Twisted Orlicz algebras and complete isomorphism to operator algebras

TL;DR

This paper demonstrates that certain twisted Orlicz algebras associated with locally compact groups can be completely isomorphic to operator algebras under specific conditions involving 2-cocycles and Young functions.

Contribution

It establishes new conditions under which twisted Orlicz algebras are completely isomorphic to operator algebras, extending previous results to broader classes of 2-cocycles and groups.

Findings

Algebras are completely isomorphic to operator algebras when $L^ om{\Phi}(G) \subseteq L^2(G)$ and 2-cocycles satisfy certain $L^2$ conditions.

Results apply to groups of polynomial growth, broadening the scope of operator algebra representations.

Provides new classes of 2-cocycles that yield operator algebra structures on twisted Orlicz 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. It is shown in \cite{OS2} that $(L^\Phi(G),\circledast)$ becomes an Arens regular dual Banach algebra if \begin{align}\label{Eq:2-cocycle bdd sum-abstract} |\Omega(s,t)|\leq u(s)+v(t) \ \ \ (s,t\in G) \end{align} for some $u,v\in \mathcal{S}^\Psi(G)$. We prove if $L^\Phi(G)\subseteq L^2(G)$ and $u,v$ can be chosen to belong to $L^2(G)$, then $(L^\Phi(G),\circledast)$ with the maximal operator space structure is completely isomorphic to an operator algebra. We also present further classes of 2-cocycles for which one could obtain such algebras generalizing in part the results of \cite{OS1}. We apply our methods to compactly generated group of polynomial growth and demonstrate that our results could be applied to variety of cases.