$\boldsymbol {C}^{*}$ -ALGEBRAS FROM $\boldsymbol {K}$ GROUP REPRESENTATIONS. (8th December 2022)

• Record Type: Journal Article
• Authors: DEACONU, VALENTIN
• Is Part Of: Journal of the Australian Mathematical Society. Volume 113:Number 3(2022)
• Page Start: 318
• Page End: 338
• Publication Date: 2022-12-08
• Subjects: 46L05; group representation -- character table -- product system -- rank k graph -- Cuntz–Pimsner algebra; Mathematics -- Periodicals; Statistics -- Periodicals; Mathematical statistics -- Periodicals; 510.5
• DOI: 10.1017/S1446788721000392
• Languages: English
• ISSNs: 1446-7887

Abstract

Abstract: We introduce certain $C^*$ -algebras and k -graphs associated to k finite-dimensional unitary representations $\rho _1, \ldots, \rho _k$ of a compact group G . We define a higher rank Doplicher-Roberts algebra $\mathcal {O}_{\rho _1, \ldots, \rho _k}$, 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 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 $\mathcal {O}_{\rho _1, \ldots, \rho _k}$ is simple and purely infinite, and with some K -theory computations.