$\boldsymbol {C}^{*}$ -ALGEBRAS FROM $\boldsymbol {K}$ 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 $\mathcal {O}_{ρ _1, …, ρ _k}$, 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 $\hat {G}$ and the edges are determined by k commuting matrices obtained from the character table of the group. We illustrate this with some examples when $\mathcal {O}_{ρ _1, …, ρ _k}$ is simple and purely infinite, and with some K -theory computations.