Twisted Orlicz algebras, I

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.