Functional calculus for certain Banach function algebras

Multiplicatively range-preserving maps between Banach function algebras

Let K and X be compact plane sets such that K X. Let P (K) be the uniform closure of polynomials on K. Let R(K) be the closure of rational functions K with poles o K. Dene P (X;K) and R(X;K) to be the uniform algebras of functions in C(X) whose restriction to K belongs to P (K) and R(K), respectively. Let CZ(X;K) be the Banach algebra of functions f in C(X) such that fjK = 0. In this paper, we show that every nonzero complex homomorphism ’ on CZ(X;K) is an evaluation homomorphism ez for some z in XnK. By considering this fact, we characterize the maximal ideal space of the uniform algebra P (X;K). Moreover, we show that the uniform algebra R(X;K) is natural.

Let A and B be Banach function algebras on compact Hausdorff spaces X and Y and let ‖.‖X and ‖.‖Y denote the supremum norms on X and Y, respectively. We first establish a result concerning a surjective map T between particular subsets of the uniform closures of A and B, preserving multiplicatively the norm, i.e. ‖Tf Tg‖Y = ‖fg‖X, for certain elements f and g in the domain. Then we show that if α ∈ ℂ {0} and T: A → B is a surjective, not necessarily linear, map satisfying ‖fg + α‖X = ‖Tf Tg + α‖Y, f,g ∈ A, then T is injective and there exist a homeomorphism φ: c(B) → c(A) between the Choquet boundaries of B and A, an invertible element η ∈ B with η(Y) ⊆ {1, −1} and a clopen subset K of c(B) such that for each f ∈ A, $$ Tf\left( y \right) = \left\{ \begin{gathered} \eta \left( y \right)f\left( {\phi \left( y \right)} \right) y \in K, \hfill \\ - \frac{\alpha } {{\left| \alpha \right|}}\eta \left( y \right)\overline {f\left( {\phi \left( y \right)} \right)} y \in c\left( B \right)\backslash K \hfill \\ \end{gathered} \right. $$. In particular, if T satisfies the stronger condition R π(fg + α) = R π(Tf Tg + α), where R π(.) denotes the peripheral range of algebra elements, then Tf(y) = T1(y)f(φ(y)), y ∈ c(B), for some homeomorphism φ: c(B) → c(A). At the end of the paper, we consider the case where X and Y are locally compact Hausdorff spaces and show that if A and B are Banach function algebras on X and Y, respectively, then every surjective map T: A → B satisfying ‖Tf Tg‖Y = ‖fg‖, f, g ∈ A, induces a homeomorphism between quotient spaces of particular subsets of X and Y by some equivalence relations.

A complete description is given of the analytic structure of maximal dimension in the spectra of a wide class of concrete function algebras generated by analytic functions.A connection is also given with point derivations on such algebras.I. Introduction.Let B be a function algebra acting on its spectrum (maximal ideal space) Sp B, with Shilov boundary TB.According to a wellknown result of Rossi [15], the functions in B obey a weak maximum modulus principle on Sp B -TB.One of the principal themes in the theory of function algebras is the "explanation" of this analytic-type behavior by the existence of analytic structure in Sp B. More precisely, let us say that an analytic variety (of dimension n) in Sp B is a pair (X, r) where X is an analytic variety (of pure dimension n), r: X -► Sp B is a continuous map, and b ° r is analytic for each b e B. Then the question is: what are the nontrivial analytic varieties in SpS?Although Stolzenberg's example [16] shows that one does not have the existence of nontrivial analytic structure in all situations, a number of global and local conditions which imply the existence of analytic structure have been found by Gleason [11], Browder [8], Clayton [9] and others.A different approach, due to Bishop [3], which depends upon the existence of functions in B with finite fibers, leads to the construction of 1-dimensional analytic structure in Sp B. Bishop's ideas have been expanded and exploited by Stolzenberg [17], Bjork [5],[6], Alexander [1], [2], the author [20], and others, and have proven very useful in dealing with questions of approximation.In this paper, we shall be concerned with the classification of high-dimensional analytic structure.For a fairly wide class of concrete function algebras, those which contain a sufficiently rich collection of analytic functions, we are able to completely describe the analytic structure of maximal dimension which

A Banach Algebra of Functions With Bounded nth Differences

Automatic Continuity of Lipschitz Algebras

Preface

Let A be a function algebra, X its Silov boundary, M its maximal ideal space. One of the outstanding problems in function theory is to find a structure of complex analytic space on M such that the functions of A are analytic for this analytic structure. In some cases (algebras with few representative measures, or polynomial hulls of regular curves in ~n) , such structures can be described. (See also the recent work of R. Basener [I] and of Gamelin, Sibony and Wermer, this conference.) The first counterexample to the existence of analytic structure was found by Stolzenberg [14] who exhibited a compact K c ~2 such that the polynomial hull h(K) of K is distinct from K , and there exis~ no analytic structure on h(K) K . More explicit examples were found by Wermer and Basener for rationally comvex hulls (see [2] and also [6] for related class of examples). Moreover, if an analytic structure exists, it would have to live on the Gleason parts of the maximal ideal space. As a consequence,if f is a non constant function in a Gleason part P , then f(P) would have to be interior point to f(M), by the fact that analytic functions are open mappings.~nd that assertion is completely false in general. The interest for the research of analytic structure is that it provides regularity theorems on the functions of A , and also theorems about representative measures, bounded point derivation, approximation theory, etc. But such theorems, in particular differentiability theorems about the functions of A , can be proved if one can find in the maximal ideal space, some kind of dynamical system along the trajectories of which one can study the properties of functions. The aim of this paper is to show how such generalized dynamical systems (brownian paths or even sheets) lives in spectrum of certain function algebras even if there is no analytic structure in these spectrumnand to obtain from that regularity theorems, theorems about Jensen measures, approximation and derivations, just as if there was analytic structure.

We consider various algebras of functions on a semitopological left-group S = X × G S = X \times G , the direct product of a left-zero semigroup X and a group G. In §1 we examine various analogues to the theorem of Eberlein that a weakly almost periodic function on a locally compact abelian group is uniformly continuous. Several appealing conjectures are shown by example to be false. In the second section we look at compactifications of products S × T S \times T of semitopological semigroups with right identity and left identity, respectively. We show that the almost periodic compactification of the product is the product of the almost periodic compactifications, thus generalizing a result of deLeeuw and Glicksberg. The weakly almost periodic compactification of the product is not the product of the weakly almost periodic compactifications except in restrictive circumstances; for instance, when T is a compact group. Finally, as an application, we define and study analytic weakly almost periodic functions and derive the theorem, analogous to a classical theorem about almost periodic functions, that an analytic function which is weakly almost periodic on a single line is analytic weakly almost periodic on a whole strip.

We recently introduced a weighted Banach algebra of functions that are holomorphic on the unit disc D, continuous up to the boundary, and of the class C(n) at all points where the function G does not vanish. Here, G refers to a function of the disc algebra without zeros on D. Then we proved that all closed ideals in with at most countable hull are standard. In this paper, on the assumption that G is an outer function in C(n) having infinite roots in and countable zero set h0(G), we show that all the closed ideals I with hull containing h0(G) are standard.

Analytic functions in algebras

We study an interesting class of Banach function algebras of in nitely di erentiable functions onperfect, compact plane sets. These algebras were introduced by Honary and Mahyar in 1999, calledLipschitz algebras of in nitely di erentiable functions and denoted by Lip(X;M; ), where X is aperfect, compact plane set, M = fMng1n=0 is a sequence of positive numbers such that M0 = 1 and(m+n)!Mm+n ( m!Mm)( n!Mn) for m; n 2 N [ f0g and 2 (0; 1]. Let d = lim sup( n!Mn)1n and Xd = fz 2 C :dist(z;X)  dg. Let LipP;d(X;M; )[LipR;d(X;M; )] be the subalgebra of all f 2 Lip(X;M; )that can be approximated by the restriction to Xd of polynomials [rational functions with poles o Xd]. We show that the maximal ideal space of LipP;d(X;M; ) is cXd, the polynomially convex hullof Xd, and the maximal ideal space of LipR;d(X;M; ) is Xd, for certain compact plane sets.. Usingsome formulae from combinatorial analysis, we nd the maximal ideal space of certain subalgebrasof Lipschitz algebras of in nitely di erentiable functions.

Let X be a compact nowhere dense subset of the complex plane ℂ, let C(X) be the linear space of all continuous functions on X endowed with the uniform norm, and let dA denote two-dimensional Lebesgue (or area) measure in ℂ. Denote by R(X) the closure in C(X) of the set of all rational functions having no poles on X. It is well known that if X is sufficiently massive, then the functions in R(X) can inherit many of the properties usually associated with the analytic functions, such as unlimited degrees of differentiability and even the uniqueness property itself. Here we shall examine the extent to which some of those properties are inherited by the larger algebra H ∞ (X), which by definition is the weak-* closure of R(X) in L∞(X) = L∞(X, dA).

How is it possible to describe any analytic or harmonic function on a compact set in terms of much simpler “construction blocks” like polynomials? Or, how is it possible to know the values of a function inside a compact domain, by knowing only its values on the boundary? Well, these simplifications are possible because the “bricks” are actually organized in complicated and versatile structures. For example the B ∗ algebras. And in addition, the compact domains are certainly among the simples ones, being always reducible to finite reunions. Actually, a complicated structure like a B ∗ algebra, defined by 24 axioms (out of which 13 axioms on commutative algebras, five axioms on norm, one for completeness, and five more specific axioms) can be realized by continuous functions defined on a compact set. It is not the only example. The space l 1 of complex sequences with norm given by the sum of the modules of the terms is isomorphic with the algebra of functions whose Fourier series is absolutely convergent. Also, a compact Hausdorff space, with topology induced by distance, is homeomorphic with a compact subset of [0, 1]N. Any two separable Hilbert spaces are isomorphic, and so on. These similarities bring a unifying point of view: objects of apparently distinct nature, like Weierstrass–Stone theorem on function approximation, Wiener theorem on absolutely convergent Fourier series, spectral expansion of self-adjoint operators, the theorems of Tikhonov, Stone–Čech, or the fixed-point theorem of Brouwer, the Cauchy formula on complex functions, the Green representation theorem, the Poincaré Lemma, etc. actually provide the same fundamental truth: simplification by approximation is possible on compacts.KeywordsHarmonic FunctionCotangent BundleRepresentation FormulaExterior DerivativeCompact Hausdorff SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we will describe the general form of commuting mappings of Hochschild extension algebras and characterize the properness of commuting mappings on a special class of Hochschild extension algebras with the so-called p.