Banach function algebras and BSE norms. H. G. Dales, Lancaster

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1 Banach function algebras and BSE norms H. G. Dales, Lancaster Joint work with Ali Ülger, Istanbul Graduate course during 23 rd Banach algebra conference, Oulu, Finland July

2 Some references H. G. Dales and A. Ülger, Approximate identities in Banach function algebras, Studia Mathematica, 226 (2015), H. G. Dales and A. Ülger, Banach function algebras and BSE norms, in preparation. E. Kaniuth and A. Ülger, The Bochner Schoenberg Eberlein property for commutative Banach algebras, especially Fourier and Fourier Stieltjes algebras, Trans. American Math. Soc., 362 (2010), S.-E. Takahasi and O. Hatori, Commutative Banach algebras that satisfy a Bochner Schoenberg Eberlein-type theorem, Proc. American Math. Soc., 37 (1992),

3 Banach spaces Let E be a normed space. The closed unit ball is E [1] = {x E : x 1}. The dual space of E is E. This is the space of continuous = bounded linear functionals on E, 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 regard E as a closed subspace of E ; the canonical embedding is κ E : E E, where κ E (x), λ = x, λ (x E, λ E ). For a closed subspace F of E, the annihilator of F is F = {λ E : λ F = 0}. 3

4 Algebras All algebras are linear and associative and taken over the complex field, C. The identity of a unital algebra A is e A ; 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 A 2 = lin A [2]. An ideal in A is a linear subspace I such that AI I and IA I. 4

5 The radical We set A = A \ {0}; an element a of A is quasi-nilpotent if ze A 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 denoted 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

6 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 0. The character space of an algebra, the collection of characters on A, is denoted by Φ A. The centre of A is Z(A) = {a A : ab = ba (b A)}; A is commutative if Z(A) = A. 6

7 Banach algebras An algebra A with a norm is a Banach algebra (BA) if (A, ) is a Banach space and ab a b (a, b A). When A is unital, we also require that e A = 1. Standard non-commutative example: A = B(E), the algebra of all bounded linear operators on a Banach space E, with operator norm op. Here for S, T A. (ST )(x) = S(T x) (x E) Each character ϕ on a BA is continuous, with ϕ 1, and so Φ A A [1] ; Φ A is a locally compact subspace of (A, σ(a, 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 n a n 1/n = 0}. 7

8 Continuous functions Let K be a locally compact space. Then C b (K) is the algebra of all bounded, continuous functions on K, with the pointwise operations; C 0 (K) consists of the continuous functions that vanish at infinity; C 00 (K) consists of the continuous functions with compact support. We define f K = sup { f(x) : x K} (f C b (K)), so that K is the uniform norm on K and (C b (K), K ) is a commutative, semisimple Banach algebra; C 0 (K) is a closed ideal in C b (K); C 00 (K) is an ideal in C b (K). The topology of pointwise convergence on C b (K) is called τ p. 8

9 Function algebras A function algebra on K is a subalgebra A of C b (K) that separates strongly the points of K, in the sense that, for each x, y K with x y, there exists f A with f(x) f(y), and, for each x K, there exists f A with f(x) 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 ε x : f f(x) for some x K; equivalently, all maximal modular ideals are of the form M x = ker ε x = {f A : f(x) = 0}. 9

10 Gel fand theory Let A be a BA. Define â(ϕ) = ϕ(a) (ϕ Φ A ). Then â C 0 (Φ A ), and the Gel fand transform G : a â, (A, ) (C 0 (Φ A ), ΦA ), is a continuous linear operator that is an algebra 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 isometrically and algebraically -isomorphic to C 0 (Φ A ). 10

11 More on BFAs Henceforth K will be a non-empty, locally compact (Hausdorff) space, and usually A will be a natural BFA on K. The closure of A C 00 (K) in A is called A 0. The BFA A is Tauberian if A = A 0. The ideal J x in A consists of the functions in A C 00 (K) that are 0 on a neighbourhood of x, so that J x M x ; A is strongly regular if J x is dense in M x for each x X. 11

12 Locally compact groups Let G be a locally compact group with left Haar measure m G. Then the group algebra is (L 1 (G),, 1 ) and the measure algebra is (M(G),, ), so that L 1 (G) is a closed ideal in M(G). Both are semi-simple Banach algebras. As a Banach space, M(G) = C 0 (G), and the product µ ν of µ, ν M(G) is given by: f, µ ν = G G f(st) dµ(s) dν(t) (f C 0(G)). The product of f, g L 1 (G) is given by (f g)(t) = G f(s)g(s 1 t) dm G (s) (t G). There is always one character on L 1 (G), namely f G f dm G; its kernel is the augmentation ideal L 1 0 (G). 12

13 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 as a Banach space. A = A F Key example: M(G) is a dual Banach algebra, with predual C 0 (G). 13

14 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 = Ĝ of all continuous characters on G is an abelian group with respect to pointwise multiplication given by: (γ 1 + γ 2 )(s) = γ 1 (s)γ 2 (s) (s G, γ 1, γ 2 Γ). The topology on Γ is that of uniform covergence 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, Ẑ = T, T = Z, and R = R. 14

15 Pontryagin duality theorem For each s G, the map γ 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 original topology on Γ, so that Γ = G. Hence Ĝ = G. 15

16 Fourier transform Let G be a LCA group. The Fourier transform of f L 1 (G) is f = Ff, so that (Ff)(γ) = f(γ) = and G f(s) s, γ dm G(s) (γ Γ), A(Γ) = { f : f L 1 (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) = and G s, γ dµ(s) (γ Γ), B(Γ) = { µ : µ M(G)}, is a Banach function algebra on Γ. Of course, F : (M(G), ) (B(Γ), ) is a linear contraction that is an algebra isomorphism. 16

17 The group C -algebra Here Γ is a locally compact group. Let π be a representation of (L 1 (Γ), ), so that π : L 1 (Γ) B(H π ) is a contractive -homomorphism for some Hilbert space H π. For f L 1 (Γ), define f = sup { π(f) : π is a representation of L 1 (Γ)} so that f f 1. Then is a norm on L 1 (Γ) such that f f = f 2 (f L 1 (Γ)), and the completion of (L 1 (Γ), ) is a C - algebra, called C (Γ), the group C -algebra of Γ. 17

18 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 L 2 (Γ)}. Let Γ be a locally compact group. A function f : Γ C is positive-definite if it is continuous and if, for each n N, t 1,..., t n G, and α 1,..., α n C, we have n i,j=1 α i α j f(t 1 i t j ) 0. 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

19 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, selfadjoint 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). Usually, A(Γ) B(Γ). 19

20 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 algebra of Γ; A(Γ) is an ideal in its bidual iff Γ is discrete. 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]. 20

21 Banach sequence algebras Let S be a non-empty set, usually N. We write c 0 (S) and l (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 l (S). Thus A is Tauberian iff c 00 (S) is dense in A. For example, l p = l p (N) and A(Z) with pointwise product are Tauberian BSAs. 21

22 Biduals of Banach algebras Let A be a Banach algebra. Then there are two products and on A, the first and second Arens products, that extend the given product on A. Roughly: Take M, N A, say M = lim α a α and N = lim β b β, where (a α ) and (b β ) are nets in A (weak- limits). Then M N = lim α lim β a α b β, M N = lim β lim α a α b β. The basic theorem of Arens is that κ A : A A is an isometric algebra monomorphism of A into both (A, ) and (A, ). We shall usually write just A for (A, ). 22

23 Arens regularity A Banach algebra A is Arens regular = AR if and coincide on A. A commutative Banach algebra is AR iff (A, ) is commutative. Fact A is a dual Banach algebra iff A is AR. Let A be a C -algebra. Then A is AR and (A, ) is also a C -algebra, called the enveloping von Neumann algebra. In particular, (C 0 (K), ) is a commutative C - algebra, and so has the form C( K) for a compact space K, called the hyper-stonean envelope of K. For K = N, we have K = βn, the Stone Čech compactification of N. Advertisement: this is discussed at length with several constructions and characterizations of K in H. G. Dales, F. K. Dashiell, Jr., A. T.-M. Lau, and D. Strauss, Banach spaces of continuous functions as dual spaces, Springer,

24 Strong Arens irregularity Let A be a Banach algebra. Then the left and right topological centres are Z (l) t (A )= { M A : M N = M N (N A ) } and Z (r) t (A )= { M A : N M = N M (N A ) }, respectively. Thus the algebra A is Arens regular if and only if Z (l) t (A ) = Z t (r) (A ) = A ; A is strongly Arens irregular = SAI if Z (l) t (A ) = Z (r) t (A ) = A. In the case where A is commutative, Z (l) t (A ) = Z (r) t (A ) = Z(A ). Example Each group algebra L 1 (G) is SAI (Lau and Losert). Indeed, each measure algebra M(G) is SAI (Neufang et al). 24

25 Ideals in biduals Let A be an algebra. For a A, we define L a and R a by L a (b) = ab, R a (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 happens iff each L a and R a is a weakly compact operator. Fact Let A be a Tauberian BSA. Then L f is compact for each f A, and so A is an ideal in its bidual. 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

26 Tensor products Let E and F be Banach spaces. Then (E F, π ) is their projective tensor product. Each element z of E F can be expressed in the form z = i=1 x i y i, where x i E, y i F and i=1 x i y i <, 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 T S : E F G such that T S (x y) = S(x, y) (x E, y F ) and such that T S = S. 26

27 Duals of tensor products We have (E F ) = B(E, F ), where the isometric isomorphism T : λ T λ, (E F ) B(E, F ), satisfies the condition that y, T λ x = 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 Sánchez and Garcia: Theorem Suppose that E has the bounded approximation property (BAP). Then the natural embedding of E F into (E F ) extends to an isomorphic embedding of E F onto a closed subspace of (E F ). 27

28 Tensor products of BFAs Let A and B be algebras, and set A = A B. Then there is a unique product on A with respect to which A is an algebra and such that (a 1 b 1 )(a 2 b 2 ) = a 1 a 2 b 1 b 2 for a 1, a 2 A and b 1, b 2 B. Fact Let A and B be natural BFAs on K and L, respectively, and suppose that A has the approximation 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). General question Suppose that A and B are BFAs that are AR. Is A B AR? A criterion involving biregularity and many examples (both ways) were given by Ali Ülger, TAMS, See later. 28

29 Uniform algebras A BFA A is a uniform algebra if it is closed in (C b (K), K ), and so the norm is equivalent to the uniform norm. For example, C 0 (K) is a natural uniform algebra on K. A natural uniform algebra A on K is trivial if A = C 0 (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 Φ A, say x y if ε x ε y < 2. This is an equivalence relation that divides Φ A into equivalence classes, called Gleason parts. A strong boundary point is a singleton part, but not conversely. 29

30 Approximate identities Let A be a CBA. A net (e α ) in A is an approximate identity for A if lim α ae α = a (a A) ; an approximate identity (e α ) is bounded if sup α e α <, and then sup α e α 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 M x has a CAI for EACH x K. Obvious example Take A = C 0 (K). Then A is contractive. Are there any more contractive BFAs? See later. Group algebras have a CAI, but the augmentation ideal L 1 0 (G) has a BAI (of bound 2 - see later), not a CAI, and so L 1 (G) is not contractive. 30

31 Pointwise approximate identities We shall consider (natural) BFAs on a locally compact space K. Let A be a natural BFA on K. A net (e α ) in A is a pointwise approximate identity (PAI) if lim α e α (x) = 1 (x K) ; the PAI is bounded, with bound m > 0, if sup α e α m, and then (e α ) 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 M x has a CPAI for each x K. Also clearly a contractive BFA is pointwise contractive. But we shall give examples to show that the converse is not true. 31

32 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) ε x exk A, where K A = {λ A : λ = 1 K, λ = 1} ; (b) x is a strong boundary point; (c) M x has a BAI; (d) M x 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 M x gives an identity in M x, hence an idempotent in C( K). The latter have norm 1. So there is a CAI in M x. 32

33 Cole algebras Definition Let A be a natural uniform algebra on a compact space K. Then A is a Cole algebra 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. There are non-trivial Cole algebras (but they took some time to find). One is R(X) for a certain compact set X in C 2. Theorem A natural uniform algebra on X is pointwise contractive if and only if each set {x} is a singleton Gleason part. Standard examples now give separable uniform algebra that are pointwise contractive, but not contractive. 33

34 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 {ε x : x K} as a subset of A, and f BSE = sup { f, λ : λ L(A) [1] } (f A). Clearly K L(A) [1] A, and so [1] f K f BSE f (f A). In fact, BSE 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 f C f BSE (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

35 BSE algebras Let A be a natural BFA on locally compact K. Then M(A) = {f C b (K) : fa A}, the multiplier algebra of A. It is a unital BFA on K with respect to the operator norm op. For example, the multiplier algebra of L 1 (G) is M(G) (Wendel). This applies to all G: each two-sided multiplier on (L 1 (G), ) has the form f f µ for some µ M(G). Let A be a natural Banach function algebra on K. Then f BSE = sup { f, λ : λ L(A) [1] } (f C b (K)), and C BSE (A) = {f C b (K) : f BSE < }. The algebra A is a BSE algebra whenever M(A) = C BSE (A). (It does not necessarily have a BSE norm.) For unital algebras, the condition is that A = C BSE (A). 35

36 Basic theorem on C BSE (A) The following is in TH in Theorem Let A be a natural Banach function algebra on K. Then (C BSE (A), BSE ) is a Banach function algebra on K. Further, C BSE (A) is the set of functions f C b (K) for which there is a bounded net (f ν ) in A with lim ν f ν = f in (C b (K), τ p ); for f C BSE (A), the infimum of the bounds of such nets is equal to f BSE. Proof Certainly C BSE (A) is a linear subspace of C b (K), and BSE is a norm on C BSE (A). It is a little exercise to check that (C BSE (A), BSE ) is a Banach space. Now take f 1, f 2 C BSE (A). We show that f 1 f 2 BSE f 1 BSE f 2 BSE ; we shall suppose that f 1 BSE, f 2 BSE = 1. 36

37 Proof continued Take λ = n i=1 α i ε xi L(A) [1], and fix ε > 0. First, set µ 1 = n i=1 α i f 1 (x i )ε xi, so that µ 1 A. Then there exists g 1 A [1] with g 1, µ 1 > µ 1 ε. Next, set µ 2 = n i=1 α i g 1 (x i )ε xi, so that µ 2 A. Then there exists g 2 A [1] with g 2, µ 2 > µ 2 ε. We see that f 1, µ 2 = g 1, µ 1 and g 2, µ 2 = g 1 g 2, λ, and hence that g 2, µ 2 g 1 g 2 λ 1. We now have f 1 f 2, λ = n i=1 α i f 1 (x i )f 2 (x i ) = f 2, µ 1 µ 1 < g 1, µ 1 + ε = f 1, µ 2 + ε µ 2 + ε < g 2, µ 2 + 2ε 1 + 2ε. This holds for each λ L(A) [1] and each ε > 0, and so f 1 f 2 BSE 1, as required. 37

38 Proof concluded Take f C b (K) to be such that there is a bounded net (f ν ) in A [m] for some m > 0 such that lim ν f ν = f in (C b (K), τ p ). For each λ L(A) [1], we have f, λ = lim ν f ν, λ m, and hence f C BSE (A) [m]. Conversely, suppose that f C BSE (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 f F,ε A such that f F,ε (x) = f(x) (x F ) and f F,ε m + ε. Then the net (f F,ε ) converges to f in (C b (K), τ p ). 38

39 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 (Φ A ), τ p ). Theorem Let A be a natural BSA. Then C BSE (A) is isometrically isomorphic to the Banach algebra A /L(A), and C BSE (A) is a dual BFA, with predual L(A). Theorem Let A be a natural BFA. Then A has a BSE norm iff the subalgebra A + L(A) is closed in A. 39

40 Sample general theorems 2 Theorem Let A be a dual BFA with predual F. Suppose that the space Φ A F [1] is dense in Φ A. Then A = C BSE (A) and f = f BSE (f A). Then A has a BSE norm. Proof Take f C BSE (A), with f BSE = m, say. Then there is a bounded net (f ν ) in A [m] with lim ν f ν = f in (C b (K), τ p ). Let g be an accumulation point of this net in (A, σ(a, F )). Then g(ϕ) = f(ϕ) (ϕ Φ A F [1] ), and so g = f. Thus f A [m] with f = f BSE, showing that A = C BSE (A). Corollary Let G be a compact group. M(G) has a BSE norm. Then Theorem A BSE algebra has a BSE norm iff it is closed in (M(A), op ). 40

41 Sample general theorems 3 The l 1 -norm on L(A) is given by n i=1 α i ε xi = 1 n i=1 α i. Theorem Let A be a BFA on K. Then C BSE (A) = C b (K) iff the usual norm on L(A) is equivalent to the l 1 -norm. 41

42 Ideals in biduals Theorem Let (A, ) be a natural BFA on K such that A is an ideal in its bidual. Then A is an ideal in C BSE (A) and on A. K op BSE 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. Theorem Let A be a dual BFA that is an ideal in its bidual. Then A = C BSE (A) is AR, A has a BSE norm, and A = A L(A). Theorem (*) Let A be a BFA that is an ideal in its bidual, is AR, and has a BAI. Then A is a BFA and has BSE norm. 42

43 Easy examples of BSAs Here all algebras have coordinatewise products. Example 1 Look at c 0. Here c o = l, so c 0 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 ) = l = C BSE (c 0 ). Example 2 Look at l 1, a Tauberian BSA, so that l 1 is an ideal in its bidual; it is a dual BSA with predual c 0. Here (l 1 ) = l = C(βN) and (l 1 ) = M(βN). Further, l 1 = C BSE (l 1 ) is AR, and M(βN) = l 1 M(N ), with the product (α, µ) (β, ν) = (αβ, 0) (α, β l 1, µ, ν M(N )). No BPAI, so not a BSE algebra; since L(l 1 ) [1] is weak- dense in (l 1 ) [1], the BSA l 1 has a BSE norm. Example 3 Look at l p, where 1 < p <. This is a Tauberian BSA and is a reflexive Banach space, and so l p is an ideal in its bidual and a dual algebra. It has a BSE norm, but it is not a BSE algebra. 43

44 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 examples. Our work seeks to give an underlying general theory, and applications to more examples. 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

45 Contractive results Theorem A contractive BFA with a BSE norm is a Cole algebra. Theorem Let A be a pointwise contractive BFA with a BSE norm. Then the norms K and BSE on A are equivalent, and A is a uniform algebra for which each singleton in Φ A is a one-point Gleason part. Further, A is a BSE algebra if and only if A = C(K). 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

46 Queries for uniform algebras Caution It is not true that every natural uniform algebra on a compact K is a BSE algebra - 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 C BSE (A) for a uniform algebra A? How do we characterize the uniform algebras that are BSE algebras? Query For example, what is C BSE (R(K)) for compact K C? Look at a Swiss cheese K. 46

47 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 = (α k ) C N, set p n (α) = 1 n n k=1 k α k+1 α k, p(α) = sup {p n (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 α = α N + p(α) (α A). Then A is a natural; A 2 = A 2 0 = A 0 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. 47

48 Tensor products of BSAs 1 Guess Suppose that A and B are BFAs that are BSE algebras/have BSE norms. Then A 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 is a natural BSA on S T. Example 1 Take p and q with 1 < p, q <, and set A = l p l q, so that A = B(l p, l q ). Then A is a Tauberian BSA on N N, and so an ideal in its bidual; it is the dual of K(l p, l q ); it is AR. It is reflexive iff pq > p + q (Pitt) (this fails for p = q = 2). Here A = C BSE (A), so A has a BSE norm, but A is not a BSE algebra. 48

49 Tensor products of BSAs 2 Example 2 Let A = c 0 c 0, so A is a Tauberian 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) = C BSE (A) = A. By Theorem (*), A has a BSE norm. Of course c 0 = l = 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

50 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, f 1,..., f n C(K), and g 1,..., g n C(L), we must find S L(V ) [1] such that g, (T S)f < ε whenever f {f 1,..., f n } and g {g 1,..., g n }. 50

51 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. Question Is V = V (K, L) a BSE algebra? For this, we would have to show that C BSE (V ) = V. At least we know that C BSE (V ) C(K L), using a result in the book of Helemskii. 51

52 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 algebra A on a compact space K is never complemented in C(K). The result is true in the special case where A and B are the disc algebra, as shown by Bourgain Theorem Let A := A(D) to be the disc algebra. Then A A is a closed subalgebra of V (D, D), and so A A has a BSE norm. Query What happens for different uniform algebras? Is A(D) A(D) a BSE algebra? 52

53 Group and measure algebras Let G be an infinite, LCA group with dual Γ. Theorem (i) L 1 (G) is a BSE algebra, and M(L 1 (G)) = C BSE (L 1 (G)) = M(G). (ii) µ = µ BSE (µ M(G)), and so M(G) and L 1 (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 C BSE (M(G)) that are not in M(G) when G is not discrete. 53

54 Compact abelian groups Take G to be a compact, abelian group. For 1 p, (L p (G), ) is a semi-simple CBA. For 1 < p <, F(L p (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 bidual. Theorem For 1 < p, F(L p (G)) has a BSE norm, but it is not a BSE algebra. 54

55 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 l 1 (Z, ω) is the space of functions f = f(n)δ n such that f ω = f(n) ω(n) <. This is a commutative Banach algebra for convolution. Via the Fourier transform, l 1 (Z, ω) is a BFA on the circle or an annulus in C. The algebra l 1 (Z, ω) is a dual BFA, with predual c 0 (Z, 1/ω); it is not an ideal in its bidual. Examples show that l 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

56 Beurling algebras as BSE algebras Theorem Beurling algebras A ω are BSE algebras with a BSE norm for most, may be all, weights. Proof This works when Φ Aω c 0 (Z, 1/ω) [1] is dense in Φ Aω. Trouble for weights ω with lim sup n ω(n) = and lim inf n ω(n) = 1; they exist. 56

57 Figà-Talamanca Herz algebras Let Γ be a locally compact group.and take p with 1 < p <. The Figà-Talamanca Herz (FTH) algebra is A p (Γ). Formally, A p (Γ) is the collection of sums f = n=1 g n ȟ n where g n L p (Γ) and h n L p (Γ) for each n N and n=1 g n p h n p <, and f is the infimum of such sums. Thus A p (Γ) is a self-adjoint, Tauberian, natural, strongly regular Banach function algebra on Γ. [See papers of Herz, a book and lectures of Derighetti.] 57

58 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) A p (Γ) has a BAI; (c) A p (Γ) has a BPAI; (d) A p (Γ) has a CAI. 58

59 Arens regularity of Fourier algebras Theorem (Lau Wong) Let Γ be a LC group, and suppose that A(Γ) is AR. Then Γ is discrete, and every amenable subgroup is finite. May be Γ must be finite. Suppose that Γ is discrete. If Γ is amenable, then A(Γ) is SAI (Lau Losert, 1988), but not if Γ contains F 2 (Losert, 2016). For the case where Γ is not discrete, and especially when Γ is compact, see the lectures of Jorge Galindo in Oulu. 59

60 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 Kaplansky s density theorem for C -algebras. Theorem (KU) A(Γ) is a BSE algebra iff Γ is amenable. 60

61 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

62 Tensor products of Fourier algebras Let Γ 1 and Γ 2 be locally compact groups. Suppose that A(Γ 1 ) A(Γ 2 ) = A(Γ 1 Γ 2 ). ( ) Then A(Γ 1 ) A(Γ 2 ) has a BSE norm, and it is a BSE algebra if and only if both Γ 1 and Γ 2 are amenable. But (*) is not always true (Losert). Guess A(Γ 1 ) A(Γ 2 ) always has a BSE norm, and is a BSE algebra if and only if both Γ 1 and Γ 2 are amenable. 62

63 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 Γ d is amenable. Then the minimum bound of a BPAI in M is also 2. In particular, A(Γ) is not pointwise contractive. Query What happens if Γ is amenable, but Γ d is not? (Eg., Γ = SO(3).) The minimum bound is > 1. Query What happens for A p (Γ) when p > 1 and p 2? 63

64 FTH algebras A p (Γ) Here Γ is a LC group and 1 < p <. Theorem (Forrest) A p (Γ) is an ideal in its bidual iff Γ is discrete. Theorem (Forrest) Suppose that A p (Γ) is AR. Then Γ is discrete and every abelian subgroup is finite. May be Γ must be finite. Apparently nothing is known of when A p (Γ) is SAI. There are varying definitions of B p (Γ). The first was by Herz. Cowling said it was M(A p (Γ)); Runde gave a definition involving representation theory; we prefer Runde s definition because it gives the previous B p (Γ) when p = 2. The definitions all agree when Γ is amenable. 64

65 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 A p (Γ) is a BSE algebra if and only if Γ is amenable. In this case, B p (Γ) = C BSE (A p (Γ)) = M(A p (Γ)), and A p (Γ) and B p (Γ) have BSE norms. Proof Uses interplay with B p (Γ d ) and results of Herz and of Derighetti. Query Does A p (Γ) have a BSE norm for each Γ? This is true for p = 2. 65

66 Segal algebras Definition Let (A, A ) be a natural Banach function algebra on a locally compact space K. A Banach function algebra (B, B ) 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, A ) and (B, B ). Classical Segal algebras are abstract Segal algebras with respect to L 1 (G). Let S be a Segal algebra with respect to L 1 (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 = L 1 (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) = C BSE (S). 66

67 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 identify the BSE norm. Indeed, for f S, we have f f Γ BSE,S = f op,s = f 1 f S, and so S has a BSE norm iff S = L 1 (G). 67

68 An example of a Segal algebra Example Let G be a non-discrete LCA group with dual group Γ. Take p 1, define S p (G) = {f L 1 (G) : f L p (Γ)}, and set f Sp = max { f 1, f } p (f S p (G)). Then (S p (G),, Sp ) is a Segal algebra with respect to L 1 (G) and a natural, Tauberian BFA on Γ. Since S p (G) 2 S p (G), S p (G) does not have a BAI. However, by a result of Inoue and Takahari, S p (R) has a CPAI. Thus S p (R) is a BSE algebra without a BSE norm. 68

69 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 Define (S t f)(s) = f(s t) (s [ 1, 1]). Ω f (t) = f S t f 1 = and I(f) = f(s) f(s t) ds Ω f (t) t α dt. Then A = {f C(T) : I(f) < } and f = f T + I(f) (f A). We see that (A, ) is a natural, unital BFA on T; it is homogeneous. 69

70 Final example continued Let e n be the trigonometric polynomial given by e n (s) = exp(iπns) (s [ 1, 1]). Then e n A, and so A is uniformly dense in C(T). But e n n α 1, and so (A, ) is not equivalent 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/n 2 α, and so 1 n 1+O(1/n 2 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. 70

71 Conclusions We have a contractive BFA not equivalent to a uniform algebra. Here the BSE norm is equal to the uniform norm, and C BSE (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