Twisted Orlicz algebras, II

TL;DR

This paper explores the algebraic and cohomological properties of twisted Orlicz algebras on locally compact groups, establishing conditions for identities, regularity, and amenability, with applications to groups of polynomial growth.

Contribution

It provides new characterizations of when twisted Orlicz algebras are unital, Arens regular, or amenable, extending the understanding of their structure and properties.

Findings

Twisted Orlicz algebra has a bounded approximate identity iff it is unital and G is discrete.

Under certain conditions, the algebra is Arens regular and a dual Banach algebra.

Amenability and Connes-amenability are rare in these algebras.

Abstract

Let G be a locally compact group, let $\Omega:G\times G\to \mathbb{C}^*$ be a 2-cocycle, and let ($\Phi$,$\Psi$) be a complementary pair of strictly increasing continuous Young functions. We continue our investigation of the algebraic properties of the Orlicz space $L^\Phi(G)$ with respect to the twisted convolution $\circledast$ coming from $\Omega$. We show that the twisted Orlicz algebra $(L^\Phi(G),\circledast)$ posses a bounded approximate identity if and only if it is unital if and only if $G$ is discrete. On the other hand, under suitable condition on $\Omega$, $(L^\Phi(G),\circledast)$ becomes an Arens regular, dual Banach algebra. We also look into certain cohomological properties of $(L^\Phi(G),\circledast)$, namely amenability and Connes-amenability, and show that they rarely happen. We apply our methods to compactly generated group of polynomial growth and demonstrate that our results could be applied to variety of cases.