指数李群的群代数中的L1-determined理想
L1-determined ideals in group algebras of exponential Lie groups
摘要：
一个局部紧群$G$称为$\ast$ -正则的，如果自然映射$Ψ:\Prim C^\ast(G)\to\Prim_{\ast} L^1(G)$是关于本原理想空间$\Prim C^\ast(G)$和$\Prim_{\ast} L^1(G)$上的Jacobson拓扑的同胚. 1980年J. Boidol用一个纯代数条件刻画了所有指数李群中的$\ast$ -正则群。本文引入了$L^1$ -确定理想的概念，以讨论本原$\ast$ -正则性的较弱性质。给出了$C^\ast(G)$的闭理想$I$是$L^1$ -决定的两个充分判据.由此我们推出一个策略来证明一个给定的指数李群是本原$\ast$ -正则的。作者在论文中证明了所有维数为$\le 7$的指数李群都具有这一性质。到目前为止，还没有已知的反例。这里我们讨论示例$G=B_5$，它是维度$\le 5$中唯一的关键维度。
英文摘要：
A locally compact group $G$ is said to be $\ast$-regular if the natural map $Ψ:\Prim C^\ast(G)\to\Prim_{\ast} L^1(G)$ is a homeomorphism with respect to the Jacobson topologies on the primitive ideal spaces $\Prim C^\ast(G)$ and $\Prim_{\ast} L^1(G)$. In 1980 J. Boidol characterized the $\ast$-regular ones among all exponential Lie groups by a purely algebraic condition. In this article we introduce the notion of $L^1$-determined ideals in order to discuss the weaker property of primitive $\ast$-regularity. We give two sufficient criteria for closed ideals $I$ of $C^\ast(G)$ to be $L^1$-determined. Herefrom we deduce a strategy to prove that a given exponential Lie group is primitive $\ast$-regular. The author proved in his thesis that all exponential Lie groups of dimension $\le 7$ have this property. So far no counter-example is known. Here we discuss the example $G=B_5$, the only critical one in dimension $\le 5$.
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