On closed Lie ideals and center of generalized group algebras

TL;DR

This paper characterizes the closed Lie ideals and the center of generalized group algebras $L^1(G,A)$, revealing their structure in terms of group actions and center-valued functions, with applications to tensor products and finite groups.

Contribution

It provides a new characterization of closed Lie ideals and the center of $L^1(G,A)$, extending previous results to generalized group algebras and specific classes of groups and Banach algebras.

Findings

Closed Lie ideals characterized via group and algebra actions.

Center of $L^1(G,A)$ consists of conjugacy class constant functions for ${f [SIN]}$ groups.

Center of tensor products equals tensor product of centers for certain groups and algebras.

Abstract

For any locally compact group $G$ and any Banach algebra $A$, a characterization of the closed Lie ideals of the generalized group algebra $L^1(G,A)$ is obtained in terms of left and right actions by $G$ and $A$. In addition, when $A$ is unital and $G$ is an ${\bf [SIN]}$ group, we show that the center of $L^1(G,A)$ is precisely the collection of all center valued functions which are constant on the conjugacy classes of $G$. As an application, we establish that $\mathcal{Z}(L^1(G) \otimes^{\gamma} A)= \mathcal{Z}(L^1(G)) \otimes^{\gamma} \mathcal{Z}(A)$, for a class of groups and Banach algebras. And, prior to these, for any finite group $G$, the Lie ideals of the group algebra $\mathbb{C}[G]$ are identified in terms of some canonical spaces determined by the irreducible characters of $G$.