Multiplicatively range-preserving maps between Banach function algebras

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.

On the existence of continuous (approximate) roots of algebraic equations

In this paper we study the symbolic calculus for a Banach function algebra with certain conditions. First we define a class of Banach function algebras which contains the class of all function algebras and the class of all ultraseparating Banach function algebras. Our purpose is to prove the theorem asserting that if A is a non-trivial Banach function algebra in the class, then only analytic functions can operate on A. It is a generalization of theorems of de Leeuw and Katznelson [6], Bernard [2] and Bernard and Dufresnoy [3].

Let X, Y be compact Hausdorff spaces and B£(X), B*(Y)f 0 < a, |3 < 12 (the first uncountable ordinal), the associated Banach spaces of bounded real-valued Baire functions of classes a and 0. If B*(X) =£ B*AX) (which is the case if a =£ 0 and X is not dispersed), then B*(X) is neither linearly isometric to BQ(Y) nor equivalent to B^(Y) in several other ways. B^(X) is linearly isometric to B%(Y) if and only if X is Baire isomorphic to Y. For 1 < a < ft the maximal ideal space of B£(X) for a nondispersed compact space X As not an F-space. 1. Let I bea compact (more generally, completely regular) Hausdorff space and C(X) the space of continuous real-valued functions on X. Let B0(X) = C(X)9 and inductively define Ba(X) for each ordinal a < Q, (Q denotes the first uncountable ordinal) to be the space of pointwise limits of sequences of functions in U | < a B^(X). Let B*(X) be the space of bounded functions contained in Ba(X). With the pointwise operations Ba(X) and B*(X) are lattice-ordered algebras. With the supremum norm B*(X) is a Banach algebra (see [4, §41]). The Baire sets of X of multiplicative class a, denoted by Za(X)9 are defined to be the zero sets of functions in B*(X). Those of additive class a, denoted by CZa(X), are defined as the complements of sets in Za(X). Finally, those of ambiguous class a, denoted by Aa(X), are the sets which are simultaneously in Za(X) and CZa(X). With the set-theoretic operations of union and intersection, Aa(X) is a Boolean algebra for each a < 12. The sets of exactly ambiguous class a, denoted by EAa(X are those in ^ a ( Z ) U ^ < a Ac(X). The sets of exactly additive and exactly multiplicative class a are defined analogously. The class of all Baire subsets of X is AMS (MOS) subject classifications (1970). Primary 06A65, 26A21, 28A05, 46E1S, 46E30, 46J10, 54C50, 54H05; Secondary 04A15.

Let X be a compact Hausdorff space and C(X) the Banach space of all real-valued continuous functions on X under the sup norm. If H is a linear subspace of C(X) which separates the points of X and contains the constant functions, then it is well known that there exists a smallest closed subset of X, called the Silov boundary of X relative to H and denoted by OHX, with the property that each hCH attains its maximum value on afHX. A point xeX is called an H-extremal point if the only positive linear functional u on C(X) such that u(h) = h(x) for all hCH is the evaluation functional 4. where O.,(f) =f(x) for all fE C(X). The set of H-extremal points of X is called the Choquet boundary of X relative to H and will be denoted by VHX. H. Bauer has shown [1, 2.1 ] that the Choquet boundary of X relative to H is nonempty and the Silov boundary of X relative to H is equal to the closure of the Choquet boundary. Bauer has also introduced (see [1 ]) an abstract Dirichlet problem for the above setting. If S:DHX is closed and xCX, then MS denotes the set of positive linear functionals u on C(S) such that u(h I S) =h(x) for all hCH. MS is always nonempty. The measures in MX'HX are called the H-harmonic measures belonging to x. A function f in C(X) is said to be H-harmonic if for every xEX and every u M we have u(f) =f(x). The set of H-harmonic functions is denoted by H. The Dirichlet problem is said to be solvable for X relative to H if 1^ j aHX = c(aHX). The Dirichlet problem is solvable if and only if to each x in X belongs exactly one H-harmonic measure (see [1, Satz 9]). Let X1, X2 be compact Hausdorff spaces and H1, H2 separating linear subspaces of C(X1), C(X2) respectively, which contain the constant functions. Equipped with the usual product topology, the cartesian product X1 X X2 is a compact Hausdorff space. In this paper we show that a subfamily H1+H2 of C(XIXX2) can be defined, in a natural way, for which VH1+H2Xl X X2 exists and VH1+H2Xl X X2 = VH1X1X VH2X2. We then give two theorems on the relation of the

Function-space compactifications of function spaces

Automatic Continuity of Lipschitz Algebras

Almost compactness and near compactness in fuzzy topological spaces

Focusing on the role that automorphisms and equivalence relations play in the algebraic theory of minimal sets provides an original treatment of some key aspects of abstract topological dynamics. Such an approach is presented in this lucid and self-contained book, leading to simpler proofs of classical results, as well as providing motivation for further study. Minimal flows on compact Hausdorff spaces are studied as icers on the universal minimal flow M. The group of the icer representing a minimal flow is defined as a subgroup of the automorphism group G of M, and icers are constructed explicitly as relative products using subgroups of G. Many classical results are then obtained by examining the structure of the icers on M, including a proof of the Furstenberg structure theorem for distal extensions. This book is designed as both a guide for graduate students, and a source of interesting new ideas for researchers.

Linear extension operators of bounded norms

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.

For a topological space $X$, let $X_\delta$ be the space $X$ with $G_\delta$-topology of $X$. For an uncountable cardinal $\kappa$, we prove that the following are equivalent: (1) $\kappa$ is $\omega_1$-strongly compact. (2) For every compact Hausdorff space $X$, the Lindelof degree of $X_\delta$ is $\le \kappa$. (3) For every compact Hausdorff space $X$, the weak Lindelof degree of $X_\delta$ is $\le \kappa$. This shows that the least $\omega_1$-strongly compact cardinal is the supremum of the Lindelof and the weak Lindelof degrees of compact Hausdorff spaces with $G_\delta$-topology. We also prove the least measurable cardinal is the supremum of the extents of compact Hausdorff spaces with $G_\delta$-topology. For the square of a Lindelof space, using weak $G_\delta$-topology, we prove that the following are consistent: (1) the least $\omega_1$-strongly compact cardinal is the supremum of the (weak) Lindelof degrees of the squares of regular $T_1$ Lindelof spaces. (2) The least measurable cardinal is the supremum of the extents of the squares of regular $T_1$ Lindelof spaces.

We assume a basic knowledge of general topology and integration theory (as found, for example, in [91], [104], [130], and [253]). The purpose of this chapter is to present some special results which are not necessarily found in general references. In section 4 we prove an interpolation theorem and investigate when compact Hausdorff spaces can be mapped continously onto the closed unit interval [0, 1]. A brief development of dispersed spaces and their relationship to spaces of ordinal numbers is given in section 5. Section 6 is devoted to a study of the Cantor set and section 7 is concerned with extremally disconnected compact Hausdorff spaces and their role as projectives (in the category of compact Hausdorff spaces and continuous maps). In section eight we briefly develop the theory of regular Borel measures and prove representation theorems for C(T,ℝ)* (and C(T,ℂ)*).

The hyperspace of a compact space, I

Compact spaces of diversity two