Functional and Harmonic Analysis
Org:
Benjamin AndersonSackaney and
Ebrahim Samei (University of Saskatchewan)
[
PDF]
 BENJAMIN ANDERSONSACKANEY, University of Saskatchewan
Tracial States on Quantum Group C*algebras [PDF]

When working with the tracial states on a group $C^*$algebra $C^*_\pi(G)$ of a group $G$, an indispensable fact is the observation that the tracial states on $C^*_\pi(G)$ are exactly the states that are invariant with respect to the conjugation action of $G$ on $C^*_\pi(G)$. An analogous observation for discrete quantum groups had been missing until quite recently: it was established for unimodular discrete quantum groups in a recent paper by Kalantar, Kasprzak, Skalski, and Vergnioux. In this talk we will present a generalization of this result for arbitrary discrete quantum groups and discuss various consequences of this result on the reduced $C^*$algebras of discrete quantum groups.
 FINLAY RANKIN, Carleton
Quantum automorphisms of commuting squares [PDF]

Banica defined a compact quantum group of automorphisms for an inclusion of finitedimensional \(C^\ast\)algebras and determined its representation theory in certain cases. We generalize Banica's work and assign a compact quantum group of automorphisms to a nondegenerate commuting square consisting of finitedimensional \(C^\ast\)algebras and show that it can be realized as a generalized Drinfeld double. Finally, we discuss the representation theory in special cases.
 PAWEL SARKOWICZ, University of Waterloo
Embeddings of unitary groups [PDF]

We discuss unitary groups of C*algebras with a focus on group homomorphisms between them, and how such homomorphsisms give relationships between the Ktheory and traces. With this information, one can use the stateoftheart Ktheoretic classification of embeddings to conclude that there are certain embeddings between C*algebras if and only if there are appropriate embeddings between their unitary groups.
 ERIK SEGUIN, University of Waterloo
Amenability and stability for discrete groups [PDF]

The notion of a representation of a group $G$ on a Hilbert space $\mathcal{H}$ can be generalized to that of an “approximate representation”, in which the usual homomorphism condition $\varphi(xy)=\varphi(x)\,\varphi(y)$ is replaced by some upper bound on $\lVert\varphi(xy)\varphi(x)\,\varphi(y)\rVert$. The supremum over all $x,y\in G$ of this quantity is referred to as the “defect” of the map $\varphi$ and measures how far $\varphi$ is from being a genuine representation. It is natural to ask about the stability of this class of maps: namely, when the defect of $\varphi$ is small, under what conditions is it wellapproximated by a genuine representation of $G$? We discuss the connection between amenability and stability of approximate representations for discrete groups.