Weighted Orlicz $\ast$-algebras on locally elliptic groups

Abstract

Let $G$ be a locally elliptic group, $(\Phi ,\Psi )$ a complementary pair of Young functions, and $\omega :G\to [1,\infty )$ a weight function on $G$ such that the weighted Orlicz space $L^{\Phi }(G,\omega )$ is a Banach $\ast$-algebra when equipped with the convolution product and involution $f^{\ast }(x):=\overline{f(x^{-1})}$ ($f\in L^{\Phi }(G,\omega )$). 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 $\ast$-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 $\ast$-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.

33 pages. Some minor changes have been made to the exposition. Accepted for publication in Studia Mathematica