Noncommutative Analysis

Month: November, 2020

Seminar talk by Dor-On: Quantum symmetries in the representation theory of operator algebras

NOTE: THE SEMINAR WAS POSTPONED TO DECEMBER 10.

On next Thursday the Operator Algebras and Operator Seminar will convene for a talk by Adam Dor-On.

Title: Quantum symmetries in the representation theory of operator algebras

Speaker: Adam Dor-On (University of Illinois, Urbana-Champaign)

Time: AFTERNOON Thursday Dec. 10, 2020 (NOTE: THE SEMINAR WAS POSTPONED BY ONE WEEK FROM ORIGINAL DATE).

(Zoom room will open about ten minutes earlier, and the talk will begin at 15:30)

Zoom link: email me.

Abstract:

We introduce a non-self-adjoint generalization of Quigg’s notion of coaction of a discrete group G on a C*-algebra. We call these coactions “quantum symmetries” because from the point of view of quantum groups, coactions on C*-algebras are just actions of a quantum dual group of G on the C*-algebra. We introduce and develop a compatible C*-envelope, which is the smallest C*-coaction system which contains a given operator algebra coaction system, and we call it the cosystem C*-envelope.

It turns out that the new point of view of quantum symmetries of non-self-adjoint algebras is useful for resolving problems in both C*-algebra theory and non-self-adjoint operator algebra theory. We use quantum symmetries to resolve some problems left open in work of Clouatre and Ramsey on finite dimensional approximations of representations, as well as a problem of Carlsen, Larsen, Sims and Vitadello on the existence of a co-universal C*-algebra for product systems over arbitrary right LCM semigroup embedded in groups. This latter problem was resolved for abelian lattice ordered semigroups by the speaker and Katsoulis, and we extend this to arbitrary right LCM semigroups. Consequently, we are also able to extend the Hao-Ng isomorphism theorems of the speaker with Katsoulis from abelian lattice ordered semigroups to arbitrary right LCM semigroups.

*This talk is based on two papers. One with Clouatre, and another with Kakariadis, Katsoulis, Laca and X. Li.

New paper: Distance between reproducing kernel Hilbert spaces and geometry of finite sets in the unit ball

Danny Ofek, Satish Pandey and I just uploaded our new paper “Distance between reproducing kernel Hilbert spaces and geometry of finite sets in the unit ball” to the arxiv. This papers fits into my ongoing work on the isomorphism problem for complete Pick algebras, but it raises a very fundamental question that I think is worth highlighting.

As in other subjects of mathematics, when working on Hilbert function spaces, one sometimes asks very basic questions, such as: when are two Hilbert function spaces the same? what is the “true” set on which the functions in a RKHS are defined? (see Section 2 in this paper) or what information is encoded in a space or its multiplier algebra? (see the “road map” here). The underlying questions behind our new paper are when are two Hilbert function spaces “almost” the same and what happens if you change a Hilbert function space “just a little bit”? If these sound like interesting questions, then I suggest you take a look at the paper’s introduction.

Here is the abstract:

In this paper we study the relationships between a reproducing kernel Hilbert space, its multiplier algebra, and the geometry of the point set on which they live. We introduce a variant of the Banach-Mazur distance suited for measuring the distance between reproducing kernel Hilbert spaces, that quantifies how far two spaces are from being isometrically isomorphic as reproducing kernel Hilbert spaces. We introduce an analogous distance for multiplier algebras, that quantifies how far two algebras are from being completely isometrically isomorphic. We show that, in the setting of finite dimensional quotients of the Drury-Arveson space, two spaces are “close” to one another if and only if their multiplier algebras are “close”, and that this happens if and only if the underlying point-sets are “almost congruent”, meaning that one of the sets is very close to an image of the other under a biholomorphic automorphism of the unit ball. These equivalences are obtained as corollaries of quantitative estimates that we prove.