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…]]>

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.

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…

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**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:

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.

**Abstract:**

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

For general reproducing kernel Hilbert spaces, the multiplier norm can be computed by testing positivity of matrices analogous to the classical Pick matrix. For , suffices. I will talk about when it suffices to consider matrices of bounded size . 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

]]>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.

]]>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.

**Title: Combinatorial and operator algebraic aspects of proximal actions**

**Speaker: Guy Salomon** (Weizmann Institute)

**Time: **15:30-16:30,Thursday Nov. 12, 2020

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

**Zoom link:** email me.

**Abstract:**

An action of a discrete group on a compact Hausdorff space is called * proximal* if for every two points and of there is a net such that , and

In this talk I will present relations between some fundamental operator theoretic concepts to proximal and strongly proximal actions, and hence to amenable and strongly amenable groups. In particular, I will focus on the C*-algebra of continuous functions over the universal minimal proximal -flow and characterize it in the category of -operator-systems.

I will then show that nontrivial proximal actions of can arise from partitions of into a certain kind of “large” subsets. If time allows, I will also present some relations to the Poisson boundaries of . The talk is based on a joint work with Matthew Kennedy and Sven Raum.

]]>The two excellent students that worked with us – Danny Ofek and Gilad Sofer – got some nice results. They almost solved to a large extent the main problems mentioned in my earlier post. See this poster for a concise summary of the main results:

Danny and Gilad summarized their results in the following paper. Just take a look. They have some new results that I thought were true, they have some new results that I didn’t guess were true, and they also have some new and simplified proofs for a couple of known results. Their work fits in the long term research project to discover how the structure of Hilbert function spaces and their multiplier algebras encodes the underlying structures, and especially the geometry of sets in the unit disc or the unit ball. More on that soon!

]]>The breakthrough result that every complete Nevanlinna-Pick space has the column-row property is one of his latest results and has appeared on the arxiv this May. Besides solving an interesting open problem, it is a really elegant and strong paper.

It is satisfying to see a young and very talented mathematician get recognition!

**Full disclosure ** Michael is a sort of mathematical relative (he was a PhD student of my postdoc supervisor Ken Davidson), a collaborator (together with Ken Davidson we wrote the paper Multipliers of embedded discs) and a friend. I have to boast that from the moment that I heard about him I knew that he will do great things – in his first paper, which he wrote as a masters student, he ingeniously solved an open problem of Davidson, Ramsey and myself. Since then he has worked a lot on some problems that are close to my interests, and I have been following him with admiration.

Congratulations Michael!

]]>Suppose that is a reproducing kernel Hilbert space on a space (I will assume knowledge in reproducing kernel Hilbert spaces (aka Hilbert function spaces) – see this old post for a crash introduction to the subject. ) The theory of reproducing kernel Hilbert spaces makes a connection between function theory, on the one hand, and Hilbert spaces and operator theory, on the other. The function theory captures features of the space , while the Hilbert space allows us to attach various functional analytic structures. The goal of the project is to explore the relationship between the various ingredients appearing above: the geometry/topology of and its subsets, the function theory on , the Hilbert function space structure of subspaces of and of , and maybe also operator algebras acting on these subspaces.

**Circle of problems 1: When are Hilbert function spaces “the same”?**

Let be a sequence of positive numbers, and consider the space of power series

.

One can show that (under some mild assumption) every element in is an analytic function in the unit disc, and that when one defines the inner product

then is a RKHS. If for all then we get the Hardy space , if then we get the Bergman space .

The sequence determines the space , and one function theoretic aspect encoded by is the rate of convergence of the coefficients, and in turn the “regularity” of functions in the space at the boundary.

As a warmup one can try to understand for which sequences does this construction work, and what is the reproducing kernel. The main problems to treat here are then to find when these spaces are “the same” in some sense. We need to explain what we mean by “the same”.

All Hilbert spaces (of the same dimension) are the same: there is alway a unitary – a linear isomorphism that preserves inner products – between them. However, when speaking of Hilbert function spaces, we wish our linear isomorphisms to preserve also the function-space structure. So, we will consider “natural” maps between Hilbert spaces (and also between their multiplier algebras).

If () are Hilbert spaces on sets with kernels , then there are several kinds of natural linear maps that in some sense preserve the structure of these spaces as RKHSs.

In the case that , one kind of natural map is multiplication: , where is a function. Another natural map is a composition operator , where is a map. It is not a trivial matter to decide which maps give rise to well defined composition operators, and then it is also quite a serious problem to be able to determine properties of the composition operator from the properties of the map . Composition operators can also be considered between multiplier algebras, and it is an interesting problem, with no known general answer, whether composition operators between the RKHS correspond to composition operators between the multiplier algebras.

A somewhat more flexible kind of “natural” operator acting between Hilbert function spaces is a * weighted composition operator*, which is nothing but the composition of a a composition operator and a multiplication operator, that is, a map that has the form

.

Recalling that an RKHS on a set is determined by the kernel functions for , and is the closed linear span of these kernel functions, it is natural to look at maps that send kernel functions to kernel functions, or at least to scalar multiples of kernel functions. So, we consider “diagonal” operators of the form

(*)

where is a family of scalars, and . Let us say that a linear map is a an ** isomorphism of Hilbert function spaces** if it is a bounded bijective linear map of the form (*). If the map is isometric, then we say that it is an

I leave it to the reader to figure out what is the relation between weighted composition operators and operators of the form given by equation (*).

**Concrete problems:**

- For which sequences is a RKHS? What is the kernel? One can also ask, when does the Hilbert function space consist of continuous functions on the closed disc?
- For which of those sequences , are the composition operators (for ) automorphisms of (isometric or not)? DOes the situation change if we allow for weighted composition operators?
- Determine, given sequences and , whether and isometric as Hilbert function spaces? In particular, for what sequences is the generalized Hardy space isomorphic to the classical Hardy space?
- Questions 2 and 3 can also be asked for the multiplier algebras: when do the automorphisms of the disc give rise to automorphisms of the multiplier algebras? And under what circumstances are the multiplier algebras and (isometrically) isomorphic?

**Circle of problems 2: Geometry and Hilbert spaces structure.**

Let be a subset of (a set on which the RKHS with kernel lives). We can form . Our second circle of problems is around the question: *how does the geometry of reflect in the structure of *. We will focus on the case that is a finite set.

Sometimes, the set has some structure to begin with. For example, if is the Hardy space , then . has the natural Euclidean metric defined on it

as well as the *pseudohyperbolic metric*

which is in some sense more natural – for example, an analytic self map of the disc is a conformal automorphism if and only if it is an isometry with respect to the pseudohyperbolic metric.

However, whether or not carries some structure to begin with, the Hilbert function space induces various metrics on it. For example, we can define

.

Another, slightly different metric is given by

,

where is the orthogonal projection on the one dimensional subspace spanned by . It can be shown (really can!) that

.

It is interesting to note that in the case of the Hardy space, we get

(see this paper by Arcozzi, Rochberg, Sawyer and Wick for hints about the above two computation jumps). So this shows that was a useful choice of metric, or looking at it the other way around, that the pseudohyperbolic metric is a natural one for when studying the Hardy space.

**Concrete problems:**

- Determine when, given , are and isometrically isomorphic. Do the same for multiplier algebras.
- When and are finite, it is clear that and are isomorphic if and only if and have the same number of points in them. However, when and are infinite, then determining when the Hilbert function spaces are isomorphic is an interesting and difficult problem.
- The above two problems might have a more substantial answer if we consider a certain space, such as the Hardy space or the Bergman space . We actually know the answer for finite in the case of the Hardy space, but we think that it could be a great problem to start the project with. The case of the Bergman space, as far as I know, is open.
- There are higher dimensional versions of the above problems, where the hardy space on the disc is replaced by the
space on the unit ball in (see these two older posts: one, two), or where the Bergman space on the unit disc is replaced by the Bergman space on the unit ball.*Drury-Arveson* - Finally, there is a quantitative version of the above problems, that we may describe during the week, depending on the progress made.

That’s all for now. The plan is the the students will think about these problems, choose one or two, or something related, and do their best in solving them. I hope to report on interesting discoveries after the week is over.

]]>Here is a link to the talk:

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