Copenhagen
Groups and Operator Algebras Seminar

Abstract:  Given a partial dynamical system, one can construct its transformation groupoid. We characterize when two transformation groupoids are isomorphic in terms of the continuous orbit equivalence of their corresponding partial dynamical systems. Under certain additional conditions, the isomorphism between these groupoids is also characterized via the isomorphism of their corresponding crossed products. Furthermore, we provide several necessary and sufficient conditions for a groupoid to be isomorphic to a transformation groupoid, and apply our results to the case of orthogonal and saturated partial actions. (Joint work with Eun Ji Kang)

Abstract:  Tracially complete C*-algebras are a generalization of Ozawa's W*-bundles introduced by J.Carrión--Castillejos--Evington--Gabe--Schafhauser--Tikuisis--White as a vehicle for transferring ideas and results about (tracial) von Neumann algebras to C*-algebras.  This transfer can be achieved in a uniform way, and that is what this talk will be about. This is a joint work with Hart, Hirshberg, Schafhauser, Tikuisis, and Vaccaro.

Abstract:  Recently, strong asymptotic freeness, or selflessness, in C*-algebras has emerged as a powerful technique to prove important regularity properties including simplicity, unique trace, stable rank 1, and strict comparison. In particular, in Fall 2024, Amrutam, Gao, Kunnwalkam Elayavalli, and I showed that the reduced group C*-algebras of all hyperbolic groups with trivial finite radical are selfless, which resolved the open problem of strict comparison for the reduced group C*-algebra of the free group on two generators. Since then, our result has been expanded to include a much larger class of groups. Work has also begun on isolating selflessness for C*-algebras not arising from groups, including the result of Hayes, Kunnawalkam Elayavalli, and Robert on selflessness of the reduced free product of a large class of C*-algebras. I will introduce the general notion of a selfless inclusion of C*-algebras,with which we will see the selflessness of the reduced unitary compact matrix quantum groups. This work is joint with Ben Hayes, Srivatsav Kunnawalkam Elayavalli, and Leonel Robert.

Abstract:  Wavelets and the wavelet transform were formulated in the 1980s; they brought together in one concept ideas from many fields, including quantum mechanics, harmonic analysis of singular operators, geophysics, computer vision, and electrical engineering. The synthesis of these ideas into one family of tools led to applications in many directions, ranging from astrophysics to medical imaging and sports transmissions. The presentation will review the history as well as some of the applications.

Abstract:  I will describe a general setup that allows expressing the value of a quantum-to-quantum non-local game with respect to a certain quantum resource as the norm of an associated game tensor when the latter is viewed as an element of a canonical operator space based on the trace class operators, induced by the given resource. I will apply the setup in particular to the case of the resource of all quantum commuting no-signalling correlations, which will permit obtaining formulas for the quantum commuting value of a game as a maximal operator space tensor product norm. The talk is based on a joint work with J. Crann, R. Levene and L. Turowska.

Abstract:  In this talk, we discuss the problem of characterizing of the Sobolev wavefront set of a tempered distribution in terms of its continuous wavelet transform, for higher dimensional continuous wavelet transforms constructed using square-integrable representations of a given semidirect product group ${\mathbb R}^n\rtimes H$ where $H$ can be any suitably chosen dilation group. Two important cases will be examined, where 1) the mother wavelet is compactly supported, and 2) the mother wavelet has compactly supported Fourier transform. This talk is based on joint work with Hartmut Fuhr.

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Abstract:  A group G is called W*-superrigid if it is completely remembered by its group von Neumann algebra in the following sense: any other group with the same group von Neumann algebra must be isomorphic to G. By now, there are many examples of W*-superrigid groups. Of course, a group von Neumann algebra is also a compact quantum group, hence one can ask whether superrigidity also holds in this larger class. Surprisingly, many of the typical W*-superrid groups in the literature fail to remain superrigid as quantum groups. I will present a recent joint work with Stefaan Vaes where we prove certain (cohomological) obstructions to `quantum W*-superrigidity'. We also build on previous work to find the first examples of discrete groups and (non-cocommutative) compact quantum groups which are quantum W*-superrigid.

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