Topological group actions by group automorphisms and Banach representations

Speaker
Michael Megrelishvili, BIU
Date
22/05/2022 - 13:00 - 12:00Add to Calendar 2022-05-22 12:00:00 2022-05-22 13:00:00 Topological group actions by group automorphisms and Banach representations   To every Banach space V one may associate a continuous dual action of the topological group Iso(V)  of all linear isometries on the weak-star compact unit ball B* of the dual space V*.   Which actions G x X --> X are "subactions" of Iso(V)  x B* --> B* for nice Banach spaces V ?  We study Banach representability for actions of topological groups on groups by automorphisms;  in particular, an action on itself by conjugations.  The natural question is to examine when we can find representations on low complexity Banach spaces.   In contrast to the standard left action of a locally compact second countable group G on itself,  the conjugation action need not be reflexively representable even for SL_2(R).   The conjugation action of SL_n(R) is not Asplund representable for every n > 3. The linear action of GL_n(R) on R^n, for every n  > 1, is not representable on Asplund Banach spaces.  On the other hand, this action is representable on a Rosenthal Banach space (not containing an isomorphic copy of l_1).   The conjugation action of a locally compact group need not be Rosenthal representable (even for Lie groups). This is unclear for SL_2(R). As a byproduct we obtain some counterexamples about Banach representations of homogeneous G-actions G/H.  Seminar rm, Math Bldg 216 אוניברסיטת בר-אילן - Department of Mathematics mathoffice@math.biu.ac.il Asia/Jerusalem public
Place
Seminar rm, Math Bldg 216
Abstract

 

To every Banach space V one may associate a continuous dual action of the topological group Iso(V) 
of all linear isometries on the weak-star compact unit ball B* of the dual space V*.  
Which actions G x X --> X are "subactions" of Iso(V)  x B* --> B* for nice Banach spaces V ? 
We study Banach representability for actions of topological groups on groups by automorphisms; 
in particular, an action on itself by conjugations. 
The natural question is to examine when we can find representations on low complexity Banach spaces.  
In contrast to the standard left action of a locally compact second countable group G on itself, 
the conjugation action need not be reflexively representable even for SL_2(R).  
The conjugation action of SL_n(R) is not Asplund representable for every n > 3.
The linear action of GL_n(R) on R^n, for every n  > 1, is not representable on Asplund Banach spaces. 
On the other hand, this action is representable on a Rosenthal Banach space (not containing an isomorphic copy of l_1).  
The conjugation action of a locally compact group need not be Rosenthal representable (even for Lie groups). This is unclear for SL_2(R).
As a byproduct we obtain some counterexamples about Banach representations of homogeneous G-actions G/H. 

Last Updated Date : 18/05/2022