Weighted Orlicz $*$-algebras on locally elliptic groups

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

💡 Research Summary

This paper investigates the harmonic analysis of weighted Orlicz *‑algebras on locally elliptic groups, a class of totally disconnected locally compact (tdlc) groups that can be written as an increasing union of compact open subgroups. The authors fix a complementary pair of Young functions ((\Phi,\Psi)) and a weight (\omega:G\to