Search

For a Banach function algebra A, we consider the problem of representing a continuous d-homogeneous polynomial P : A → X , where X is an arbitrary Banach space, that satisfies the property P ( f + g ) = P ( f ) + P ( g ) whenever f , g ∈ A are such that supp ( f ) ∩ supp ( g ) = ∅ . We show that such a polynomial can be represented as P ( f ) = T ( f d ) ( f ∈ A ) for some continuous linear map T : A → X for a variety of Banach function algebras such as the algebra of continuous functions C 0 ( Ω ) for any locally compact Hausdorff space Ω, the algebra of Lipschitz functions lip α ( K ) for any compact metric space K and α ∈ ] 0 , 1 [ , the Figa–Talamanca–Herz algebra A p ( G ) for some locally compact groups G and p ∈ ] 1 , + ∞ [ , the algebras A C ( [ a , b ] ) and B V C ( [ a , b ] ) of absolutely continuous functions and of continuous functions of bounded variation on the interval [ a , b ] . In the case where A = C n ( [ a , b ] ) , P can be represented as P ( f ) = ∑ T ( n 1 , … , n d ) ( f ( n 1 ) ⋯ f ( n d ) ) , where the sum is taken over ( n 1 , … , n d ) ∈ Z d with 0 ≤ n 1 ≤ … ≤ n d ≤ n , for appropriate continuous linear maps T ( n 1 , … , n d ) : C n − n d ( [ a , b ] ) → X .

Let 𝑋 and 𝐾 be compact plane sets with 𝐾 ⊆ 𝑋 . We define 𝐴 ( 𝑋 , 𝐾 ) = { 𝑓 ∈ 𝐶 ( 𝑋 ) ∶ 𝑓 | 𝐾 ∈ 𝐴 ( 𝐾 ) } , where 𝐴 ( 𝐾 ) = { 𝑔 ∈ 𝐶 ( 𝑋 ) ∶ 𝑔 is analytic on i n t ( 𝐾 ) } . For 𝛼 ∈ ( 0 , 1 ] , we define L i p ( 𝑋 , 𝐾 , 𝛼 ) = { 𝑓 ∈ 𝐶 ( 𝑋 ) ∶ 𝑝 𝛼 , 𝐾 ( 𝑓 ) = s u p { | 𝑓 ( 𝑧 ) − 𝑓 ( 𝑤 ) | / | 𝑧 − 𝑤 | 𝛼 ∶ 𝑧 , 𝑤 ∈ 𝐾 , 𝑧 ≠ 𝑤 } < ∞ } and L i p 𝐴 ( 𝑋 , 𝐾 , 𝛼 ) = 𝐴 ( 𝑋 , 𝐾 ) ∩ L i p ( 𝑋 , 𝐾 , 𝛼 ) . It is known that L i p 𝐴 ( 𝑋 , 𝐾 , 𝛼 ) is a natural Banach function algebra on 𝑋 under the norm | | 𝑓 | | L i p ( 𝑋 , 𝐾 , 𝛼 ) = | | 𝑓 | | 𝑋 + 𝑝 𝛼 , 𝐾 ( 𝑓 ) , where | | 𝑓 | | 𝑋 = s u p { | 𝑓 ( 𝑥 ) | ∶ 𝑥 ∈ 𝑋 } . These algebras are called extended analytic Lipschitz algebras. In this paper we study unital homomorphisms from natural Banach function subalgebras of L i p 𝐴 ( 𝑋 1 , 𝐾 1 , 𝛼 1 ) to natural Banach function subalgebras of L i p 𝐴 ( 𝑋 2 , 𝐾 2 , 𝛼 2 ) and investigate necessary and sufficient conditions for which these homomorphisms are compact. We also determine the spectrum of unital compact endomorphisms of L i p 𝐴 ( 𝑋 , 𝐾 , 𝛼 ) .

In 2003 Dawson and Feinstein asked whether or not a Banach function algebra with dense invertible group can have a proper Shilov boundary. We give an example of a uniform algebra showing that this can happen, and investigate the properties of such algebras. We make some remarks on the topological stable rank of commutative, unital Banach algebras. In particular, we prove that $ \mathrm{tsr}(A) \geq \mathrm{tsr}(C(\Phi_A))$ whenever $ A$ is approximately regular.

We exhibit a singly generated, semisimple commutative operator algebra with a contractive approximate identity, such that the spectrum of the generator is a null sequence and zero, but the algebra is not the closed linear span of the idempotents associated with the null sequence and obtained from the analytic functional calculus. Moreover the multiplication on the algebra is neither compact nor weakly compact. Thus we construct a `large' operator algebra of orthogonal idempotents, which may be viewed as a dense subalgebra of c0., Comment: 28 pages, to appear Transactions American Math. Society

We investigate the completeness and completions of the normed algebras (D(1)(X),∥•∥) for perfect, compact plane sets X. In particular, we construct a radially self-absorbing, compact plane set X such that the normed algebra (D(1)(X),∥•∥) is not complete. This solves a question of Bland and Feinstein. We also prove that there are several classes of connected, compact plane sets X for which the completeness of (D(1)(X),∥•∥) is equivalent to the pointwise regularity of X. For example, this is true for all rectifiably connected, polynomially convex, compact plane sets with empty interior, for all star-shaped, compact plane sets, and for all Jordan arcs in ℂ. In an earlier paper of Bland and Feinstein, the notion of an F-derivative of a function was introduced, where F is a suitable set of rectifiable paths, and with it a new family of Banach algebras D ((1))/F corresponding to the normed algebras (D(1)(X),∥•∥). In the present paper, we obtain stronger results concerning the questions when (D(1)(X),∥•∥) and D ((1))/F (X) are equal, and when the former is dense in the latter. In particular, we show that equality holds whenever X is ‘F-regular'. An example of Bishop shows that the completion of (D(1)(X),∥•∥) need not be semisimple. We show that the completion of (D(1)(X),∥•∥) is semisimple whenever the union of all the rectifiable Jordan arcs in X is dense in X. We prove that the character space of D(1)(X) is equal to X for all perfect, compact plane sets X, whether or not (D(1)(X),∥•∥) is complete. In particular, characters on the normed algebras (D(1)(X),∥•∥) are automatically continuous.

Functions in Op ( I D , Re ⁡ A + L ) {\text {Op}}\left ( {{I_D},\operatorname {Re} A + L} \right ) are harmonic on D D for a closed subalgebra A A of C 0 ( Y ) {C_0}\left ( Y \right ) , an ideal I I of A A and a linear subspace L L of finite dimension in C 0 , R ( Y ) {C_0}{,_R}\left ( Y \right ) unless the uniform closure of I I is selfadjoint.

Let A be a Banach function algebra on a compact Hausdorff space X. In this paper we consider some relations between the maximal ideal space, the Shilov boundary and the Choquet boundary of A and its uniform closure A. As an application we determine the maximal ideal space, the Shilov boundary and the Choquet boundary of algebras of infinitely differentiable functions which were introduced by Dales and Davie in 1973. For some notations, definitions, elementary and known results, one can refer to [2] and [3]. Let X be a compact Hausdorff space and let C(X) denote the space of all continuous complex valued functions on X. A function algebra on X is a subalgebra of C(X) which contains the constants and separates the points of X. If there is an algebra norm on A so that Ilf gII < IlfH *HgII for all f, g E A, then A is called a normed function algebra. A complete normed function algebra on X is called a Banach function algebra on X. If the norm of a Banach function algebra is the uniform norm on X; i.e. Ilf IX = supxEX If(x)I, it is called a uniform algebra on X. If A is a function algebra on X, then A, the uniform closure of A, is a uniform algebra on X. If (A, 11 11) is a Banach function algebra on X, for every x E X the map ox: A -* C, defined by Ox(f) = f(x), is a nonzero continuous complex homomorphism on A and so Ox E MA, where MA is the maximal ideal space of A. We call ox the evaluation homomorphism at x. Clearly for every || fKx = sup If(x)I = sup IO'(f)I < sup 10(f)l = 1flM ? llfll xEX XEX OEMA where E is the Gelfand transform of f. The Banach function algebra A on X is called natural, if every 0 E MA is given by an evaluation homomorphism ox at some x E X; or, in other Received by the editors February 9, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 46J10; Secondary 46J20.

We give an example of a natural Banach function algebra on the unit disc such that a smaller disc is a peak set for the algebra, but it does not contain any peak point. A Banach function algebra on a compact Hausdorff space X is a Banach algebra A consisting of continuous functions on X, such that A separates points of X and contains the constant functions. If the norm of the algebra A coincides with the sup norm on X , it is called a uniform algebra. If any linear and multiplicative functional on A is of the form f 7→ f(x) for some x ∈ X, the algebra is called natural. A subset K of X is a peak set for A if there is an f ∈ A such that f ≡ 1 on K and |f(x)| < 1 for x / ∈ K; if K = {x0} , then x0 is a peak point. If no proper subset of K is a peak set we call it a minimal peak set for A. It is well known [2] that if A is a uniform algebra on a metrizable set X then any peak set contains a peak point. In [1] H. G. Dales constructed a natural Banach function algebra on a compact subset of C having a peak set without any peak point. T. G. Honary [3] provided an example in R, however his algebra is not natural. In this note we give a very simple example of a natural Banach function algebra on the unit disc with peak sets not containing any peak point. Let K be an open nonempty subset of the complex plane C. By C(K) we denote the algebra of all bounded complex valued functions on K with continuous and bounded first order partial derivatives on K. C(K) is a Banach function algebra on K if equipped with the norm ‖f‖ = ‖f‖∞ + ‖fx‖∞ + ‖fy‖∞ , where ‖·‖∞ is the sup norm. By A(K) we denote the uniform algebra of all continuous functions on K which are analytic on K. We put Dr = {z ∈ C : |z| < r} and C− = {z : Re z < 0}. Theorem 1. Let A be a subalgebra of C(D1) consisting of all functions which are analytic on D 1 2 . Then D 1 2 is a minimal peak set for A; in particular D 1 2 does not contain any peak point. Lemma 2. There is no h ∈ A(D1) such that h(1) = 0 and |1 + (z − 1)h(z)| < 1 for z 6= 1. Proof of the lemma. Assume that such a function does exist. If we compose 1 + (z − 1)h(z) with a suitable fractional linear transformation we get an f ∈ A (C−) Received by the editors November 8, 1995. 1991 Mathematics Subject Classification. Primary 46J10. c ©1997 American Mathematical Society

Let X be a compact subset of the complex plane C, and let 0 < a < 1 . We show that the maximal ideal space of liPA (X, a) is X. Let F be a closed nonempty set in RI, and take ca with 0 < ca < 1. Then Lip(F, ca) is the algebra of bounded complex-valued functions f on F such that pa(f) = SUp {I f(x) jf(Y) I: XYEF.x Y} is finite, and lip(F, ca) is the subalgebra of functions f such that IfPx) f(Y) I,0 as Ix yJ '?. For f E Lip(X, c), set lIfila = i1f lIF +Pa(f), where I lF is the uniform norm on the closed set F . Then (Lip(F, ca), I I *a) is a Banach function algebra on F, and lip(F, ca) is a closed subalgebra of Lip(F, ca). These Lipschitz algebras were first studied by Sherbert [3]. Let X be a compact subset of the complex plane C. We follow established custom in denoting by C(X) the algebra of all continuous complex-valued functions on X and denoting by A(X) the subalgebra of functions analytic on int(X). We further define LiPA(X, ca) = Lip(X, ca) n A(X) and lipA(X, ca) = lip(X, ca) n A(X), so that LiPA(X, ca) and lipA(X, ca) are closed subalgebras of Lip(X, ca). This paper concerns the maximal ideal space of the algebra liPA(X, ca). It is clear that when the interior of X is empty, lipA(X, ca) is the algebra lip(X, ca) and its maximal ideal space is X [3]. So suppose that the interior of X is nonempty. We shall show that the maximal ideal space of lipA(X, ca) is X. The proof of this result depends on an extension theorem and some approximation lemmas. An interesting extension theorem for the Lipschitz algebra Lip(X, ca) is given in [4, Chapter VI, Theorem 3]. Here we modify the proof of that result to prove the following extension theorem for lip(X, ca). Received by the editors December 9, 1992. 1991 Mathematics Subject Classification. Primary 46J 10; Secondary 46J1 5, 46J20.

We study the range transformations Op ⁡ ( A D , Re ⁡ B ) \operatorname {Op} ({A_{D,}}\operatorname {Re} B) and Op ⁡ ( A D , B ) \operatorname {Op} ({A_D},B) for Banach function algebras A A and B B . As a special instance, the harmonicity of functions in Op ⁡ ( A D , Re ⁡ A ) \operatorname {Op} ({A_D},\operatorname {Re} A) for a nontrivial function algebra A A is established and is compared with previous investigations of Op ⁡ ( A D , A ) \operatorname {Op} ({A_D},A) and Op ⁡ ( ( Re ⁡ A ) I , ( Re ⁡ A ) ) \operatorname {Op} ({(\operatorname {Re} A)_I},(\operatorname {Re} A)) for an interval I I . In § 2 \S 2 we present some results on Op ⁡ ( A D , B ) \operatorname {Op} ({A_D},B) and use them to show that functions in Op C ( A D , B ) {\operatorname {Op} ^C}({A_D},B) are analytic for certain Banach function algebras.

H. G. Dales has proved in [1] that if A A is a Banach function algebra on a compact metrizable space X X , then S ¯ 0 ( A , X ) = Γ ( A , X ) {\bar S_0}(A,X) = \Gamma (A,X) , where S 0 ( A , X ) {S_0}(A,X) is the set of peak points of A A (w.r.t. X X ) and Γ ( A , X ) \Gamma (A,X) is the Shilov boundary of A A (w.r.t. X X ). Here, by considering the relation between peak sets and peak points of a Banach function algebra A A and its uniform closure A ¯ \bar A , we present an elementary and constructive proof of the density of peak points in the Shilov boundary.