| Date: | June 18, 2020 |
| Time: | 4:00pm |
| Location: | online |
| Speaker: | Alejandro Chavez-Domınguez, University of Oklahoma |
| Title: | Asymptotic dimension and coarse embeddings in the quantum setting |
| Abstract: | We generalize the notions of asymptotic dimension and coarse embeddings from metric spaces to quantum metric spaces in the sense of Kuperberg and Weaver. We show that quantum asymptotic dimension behaves well with respect to metric quotients, direct sums, and quantum coarse embeddings. Moreover, we prove that a quantum metric space that equi-coarsely contains a sequence of reflexive quantum expanders must have infinite asymptotic dimension. This is done by proving a vertex-isoperimetric inequality for quantum expanders, based upon a previously known edge-isoperimetric one due to Temme, Kastoryano, Ruskai, Wolf, and Verstraete. Joint work with Andrew Swift. |
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| Date: | July 22, 2020 |
| Time: | Noon |
| Location: | online |
| Speaker: | Peter Kuchment, Texas A&M University |
| Title: | Extensions of group representations and functor Ext^1 in the category of Banach spaces |
| Abstract: | Let one have a representation T of a group G in a HIlbert space H and E be a closed invariant subspace. Then T generates a sub-representation T_1 in E, as well as quotient-representation T_2 in the quotient space F:=H/E. What additional information, besides T_1 and T_2 is needed to recover the presentation T? The answer in the finite-dimensional or in HIlbert spaces case is well known: one needs a group cohomology class h in H^1_G(L(F,E)), where L(F,E) is the space of bounded linear operators from F to E. This result hinges upon existence of a complement to the space E in H. However, it is known that in any Banach space that is not isomorphic to Hilbert one, there exist non-complemented subspaces. This makes the problem (formulated 50 years ago by A. Kirillov in his group representation textbook) non-trivial even for trivial group actions.
The speaker announced without proof 45 years ago a solution of this problem. It also entailed studying the functor Ext^1 in the category of Banach spaces. The author's work on this, besides the Kirillov's problem, was triggered by the wonderful paper by Enflo, Lindenstrauss, and Pisier on extensions of Hilbert spaces. A few years later, starting with the work by N. Kalton, such a study, named "the three-space problem" started flourishing and has been developed significantly since. However, approaches to and results on Ext^1 of the speaker's paper mostly have not been rediscovered, to the best of the author's knowledge.
The talk will contain the results and sketches of the proofs. (A preprint is available on arXiv.) |
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