Twisted Orlicz algebras, II

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.