Topological group actions by group automorphisms and Banach representations
Seminar
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,
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.
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.
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
