Regularity and Algebras of Analytic Functions in Infinite Dimensions

Abstract

Abstract. A Banach space E is known to be Arens regular if every continuous linear mapping from E to E is weakly compact. Let U be an open subset of E, and let H b (U ) denote the algebra of analytic functions on U which are bounded on bounded subsets of U lying at a positive distance from the boundary of U. We endow H b (U ) with the usual Fréchet topology. M b (U ) denotes the set of continuous homomorphisms φ : H b (U ) → C. We study the relation between the Arens regularity of the space E and the structure of M b (U ).

Supersymmetric Polynomials on the Space of Absolutely Convergent Series

“…Investigations of the spectrum of H b (X) were started by Aron, Cole and Gamelin in their fundamental work [2]. Note that, in the general case, M b = M(H b (X)) has complicated topological and algebraic structures (see [5,25]) which can be described only implicitly involving such tools as the Aron-Berner extension, topological tensor products, StoneCech compactification, ect. On the other hand, it is convenient for applications to have algebras of analytic functions of infinite many variables whose spectra admit explicit descriptions.…”

Note on Arens regularity of symmetric tensor products

“…∑ 4 ℓ 2 is not symmetrically regular. Note that in [3] it is shown that the complete projective tensor product ℓ 2 ⊗ π ℓ 2 is not symmetrically regular.…”

“…X is regular if each bilinear form on X × X is regular. X is symmetricaly regular if each symmetric bilinear form on X is regular (see [3]). If A is a Banach algebra, then A is called Arens regular if the bilinear map associated with the algebra product (x, y) → xy is regular.…”

The Spectra of Algebras of Group-Symmetric Functions

“…Algebras of general holomorphic functions (which can be viewed as the case G = {I}) and their spectra, on the other hand, have been extensively studied, by Aron et al [5] and by Aron et al, among others [3]. A central element in the theory is the Aron-Berner extension [2]: every holomorphic function f over an open subset of E can be extended to f , holomorphic on an open subset of the bidual E of E, giving rise to evaluation characters outside the space E, and bringing into play the bidual and its properties.…”