C^*-algebras from k group representations
Valentin Deaconu

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 -algebras and -graphs associated to finite dimensional unitary representations of a compact group . We define a higher rank Doplicher-Roberts algebra , constructed from intertwiners of tensor powers of these representations. Under certain conditions, we show that this -algebra is isomorphic to a corner in the -algebra of a row finite rank graph with no sources. For finite and faithful of dimension at least , this graph is irreducible, it has vertices and the edges are determined by commuting matrices obtained from the character table of the group. We illustrate with some examples when is simple and purely infinite, and with some -theory computations.
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Taxonomy
TopicsAdvanced Operator Algebra Research · Advanced Topics in Algebra · Noncommutative and Quantum Gravity Theories
