L1-determined ideals in group algebras of exponential Lie groups

TL;DR

This paper introduces the concept of L^1-determined ideals in group algebras of exponential Lie groups, providing criteria to analyze primitive -regularity and applying this to specific groups.

Contribution

It defines L^1-determined ideals and offers criteria to establish primitive -regularity in exponential Lie groups, extending previous algebraic characterizations.

Findings

All exponential Lie groups of dimension  or less are primitive -regular.

Criteria for L^1-determined ideals are established.

The example G=B_5 is discussed as a critical case.

Abstract

A locally compact group $G$ is said to be $\ast$-regular if the natural map $\Psi:\Prim C^\ast(G)\to\Prim_{\ast} L^1(G)$ 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 $\le 5$.