Weighted Orlicz $*$-algebras on locally elliptic groups

Abstract

Let $G$ be a locally elliptic group, $(\Phi,\Psi)$ a complementary pair of Young functions, and $\omega: G \rightarrow [1,\infty)$ a weight function on $G$ such that the weighted Orlicz space $L^\Phi(G,\omega)$ is a Banach $*$-algebra when equipped with the convolution product and involution $f^*(x):=\overline{f(x^{-1})}$ ($f \in L^\Phi(G,\omega)$). Such a weight always exists on $G$ and we call it an $L^\Phi$-weight. We assume that $1/\omega \in L^\Psi(G)$ so that $L^\Phi(G,\omega) \subseteq L^1(G)$. This paper studies the spectral theory and primitive ideal structure of $L^\Phi(G,\omega)$. In particular, we focus on studying the Hermitian, Wiener and $*$-regularity properties on this algebra, along with some related questions on spectral synthesis. It is shown that $L^\Phi(G,\omega)$ is always quasi-Hermitian, weakly-Wiener and $*$-regular. Thus, if $L^\Phi(G,\omega)$ is Hermitian, then it is also Wiener. Although, in general, $L^\Phi(G,\omega)$ is not always Hermitian, it is known that Hermitianness of $L^1(G)$ implies Hermitianness of $L^\Phi(G,\omega)$ if $\omega$ is sub-additive. We give numerous examples of locally elliptic groups $G$ for which $L^1(G)$ is Hermitian and sub-additive $L^\Phi$-weights on these groups. In the weighted $L^1$ case, even stronger Hermitianness results are formulated.