Weighted Orlicz $*$-algebras on locally elliptic groups
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914388860993536 |
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| author | Carter, Max |
| author_facet | Carter, Max |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_20735 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weighted Orlicz $*$-algebras on locally elliptic groups Carter, Max Functional Analysis Group Theory Operator Algebras Representation Theory 22D15, 22D20, 43A15, 43A20, 43A45, 46H05, 46H30 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. |
| title | Weighted Orlicz $*$-algebras on locally elliptic groups |
| topic | Functional Analysis Group Theory Operator Algebras Representation Theory 22D15, 22D20, 43A15, 43A20, 43A45, 46H05, 46H30 |
| url | https://arxiv.org/abs/2503.20735 |