正规子群不变特征标的协变函数代数
Covariant Function Algebras of Invariant Characters of Normal Subgroups
AI总结：
本文建立局部紧群闭正规子群不变特征标协变函数空间的卷积代数与 Banach 模理论，并推广到半直积群及示例。
摘要：
本文为正规子群不变特征标的协变函数代数结构提供抽象调和分析基础。设 G 为局部紧群，N 为 G 的闭正规子群。令 ξ:N→T 为连续的 G-不变特征标，1≤p<∞，并令 L_ξ^p(G,N) 为 G 上 ξ 的所有协变函数构成的 L^p 空间。我们研究 L_ξ^p(G,N) 中协变卷积的结构。证明了 L_ξ^1(G,N) 是 Banach *-代数，而 L_ξ^p(G,N) 是 Banach L_ξ^1(G,N)-模。随后研究半直积群中典范正规子群特征标情形的协变卷积理论。最后通过不同示例实现该理论。
英文摘要：
This paper presents abstract harmonic analysis foundations for structure of covariant function algebras of invariant characters of normal subgroups. Suppose that $G$ is a locally compact group and $N$ is a closed normal subgroup of $G$. Let $ξ:N\to\mathbb{T}$ be a continuous $G$-invariant character, $1\le p<\infty$, and $L_ξ^p(G,N)$ be the $L^p$-space of all covariant functions of $ξ$ on $G$. We study structure of covariant convolution in $L^p_ξ(G,N)$. It is proved that $L^1_ξ(G,N)$ is a Banach $*$-algebra and $L^p_ξ(G,N)$ is a Banach $L^1_ξ(G,N)$-module. We then investigate the theory of covariant convolutions for the case of characters of canonical normal subgroups in semi-direct product groups. The paper is concluded by realization of the theory in the case of different examples.
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