[Settfa] Tuesday 13/06, Michal Doucha

[kubis at math.cas.cz]

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 Tuesday 13th June, 10:00am 
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 Speaker:Michal Doucha, IM CAS
 Title: Generic unitary representations of groups and C*-algebras

 Abstract  
For a fixed countable group G we consider the Polish space of all unitary representations of G (in a separable infinite-dimensional Hilbert space). We are interested in "generic" properties of elements of this space. In particular, we are interested whether there is a unitary representation of G whose equivalence class is comeager.

We show that whenever G has the Haagerup property and its full group C*-algebra is residually finite-dimensional, then equivalence classes of all unitary representations of G are meager. This extends the known results for free groups. We also generalize this result to several classes of C*-algebras, i.e. show the equivalence classes of representations of these C*-algebras are meager.

On the other hand, when G has the Kazhdan's property T and its full group C*-algebra is residually finite-dimensional we show that under some unproved but plausible condition on irreducible representations of G, G has a comeager equivalence class. We show that in this case that however leads to the conclusion that G is finite. That would negatively answer a certain question of Lubotzky and Shalom about Kazhdan groups.
This is joint work in progress with Maciej Malicki.
 

 For more information see the seminar web page at 
 https://calendar.math.cas.cz/set-theory-and-analysis-actual .