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