Approximation properties of discrete quantum groups

Abstract: It is a classical result in abstract harmonic analysis, that discrete group G is amenable if and only if its group von Neumann algebra vN(G) has weak* CPAP (completely positive approximation property). There is also a variant of this result for weak amenability: G is weakly amenable if and only if vN(G) has weak* CBAP (completely bounded approximation property). These equivalences remain true also for unimodular discrete quantum groups, which form a class of objects strictly containing discrete groups. It is however an open question, whether approximation properties of vN(G) imply analogous one for G, if G is a non-unimodular quantum group. During the talk I will discuss how one can obtain positive results by considering vN(G) not just as a von Neumann algebra, but as an operator module over $L^1(\hat{G})$. If time permits, I will also discuss a recent result about multiplicativity of Cowling-Haagerup (weak amenability) constant.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( paper | slides | video )

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Noncommutative geometry in NYC

Series comments: Noncommutative Geometry studies an interplay between spatial forms and algebras with non-commutative multiplication. Our seminar welcomes talks in Number Theory, Geometric Topology and Representation Theory linked to the context of Operator Algebras. All talks are kept at the entry-level accessible to the graduate students and non-experts in the field. To join us click meet.google.com/zjd-ehrs-wtx (5 min in advance) and igor DOT v DOT nikolaev AT gmail DOT com to subscribe/unsubscribe for the mailing list, to propose a talk or to suggest a speaker. Pending speaker's consent, we record and publish all talks at the hyperlink "video" on speaker's profile at the "Past talks" section. The slides can be posted by providing the organizers with a link in the format "myschool.edu/~myfolder/myslides.pdf". The duration of talks is 1 hour plus or minus 10 minutes.

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