Per il ciclo “Crossings - Math Seminars”, Michael Cowling (University of New South Wales) terrà i seguenti seminari:
#1
Titolo: Admissibility of uniformly bounded representations of $SL(2,R)$ on Hilbert spaces
Data e ora: Lunedì 10 Giugno, ore 14:30
Luogo: Edificio U1-01 (Piano -1), Quadrilatero della Scienza, Università di Milano-Bicocca
Seguirà un coffee break
#2
Titolo: What else do we know about uniformly bounded representations on Hilbert spaces?
Data e ora: Giovedì 13 Giugno, ore 14:30
Luogo: Edificio U3-05 (Piano -1), Quadrilatero della Scienza, Università di Milano-Bicocca
These presentations are based on work in collaboration with Francesca Astengo (Genova) and Bianca di Blasio (Milano Bicocca). This work will be integrated into its historical and mathematical context.
From about 1950, there has been considerable study of the representations $\pi$ of the group $G := SL(2,R)$, andmore generally those of semisimple Lie groups, on Hilbert spaces $\cH$. Some of these representations areisometric, that is, $\| \pi(g)\xi \| = \| \xi \|$ for all $g \in G$ and $\xi \in \cH$. Others are uniformly bounded, that is, theoperator norms of the $\pi(g)$ are uniformly bounded in $G$.
An early question (1950) of Dixmier was whether every uniformly bounded representation of a locally compact groupis similar to a unitary representation; a counterexample was found by Ehrenpreis and Mautner in 1955. During thissame period, Harish-Chandra introduced the concept of admissible representation (which means essentiallyaccessible with the tools of algebra) and showed that irreducible unitary representations are admissible. In 1988 Soergel constructed an example of a nonadmissible isometric representation of $G$ on a Banach space, using thenegative solution of the so-called invariant subspace problem in Banach spaces (due to Enflo and Read). Recentlytwo proposed solutions of the same problem in Hilbert spaces (by Enflo and by Neville have appeared; these bothclaim that the invariant subspace problem has a positive solution.
In the first talk, we show that if the proposed solutions to the invariant subspace problem in Hilbert spaces are valid, then uniformly bounded representations of $SL(2,R)$ are admissible.
In the second, we ask what uniformly bounded representations are good for. We present a summary of known resultsabout them and mention a few of their applications (one of which is still conjectural).
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