Topological convolution algebras

TL;DR

This paper introduces a new class of topological convolution algebras on semi-groups within locally compact groups, establishing key inequalities, and explores their functional calculus and spectrum with applications to stochastic distributions and Dirichlet series.

Contribution

It defines a novel family of topological convolution algebras with specific inequalities and provides conditions for their measures, advancing the understanding of their functional calculus and spectral properties.

Findings

Established a convolution inequality for the algebra elements.

Provided sufficient conditions on measures for the inequalities to hold.

Explored the spectrum and functional calculus of the algebra elements.

Abstract

In this paper we introduce a new family of topological convolution algebras of the form $\bigcup_{p\in\mathbb N} L_2(S,\mu_p)$, where $S$ is a Borel semi-group in a locally compact group $G$, which carries an inequality of the type $\|f*g\|_p\le A_{p,q}\|f\|_q\|g\|_p$ for $p > q+d$ where $d$ pre-assigned, and $A_{p,q}$ is a constant. We give a sufficient condition on the measures $\mu_p$ for such an inequality to hold. We study the functional calculus and the spectrum of the elements of these algebras, and present two examples, one in the setting of non commutative stochastic distributions, and the other related to Dirichlet series.