The Wave Equation in the Context of Reduced Groups $C^*$-Algebras

TL;DR

This paper extends the classical wave equation to reduced group $C^*$-algebras for infinite groups, establishing existence and uniqueness of solutions using automorphism groups and an analogue of the Laplacian.

Contribution

It introduces a novel framework for analyzing the wave equation on non-abelian group $C^*$-algebras, generalizing classical results.

Findings

Existence of solutions to the wave equation in this new setting

Uniqueness of solutions established

Development of an analogue Laplacian for non-abelian groups

Abstract

Motivated by the identification $C(\mathbb{T})\cong C_r^*(\mathbb{Z})$ and the wave equation on the circle, we explore the wave equation in the context of reduced group $C^*$-algebras $C_r^*(G)$ for countably infinite, possibly non-abelian groups $G$. Using a one-parameter group of $*$-automorphisms whose infinitesimal generator paves the way to an analogue of the Laplacian, we establish the existence and uniqueness of solutions to the wave equation within this framework.