Realization of minimal $C^*$-dynamical systems in terms of Cuntz-Pimsner algebras

Fernando Lledó, Ezio Vasselli
February 26, 2007
In the present paper we study tensor C*-categories with non-simple unit realised as C*-dynamical systems $(\mathcal{F},\mathcal{G},\beta)$ with a compact (non-Abelian) group $\mathcal{G}$ and fixed point algebra $\mathcal{A} := \mathcal{F}^\mathcal{G}$. We consider C*-dynamical systems with minimal relative commutant of $\mathcal{A}$ in $\mathcal{F}$, i.e. $\mathcal{A}' \cap \mathcal{F} = \mathcal{Z}$, where $\mathcal{Z}$ is the center of $\mathcal{A}$ which we assume to be nontrivial. We give first several constructions of minimal C*-dynamical systems in terms of a single Cuntz-Pimsner algebra associated to a suitable $\mathcal{Z}$-bimodule. These examples are labelled by the action of a discrete Abelian group (which we call the chain group) on $\mathcal{Z}$ and by the choice of a suitable class of finite dimensional representations of $\mathcal{G}$. Second, we present a construction of a minimal C*-dynamical system with nontrivial $\mathcal{Z}$ that also encodes the representation category of $\mathcal{G}$. In this case the C*-algebra $\mathcal{F}$ is generated by a family of Cuntz-Pimsner algebras, where the product of the elements in different algebras is twisted by the chain group action. We apply these constructions to the group $G = SU(N)$.
open access link