C^* -ALGEBRAS FROM K GROUP REPRESENTATIONS. (8th December 2022) - DEACONU, VALENTIN
Abstract: We introduce certain C^* -algebras and k -graphs associated to k finite-dimensional unitary representations ρ _1, …, ρ _k of a compact group G . We define a higher rank Doplicher-Roberts algebra O_ρ _1, …, ρ _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 Λ with no sources. For G finite and ρ _i faithful of dimension at least two, this graph is irreducible, it has vertices G and the edges are determined by k commuting matrices obtained from the character table of the group. We illustrate this with some examples when O_ρ _1, …, ρ _k is simple and purely infinite, and with some K -theory computations.