Homomorphisms on algebras of analytic functions on non-symmetrically regular spaces

TL;DR

This paper investigates the structure of homomorphisms on algebras of analytic functions over Banach spaces lacking symmetric regularity, revealing complex behaviors of evaluations and their commutativity properties.

Contribution

It characterizes evaluations in the spectrum of these algebras on non-symmetrically regular spaces and explores conditions for symmetry in multilinear operator extensions.

Findings

Evaluations in fibers of the spectrum may differ in higher duals from those in the second dual.

In certain Banach spaces like , only evaluations in the original space commute with all evaluations in the second dual.

Conditions are provided for the symmetry of extensions of symmetric multilinear operators on non-symmetrically regular spaces.

Abstract

We study homomorphisms on the algebra of analytic functions of bounded type on a Banach space. When the domain space lacks symmetric regularity, we show that in every fiber of the spectrum there are evaluations (in higher duals) which do not coincide with evaluations in the second dual. We also consider the commutativity of convolutions between evaluations. We show that in some Banach spaces $X$ (for example, $X=\ell_1$) the only evaluations that commute with every other evaluation in $X''$ are those in $X$. Finally, we establish conditions ensuring the symmetry of the canonical extension of a symmetric multilinear operator (on a non-symmetrically regular space) and present some applications.