Noncommutative Analysis

Seminar talk by Pandey – Distance between reproducing kernel Hilbert spaces and geometry of finite sets in the unit ball

In our next Operator Algebras/Operator Theory Seminar, Satish Pandey will present our recently published online paper (together with Danny Ofek and myself) “Distance between reproducing kernel Hilbert spaces and geometry of finite sets in the unit ball” (arxiv version).

Time: 15:30-16:30

Date: May 6th, 2021

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


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 one of the underlying point sets is 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.

This is joint work with Danny Ofek and Orr Shalit.

If you are interested in the zoom link, let me know.

William Arveson

I came back to this old post, and noticed that it is almost ten years since Bill Arveson passed away. It’s hard to believe.

Noncommutative Analysis

William B. Arveson was born in 1934 and died last year on November 15, 2011. He was my mathematical hero; his written mathematics has influenced me more than anybody else’s. Of course, he has been much more than just my hero, his work has had deep and wide influence on the entire operator theory and operator algebras communities. Let me quickly give an example that everyone can appreciate: Arveson proved what may be considered as the “Hahn-Banach Theorem” appropriate for operator algebras. He did much more than that, and I will expand below on some of his early contributions, but I want to say something before that on what he was to me.

When I was a PhD student I worked in noncommutative dynamics. Briefly, this is the study of actions of (one-parameter) semigroups of *-endomorphisms on von Neumann algebras (in short E-semigroups). The definitive book on this subject is…

View original post 1,794 more words

Seminar talk by Viselter – Quantum groups: constructions and lattices

Our speaker for next Thursday’s Operator Algebras/Operator Theory Seminar is Ami Viselter (Haifa University). 

Time: 15:30-16:30 Thursday, February 18, 2021

Title: Quantum groups: constructions and lattices

Abstract: We will present a few constructions of locally compact quantum groups, and relate them to structural notions such as lattices and unimodularity, as well as to property (T).
Zoom link:

Seminar talk by Hartz: How can you compute the multiplier norm?

Happy new year!

Next Thursday, January 7th, 2021, Michael Hartz will speak in our Operator Algebras and Operator Theory seminar.

Title: How can you compute the multiplier norm?

Time: 15:30-16:30

Zoom link: Email me.


Multipliers of reproducing kernel Hilbert spaces arise in various contexts in operator theory and complex analysis. A basic example is the Hardy space H^2, whose multiplier algebra is H^\infty, the algebra of bounded holomorphic functions. In particular, the norm of a multiplier on H^2 is the pointwise supremum norm. 

For general reproducing kernel Hilbert spaces, the multiplier norm can be computed by testing positivity of n \times n matrices analogous to the classical Pick matrix. For H^2, n=1 suffices. I will talk about when it suffices to consider matrices of bounded size n. Moreover, I will explain how this problem is related to subhomogeneity of operator algebras.

This is joint work with Alexandru Aleman, John McCarthy and Stefan Richter

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


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)


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

Zoom link: email me.


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.