The Wave Equation in the Context of Reduced Groups $C^{\ast }$-Algebras

Motivated by the identification $C(\mathbb{T})\cong C_r^{\ast }(\mathbb{Z})$ and the wave equation on the circle, we explore the wave equation in the context of reduced group $C^{\ast }$-algebras $C_r^{\ast }(G)$ for countably infinite, possibly non-abelian groups $G$. Using a one-parameter group of $\ast$-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.