Fixed Points of Compact Quantum Groups Actions on Cuntz Algebras

Abstract

Given an action of a Compact Quantum Group (CQG) on a finite dimensional Hilbert space, we can construct an action on the associated Cuntz algebra. We study the fixed point algebra of this action, using Kirchberg classification results and Pimsner algebras. Under certain conditions, we prove that the fixed point algebra is purely infinite and nuclear. We further identify it as a Pimsner algebra, compute its K-theory and prove a "stability property": the fixed points only depend on the CQG via its fusion rules. We apply the theory to SU q (N ) and illustrate by explicit computations for SU q (2) and SU q (3). This construction provides examples of free actions of CQG (or "principal noncommutative bundles"). Subject Classification (2010). 46L80, 19K99, 81R50 Mathematics

Actions of compact quantum groups

“…Conversely, if this identity holds for all π, then the estimates for Tr(F −1 π ) in the proof of Theorem 6. 16 show that (id Hπ ⊗ϕ X )(U * X,π U X,π ) = id Hπ .…”

Fixed Point Algebras for Easy Quantum Groups

“…In the case of easy quantum groups, we may read the fixed point algebras directly from the categories of partitions. The following statement appeared in [27,Proposition 3.4], see also [19,Lemma 2.5].…”

“…In [19], the first author isolated two conditions which turn O α into a Kirchberg algebra. This class of algebras plays a central role in Kirchberg-Phillips classification theory, which proves that Kirchberg algebras are completely classified by their K-groups -see [23,24] for the original papers, [35] for an overview of classification for nuclear simple C * -algebras and [38] for the latest developments in this area.…”

“…This provides an elementary way to recover the result of Example 3.8(a) in this case. In [19], the case of SU q (2) was studied in detail (see…”

“…To compute K * (O α ), we use the inductive system (5.7). We first evaluate K 1 (O α ): according to [19,Theorem 5.4,p. 1025] this K-group is the kernel of the map ϕ defined by the inductive system (5.7).…”