Twisted Orlicz algebras, I

TL;DR

This paper explores the algebraic and harmonic analysis properties of twisted Orlicz algebras on locally compact groups, establishing conditions for their Banach algebra structure and analyzing their spectral and regularity features.

Contribution

It introduces the concept of twisted Orlicz algebras, providing sufficient conditions for their Banach algebra structure and studying their harmonic analysis properties on groups of polynomial growth.

Findings

Conditions for twisted Orlicz spaces to be Banach algebras

Analysis of symmetry and functional calculus in these algebras

Applicability to polynomial and subexponential weights

Abstract

Let G be a locally compact group, let $\Omega:G\times G\to \mathbb{C}^*$ be a 2-cocycle, and let $\Phi$ be a Young function. In this paper, we consider the Orlicz space $L^\Phi(G)$ and investigate its algebraic property under the twisted convolution $\circledast$ coming from $\Omega$. We find sufficient conditions under which $(L^\Phi(G),\circledast)$ becomes a Banach algebra or a Banach $*$-algebra; we call it a {\it twisted Orlicz algebra}. Furthermore, we study its harmonic analysis properties, such as symmetry, existence of functional calculus, regularity, and having Wiener property, mostly for the case when $G$ is a compactly generated group of polynomial growth. We apply our methods to several important classes of polynomial as well as subexponential weights and demonstrate that our results could be applied to variety of cases.