Analytic Invariants of Semidirect Products of Symmetric Groups on Banach Spaces

Abstract

We consider algebras of polynomials and analytic functions that are invariant with respect to semidirect products of groups of bounded operators on Banach spaces with symmetric bases. In particular, we consider algebras of so-called block-symmetric and double-symmetric analytic functions on Banach spaces ℓp(Cn) and the homomorphisms of these algebras. In addition, we describe an algebraic basis in the algebra of double-symmetric polynomials and discuss a structure of the spectrum of the algebra of double-symmetric analytic functions on ℓp(Cn).

Block-Supersymmetric Polynomials on Spaces of Absolutely Convergent Series

“…Let us show that H vsup b supports characters that are not point evaluation functionals. Similar results for different algebras were obtained in [8,20,27,29,34].…”

“…Investigations of spectra of algebras H b (X) of all entire functions of bounded type on Banach spaces X were started by Aron, Cole, and Gamelin in [24], where the authors observed that the spectrum of H b (X) may have a complicated structure; in particular, it contains extended point-evaluation functionals associated with points of the second dual space X * * (see also [25,26]). Subalgebras of H b (X) of symmetric analytic functions with respect to permutations of basis vectors of X = ℓ p and their spectra were studied in [27,28] and others (see [29] and references therein), with respect to continual permutations in symmetric structures of X = L p in [6,30,31] and others, and with respect to abstract groups of operators in [7,32]. There are two important questions about the spectrum of a subalgebra H 0 of H b (X).…”