C^*-algebras from k group representations

Abstract

We introduce certain $C^{\ast }$-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 $C^{\ast }$-algebra is isomorphic to a corner in the $C^{\ast }$-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.

References in corpus (2)

• C*-algebras associated to product systems of Hilbert bimodules
• Representations of Higher Rank Graph Algebras