Réunion d'été SMC 2024
Saskatchewan, 2 mai - 3 juin 2024
Functional and Harmonic Analysis
Org: Benjamin Anderson-Sackaney et Ebrahim Samei (University of Saskatchewan)
[PDF]
Org: Benjamin Anderson-Sackaney et Ebrahim Samei (University of Saskatchewan)
[PDF]
- BENJAMIN ANDERSON-SACKANEY, University of Saskatchewan
Tracial States on Quantum Group C*-algebras [PDF]
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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]
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Banica defined a compact quantum group of automorphisms for an inclusion of finite-dimensional \(C^\ast\)-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 finite-dimensional \(C^\ast\)-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]
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We discuss unitary groups of C*-algebras with a focus on group homomorphisms between them, and how such homomorphsisms give relationships between the K-theory and traces. With this information, one can use the state-of-the-art K-theoretic 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]
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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 well-approximated by a genuine representation of $G$? We discuss the connection between amenability and stability of approximate representations for discrete groups.