C^*-algebras from k group representations

TL;DR

This paper constructs higher rank $C^*$-algebras from finite-dimensional representations of compact groups, relating them to graph algebras and analyzing their properties such as simplicity, pure infiniteness, and $K$-theory.

Contribution

It introduces a new class of $C^*$-algebras from group representations and establishes their isomorphism to graph algebra corners under certain conditions.

Findings

The $C^*$-algebras are isomorphic to corners of row finite $k$-graph $C^*$-algebras.

For finite groups with faithful representations, the associated $k$-graph is irreducible and determined by the character table.

Examples show conditions for simplicity, pure infiniteness, and compute $K$-theory.

Abstract

We introduce certain $C^*$-algebras and $k$-graphs associated to $k$ finite dimensional unitary representations $\rho_1,...,\rho_k$ of a compact group $G$. We define a higher rank Doplicher-Roberts algebra $\mathcal{O}_{\rho_1,...,\rho_k}$, constructed from intertwiners of tensor powers of these representations. Under certain conditions, we show that this $C^*$-algebra is isomorphic to a corner in the $C^*$-algebra of a row finite rank $k$ graph $\Lambda$ with no sources. For $G$ finite and $\rho_i$ faithful of dimension at least $2$, this graph is irreducible, it has vertices $\hat{G}$ and the edges are determined by $k$ commuting matrices obtained from the character table of the group. We illustrate with some examples when $\mathcal{O}_{\rho_1,...,\rho_k}$ is simple and purely infinite, and with some $K$-theory computations.