The Wave Equation in the Context of Reduced Groups $C^*$-Algebras
Fan Huang
TL;DR
This work extends the classical wave equation to reduced group $C^*$-algebras $C_r^*(G)$ for countably infinite groups by introducing a Laplacian-analogue via a one-parameter automorphism group generated by a nonzero group homomorphism $b:G\to\mathbb{R}$. The authors build Sobolev-type spaces $F^n(G)$ and define a second-order operator $H_d^{\mathcal{D}}=\delta_b^{\mathcal{C}}\circ \delta_b^{\mathcal{C}}$ with $d(g)=b(g)^2$, then derive an explicit, well-posed wave evolution with initial data $x_0\in F^2(G)$ and $y_0\in F^3(G)$, giving a closed-form Fourier-series solution. The solution's spectrum splits according to $\ker(b)$ and its complement, yielding linear growth in the kernel directions and cosine-sine oscillations in the non-kernel directions, and it reduces to the classical circle case when $G=\mathbb{Z}$ and $b(n)=n$. The framework supports potential generalizations to twisted, crossed-product, or stability analyses, linking wave dynamics on noncommutative spaces to the group's algebraic structure.
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.