ1k − ε. Next, set µ2 = i=1 αig1(xi)εxi, so 0 that µ2 ∈ A . Then there exists g2 ∈ A[1] with |hg2, µ2i| > kµ2k − ε. We see that hf1, µ2i = hg1, µ1i and hg2, µ2i = hg1g2, λi , and hence that |hg2, µ2i| ≤ kg1g2k kλk ≤ 1. We now have

n X |hf1f2, λi| = α f1(x )f2(x ) i i i i=1 = |hf2, µ1i| ≤ kµ1k < |hg1, µ1i| + ε = |hf1, µ2i| + ε ≤ kµ2k + ε < |hg2, µ2i| + 2ε ≤ 1 + 2ε .

This holds for each λ ∈ L(A)[1] and each ε > 0, and so kf1f2kBSE ≤ 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 b such that limν fν = f in (C (K), τp). For each λ ∈ L(A)[1], we have |hf, λi| = lim |hf , λi| ≤ m , ν ν 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 b (C (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): kfkBSE ≤ 1} b is closed in (C (ΦA), τp). 2

Theorem Let A be a natural BSA. Then CBSE(A) is isometrically isomorphic to the Banach al- 00 ⊥ 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 A00. 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 kfk = kfkBSE (f ∈ A). Then A has a BSE norm.

Proof Take f ∈ CBSE(A), with kfkBSE = m, say. Then there is a bounded net (fν) in A[m] b with limν fν = f in (C (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 kfk = kfkBSE, 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), k · kop). 2

40 Sample general theorems – 3

The ` 1-norm on L(A) is given by

n n X X α εx = |α | . i i i i=1 1 i=1

Theorem Let A be a BFA on K. Then b CBSE(A) = C (K) iff the usual norm on L(A) is equivalent to the ` 1-norm. 2

41 Ideals in biduals

Theorem Let (A, k · k) be a natural BFA on K such that A is an ideal in its bidual. Then A is an ideal in CBSE(A) and

| · |K ≤ k · kop ≤ k · kBSE ≤ k · k on A. 2

Theorem (KU) Let A be a BFA that is an ideal in 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 ideal in its bidual. Then A = CBSE(A) is AR, A has a BSE norm, and A00 = A ⊕ L(A)⊥. 2

Theorem (*) Let A be a BFA that is an ideal in its bidual, is AR, and has a BAI. Then A00 is a BFA and has BSE norm. 2 42 Easy examples of BSAs

Here all algebras have coordinatewise products.

00 ∞ Example 1 Look at c0. Here co = ` , so c0 is an ideal in its bidual and is AR; it is not a dual algebra. It has a BSE norm, and it is a BSE ∞ algebra because M(c 0) = ` = CBSE(c 0). 2

Example 2 Look at ` 1, a Tauberian BSA, so that ` 1 is an ideal in its bidual; it is a dual BSA 1 0 ∞ with predual c0. Here (` ) = ` = C(βN) and 1 00 1 1 (` ) = M(βN). Further, ` = CBSE(` ) is 1 ∗ AR, and M(βN) = ` nM(N ), with the product 1 ∗ (α, µ) 2 (β, ν) = (αβ, 0) (α, β ∈ ` , µ, ν ∈ M(N )) . 1 No BPAI, so not a BSE algebra; since L(` )[1] 1 0 1 is weak-∗ dense in (` )[1], the BSA ` has a BSE norm. 2

Example 3 Look at ` p, where 1 < p < ∞. This is a Tauberian BSA and is a reflexive Banach space, and so ` p is an ideal in its bidual and a dual algebra. It has a BSE norm, but it is not a BSE algebra. 2 43 General results

BSE algebras and BSE norms were introduced in 1990 by Takahasi and Hatori (TH) as an abstraction of a classical theorem of harmonic analysis, the Bochner–Schoenberg–Eberlein theorem; see later.

Quite a few papers have discussed specific ex- amples. Our work seeks to give an underlying general theory, and applications to more exam- ples.

General questions Does every dual BFA have a BSE norm? Does every (even Tauberian) BSA have a BSE norm?

In both cases, we can give positive answers with the help of modest extra hypotheses; we have no counter-examples.

We can resolve these questions for many, but not all, specific examples that we have looked at – 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 | · |K and k · kBSE 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 might suspect. Does each natural BSA on N have a BSE norm? We have a general theorem that at least covers the following example.

Example For α = (αk) ∈ C N, set 1 n X pn(α) = k αk+1 − αk , n k=1

p(α) = sup {pn(α): n ∈ N} . Define A to be {α ∈ c 0 : p(α) < ∞}, so that A is a self-adjoint BSA on N for the norm kαk = |α| + p(α)(α ∈ A) . N 2 2 Then A is a natural; A = A0 = A0 ( A; A is not Tauberian; A is non-separable; A is not an ideal in its bidual. The algebra A is not Arens regular.

This example does have a BSE norm, and it is a BSE algebra. 2 47 Tensor products of BSAs – 1

Guess Suppose that A and B are BFAs that are BSE algebras/have BSE norms. Then A ⊗b B has the corresponding property.

Let A and B be natural BSAs on S and T and suppose that A has AP as a Banach space. Then A ⊗b B is a natural BSA on S × T .

Example 1 Take p and q with 1 < p, q < ∞, p q 0 p q0 and set A = ` ⊗b ` , so that A = B(` , ` ).

Then A is a Tauberian BSA on N × N, and so 0 an 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 for p = q = 2).

Here A = CBSE(A), so A has a BSE norm, but A is not a BSE algebra. 2 48 Tensor products of BSAs – 2

Example 2 Let A = c 0 ⊗b c 0, 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 00 algebra. Here M(A) = CBSE(A) = A .

By Theorem (*), A00 has a BSE norm.

00 ∞ Of course c 0 = ` = C(βN); by an earlier result, C(βN) ⊗b C(βN) is a closed subalgebra of A00, and so also has a BSE norm.

00 We do not know if either C(βN) ⊗b C(βN) or A is a BSE algebra.

Neufang has shown that A00 is not AR - see his lecture. What about C(βN) ⊗b C(βN)?

49 Varopoulos algebra

Let K and L be compact spaces, and set

V (K,L) = C(K) ⊗b 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 0 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 |hg, (T − S)fi| < 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 B always has a BSE norm. Clearly this would follow immediately from the above if we knew that A ⊗b 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 ⊗b A is a closed subalgebra of V (D, D), and so A ⊗b A has a BSE norm. 2

Query What happens for different uniform al- gebras? Is A(D) ⊗b 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 1 1 M(L (G)) = CBSE(L (G)) = M(G) .

(ii) kµk = kµkBSE (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) . 1 Then ` (Z, ω) is the space of functions P f = f(n)δn such that X kfkω = |f(n)| ω(n) < ∞ . This is a commutative Banach algebra for con- 1 volution. Via the Fourier transform, ` (Z, ω) is a BFA on the circle or an annulus in C.

1 The algebra ` (Z, ω) is a dual BFA, with pre- dual c0(Z, 1/ω); it is not an ideal in its bidual. 1 Examples show that ` (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 Fig`a-Talamanca–Herz algebras

Let Γ be a locally compact group.and take p with 1 < p < ∞. The Fig`a-Talamanca–Herz (FTH) algebra is Ap(Γ). Formally, Ap(Γ) is the collection of sums ∞ X f = gn ? ˇhn n=1 p p0 where gn ∈ L (Γ) and hn ∈ L (Γ) for each P∞ n ∈ N and n=1 kgnkp khnkp0 < ∞, and kfk 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

kfk = kfkBSE (f ∈ B(Γ)) , and so A(Γ) and B(Γ) have a BSE norm.

Proof Since B(Γ) = C∗(Γ)0, 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) ⊗b A(Γ2) = A(Γ1 × Γ2) . (∗)

Then A(Γ1) ⊗b 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) ⊗b 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 of Fourier algebras

Let Γ be an infinite, amenable locally compact group, and let M be a maximal modular ideal of 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 such a BAI. We now consider pointwise versions of this.

Theorem Let Γ be an infinite locally compact group such that Γd is amenable. Then the minimum bound of a BPAI in M is also 2. In particular, A(Γ) is not pointwise contractive.2

Query What happens if Γ is amenable, but Γd is not? (Eg., Γ = SO(3).) The minimum bound is > 1.

Query What happens for Ap(Γ) when p > 1 and 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 was M(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, k · kA) be a natural Banach function algebra on a locally compact space K. A Banach function algebra (B, k · kB) is an abstract Segal algebra (with respect to A) if B is an ideal in A and there is a net in B that is an approximate identity for both (A, k · kA) and (B, k · kB). Classical Segal algebras are abstract Segal algebras 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.

1 The norm is equivalent to k · k1 iff S = L (G).

Always M(G) ⊂ M(S) (but not necessarily equal).

Theorem A Segal algebra S is a BSE algebra iff 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

fb ≤ kfk = kfk = kfk ≤ kfk , Γ BSE,S op,S 1 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

1 p Sp(G) = {f ∈ L (G): fb ∈ L (Γ)} , and set   kfk = max kfk , fb (f ∈ Sp(G)) . Sp 1 p Then ( ( ) ) is a Segal algebra with Sp G ,?, k · kSp respect to L1(G) and a natural, Tauberian BFA 2 on Γ. Since Sp(G) ( 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, but we 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 f by t is defined by

(Stf)(s) = f(s − t)(s ∈ [−1, 1]) . Define Z 1 Ωf (t) = kf − Stfk1 = |f(s) − f(s − t)| ds −1 and Z 1 Ωf (t) I(f) = α dt . −1 |t| Then A = {f ∈ C(T): I(f) < ∞} and kfk = |f| + I(f)(f ∈ A) . T

We see that (A, k · k) is a natural, unital BFA on 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). α−1 But kenk ∼ n , and so (A, k · k) 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) . 2−α Then we can see that I(∆n) ∼ 1/n , and so 2−α k1 − ∆nk ≤ 1+O(1/n ) = 1+o(1). Further, a calculation shows that (1 − ∆n : n ∈ N) is an approximate identity for M.

We conclude that ((1 − ∆n)/ k1 − ∆nk : 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