Twisted Orlicz algebras, II

Abstract

Let G be a locally compact group, let $\Omega :G\times G\to {\mathbb{C}}^{\ast }$ 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 $⊛$ coming from $\Omega$. We show that the twisted Orlicz algebra $(L^{\Phi }(G),⊛)$ 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 $\Omega$, $(L^{\Phi }(G),⊛)$ becomes an Arens regular, dual Banach algebra. We also look into certain cohomological properties of $(L^{\Phi }(G),⊛)$, 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.

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• Weak amenability of weighted Orlicz algebras