Noncommutative Analysis

Category: Noncommutative function theory

Souvenirs from the Red River

Last week I attended the annual Canadian Operator Symposium, better known in its nickname: COSY. This conference happens every year and travels between Canadian universities, and this time it was held in the University of Manitoba, in Winnipeg. It was organized by Raphaël Clouâtre and Nina Zorboska, who altogether did a great job.

My first discovery: Winnipeg is not that bad! In fact I loved it. Example: here is the view from the window of my room in the university residence:

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Not bad, right? A very beautiful sight to wake up to in the morning. (I got the impression, that Winnipeg is nothing to look forward to, from Canadians. People of the world: don’t listen to Canadians when they say something bad about any place that just doesn’t quite live up to the standard of Montreal, Vancouver, or Banff.) Here is what you see if you look from the other side of the building: 

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The conference was very broad and diverse in subjects, as it brings together people working in Operator Theory as well as in Operator Algebras (and neither of these fields is very well defined or compact). I have mixed feelings about mixed conferences. But since I haven’t really decided what I myself want to be working on when I grow up, I think they work for me.

I was invited to give a series of three talks that I devoted to noncommutative function theory and noncommutative convexity. My second talk was about my joint work with Guy Salomon and Eli Shamovich on the isomorphism problem for algebras of bounded nc functions on nc varieties, which we, incidentally, posted on the arxiv on the day that the conference began. May I invite you to read the introduction to that paper? (if you like it, also take a look at the previous post).

On this page you can find the schedule, abstract, and the slides of most of the talks, including mine. Some of the best talks were (as it happens so often) whiteboard talks, so you won’t find them there. For example, the beautiful series by Aaron Tikuisis was given like that and now it is gone (George Elliott remarked that a survey of the advances Tikuisis describes would be very desirable, and I agree).

1. The “resolution” of Elliott’s conjecture

Aaron Tikuisis gave a beautiful series of talks on the rather recent developments in the classification theory of separable-unital-nuclear-simple C*-algebras (henceforth SUNS C*-algebras, the algebra is also assumed infinite dimensional, but let’s make that a standing hypothesis instead of complicating the acronym). I think it is fair to evaluate his series of talks as the most important talk(s) in this conference. In my opinion the work (due to many mathematicians, including himself) that Tikuisis presented can be described as the resolution of the Elliott conjecture; I am sure that some people will disagree with the last statement, including George Elliott himself.

Given a SUNS C*-algebra A, one defines its Elliott invariant, E\ell \ell(A), to be the K-theory of A, together with some additional data: the image of the unit of A in K_0(A), the space of traces T(A) of A, and the pairing between the traces and K-theory. It is clear, once one knows a little K-theory, that if A and B are isomorphic C*-algebras, then their Elliott invariants are isomorphic, in the sense that K_i(A) is isomorphic to K_i(B) for i=0,1 (in a unit preserving way), and that T(A) is affinely homeomorphic with T(B) in a way that preserves the pairing with the K-groups. Thus, if two C*-algebras are known to have a different K-group, or a different Elliott invariant, then these C*-algebras are not isomorphic. This observation was used to classify AF algebras and irrational rotation algebras (speaking of which, I cannot help but recommend my friend Claude Schochet’s recent “Notice’s” article on the irrational rotation algebras).

In the 1990s, George Elliott made the conjecture that two SUNS C*-algebras are *-isomorphic if and only if E \ell \ell (A) \cong E \ell \ell (B). This conjecture became one of the most important open problems in the theory of operator algebras, and arguably THE most important open problem in C*-algebras. Dozens of people worked on it. There were many classes of C*-algebras that were shown to be classifiable – meaning that they satisfy the Elliott conjecture – but eventually this conjecture was shown to be false in 2002 by Rordam, who built on earlier work by Villadsen.

Now, what does the community do when a conjecture turns out to be false? There are basically four things to do:

  1. Work on something else.
  2. Start classifying “clouds” of C*-algebras, for example, show that crossed products of a certain type are classifiable within this family (i.e. two algebras within a specified class are isomorphic iff their Elliott invariants are), etc.
  3. Make the class of algebras you are trying to classify smaller, i.e., add assumptions.
  4. Make the invariant bigger. For example, K_0(A) is not enough, so people used K_1(A). When that turned out to be not enough, people started looking at traces. So if the current invariant is not enough, maybe add more things, the natural candidate (I am told) being the “Cuntz Semigroup”.

The choice of what to do is a matter of personal taste, point of view, and also ability. George Elliott has made the point that choosing 4 requires one to develop new techniques, whereas choosing 3 is kind of focused around the techniques, making the class of C*-algebras smaller until the currently known techniques can tackle them.

Elliott’s objections notwithstanding, the impression that I got from the lecture series was that most main forces in the field agreed that following the third adaptation above was the way to go. That is, they tried to prove the conjecture for a slightly more restricted class of algebras than SUNS. Over the past 15 years or so (or a bit more), they identified an additional condition – let’s call it Condition Z – that, once added to the standard SUNS assumptions, allows classification. And it’s not that adding the additional assumptions made things really easy, it only made the proof possible – still it took first class work to even identify what assumption needs to be added, and more work to prove that with this additional assumptions the conjecture holds. They proved:

Theorem (lot’s of people): If A and B are infinite dimensional SUNS C*-algebras, which satisfy the Universal Coefficient Theorem and an additional condition Z, then E\ell \ell (A) \cong E \ell \ell (B) if and only if A \cong B.

I consider this as the best possible resolution of the Elliott conjecture possible, given that it is false!

A major part of Aaron’s talks was to explain to us what this additional condition Z is. (What the Universal Coefficient Theorem though, was not explained and, if I understand correctly, it is in fact not known whether this doesn’t follow immediately for such algebras). In fact, there are two conditions that one can take for “condition Z”: (i) Finite nuclear dimension, and (ii) Z-stability. The notion of nuclear dimension corresponds to the regular notion of dimension (of the spectrum) in the commutative case. Z-stability means that the algebra in question absorbs the Jiang-Su algebra under tensor products in a very strong sense. Following a very long tradition in talks about the Jiang-Su algebra – Aaron did not define the Jiang-Su algebra. This is not so bad, since he did explain in detail what finite nuclear dimension means, and said that Z-stability and finite nuclear dimension are equivalent for infinite dimensional C*-algebras (this is the Toms-Winter conjecture).

What was very nice about Aaron’s series of talks was that he gave von Neumann algebraic analogues of the theorems, conditions, and results, and explained how the C*-algebra people got concrete inspiration from the corresponding results and proofs in von Neumann algebras. In particular he showed the parallels to Connes’s theorem that every injective type II_1 factor with separable predual is isomorphic to the hyperfinite II_1 factor. He made the point that separable predual in the von Neumann algebra world corresponds to separability for C*-algebras, hyperfiniteness corresponds to finite nuclear dimension, and factor corresponds to a simple C*-algebra. He then sketched the lines of the proof of the part of Connes’s theorem that says that injectivity of a II_1 factor M implies hyper-finiteness of M (which by Murray and von Neumann’s work implies  that M is the hyperfinite II_1 factor). After that he repeated a similar sketch for the proof that Z-stability implies finite nuclear dimension.

This lecture series was very inspiring and I think that the organizers made an excellent choice inviting Tikuisiss to give this lecture series.

 

2. Residually finite operator algebras and a new trick

Christopher Ramsey gave a short talk on “residually finite dimensional (RFD) operator algebras”. This talk is based on the paper that Chris and Raphael Clouatre recently posted on the arxiv. The authors take the notion of residual finite dimensional, which is quite well studied and understood in the case of C*-algebras, and develop it in the setting of nonselfadjoint operator algebras. It is worth noting that even a finite dimensional nonselfadjoint operator algebra might fail to be representable as a subalgebra of a matrix algebra. So it is worth specifying that an operator algebra is said to be RFD if it can be completely isometrically embedded in a direct sum of matrix algebras (and so it is not immediate that a finite dimensional algebra is RFD, though they prove that it is).

What I want to share here is a neat and simple observation that Chris and Raphael made, which seemed to have been overlooked by the community.

When we study operator algebras, there are several natural relations by which to classify them: completely isometric isomorphism, unitary equivalence, completely bounded isomorphism, and similarity. Clearly, unitary equivalence implies completely isometric isomorphism, and similarity implies completely bounded isomorphism. The converses do not hold. However, in practice, many times (for example in my recent paper with Guy and Eli) operator algebras are shown to be completely boundedly isomorphic by exhibiting a similarity between them. That happens because we are many times interested in the “multiplicity free case”.

[Added in June 11, following Yemon’s comment: We say that A \subset B(H) is similar to B \subseteq B(K) if there is an invertible T \in B(H,K) such that A = T^{-1}BT. Likewise, two maps \rho : A \to B(H) and \phi: A \to B(K) are said to be similar if there is an invertible T \in B(H,K) such that \rho(a) = T^{-1} \phi(a) T for all a \in A. Paulsen’s theorem says that if \rho : A \to B(H) is a completely bounded representation then it is similar to a completely contractive representation \phi : A \to B(H). ]

Raphael and Chris observed that, in fact, completely bounded isomorphism is the same as similarity, modulo completely isometric isomorphisms. To be precise, they proved:

Theorem (the Clouatre-Ramsey trick): If A and B are completely boundedly isomorphic, then A and B are both completely isometrically isomorphic to algebras that are similar. 

Proof: Suppose that A \subseteq B(H) and B \subseteq B(K). Let \phi : A \to B be a c.b. isomorphism. By Paulsen’s theorem, \phi is similar to a completely contractive isomorphism \psi. So we get that the map

a \mapsto a \oplus \psi(a) \mapsto a \oplus \phi(a) \in B(H) \oplus B(K)

decomposes as a product of a complete isometry and a similarity. Likewise, the completely bounded isomorphism \phi^{-1} is similar to a complete contraction \rho, and we have that

\phi^{-1}(b) \oplus b \mapsto \rho(b) \oplus b \mapsto b

decomposes as the product of a similarity and a complete isometry. Since the composition of all these maps is \phi, the proof is complete.

The perfect Nullstellensatz

Question: to what extent can we recover a polynomial from its zeros?

Our goal in this post is to give several answers to this question and its generalisations. In order to obtain elegant answers, we work over the complex field \mathbb{C} (e.g., there are many polynomials, such as x^{2n} + 1, that have no real zeros; the fact that they don’t have real zeros tells us something about these polynomials, but there is no way to “recover” these polynomials from their non-existing zeros). We will write \mathbb{C}[z] for the algebra of polynomials in one complex variable with complex coefficients, and consider a polynomial as a function of the complex variable z \in \mathbb{C}. We will also write \mathbb{C}[z_1, \ldots, z_d] for the algebra of polynomials in d (commuting) variables, and think of polynomials in \mathbb{C}[z_1, \ldots, z_d] – at least initially – as a functions of the variable z = (z_1, \ldots, z_d) \in \mathbb{C}^d

[Update June 24, 2019: contrary to what I thought, the main theorem presented below holds over arbitrary fields, not just over the complex numbers, very much by the same proof. See this post.]

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Souvenirs from San Diego

Every time that I fly to a conference, I think about the airport puzzle that I once read in Terry Tao’s blog. Suppose that you are trying to get quickly from point A to point B in an airport, and that part of the way has moving walkways, and part of it doesn’t. Suppose that you can either walk or run, but you can only run for a certain small amount of the time. Where is it better to spend that amount of time running: on the moving walkways or in between the moving walkways? Does it matter?

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Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball

Guy Salomon, Eli Shamovich and I recently uploaded to the arxiv our paper “Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball“. This paper blends in with the current growing interest in noncommutative function theory, continues and unifies several strands of my past research.

A couple of years ago, after being inspired by lectures of Agler, Ball, McCarthy and  Vinnikov on the subject, and after years of being influenced by Paul Muhly and Baruch Solel’s work, I realized that many of my different research projects (subproduct systems, the isomorphism problem, space of Dirichlet series with the complete Pick property, operator algebras associated with monomial ideals) are connected by the unifying theme of bounded analytic nc functions on subvarieties of the nc ball. “Realized” is a strong word, because many of my original ideas on this turned out to be false, and others I still don’t know how to prove. Anyway, it took me a couple of years and a lot of help, and here is this paper.

In short, we study algebras of bounded analytic functions on subvarieties of the the noncommutative (nc) unit ball :

\mathfrak{B}_d = \{(X_1, \ldots, X_d) tuples of n \times n matrices,  \sum X_i X_i < I\}

as well as bounded analytic functions that extend continuously to the “boundary”. We show that these algebras are multiplier algebras of appropriate nc reproducing kernel Hilbert spaces, and are completely isometrically isomorphic to the quotient of H^\infty(\mathfrak{B}_d) (the bounded nc analytic functions in the ball) by the ideal of nc functions vanishing on the variety. We classify these algebras in terms of the varieties, similar to classification results in the commutative case. We also identify previously studied algebras (such as multiplier algebras of complete Pick spaces and tensor algebras of subproduct systems) as algebras of bounded analytic functions on nc varieties. See the introduction for more.

We certainly plan to continue this line of research in the near future – in particular, the passage to other domains (beyond the ball), and the study of algebraic/bounded isomorphisms.

Souvenirs from Amsterdam

(I am writing a post on hot trends in mathematics in the midst of war, completely ignoring it. This seems like the wrong thing to do, but my urge to write has overcome me. To any reader of this blog: I wish you a peaceful night, wherever you are).

Last week I returned from the yearly “International Workshop on Operator Theory and Applications”, IWOTA 2014 for short (see the previous post for the topic of my own talk, or this link for the slides).

This conference was very broad (and IWOTA always is). One nice thing about broad conferences is that you are able sometimes to identify a growing trend. In this talk I got particularly excited by a series of talks on “noncommutative function theory” or “free analysis”. There was a special session dedicated to this topic, but I was mostly inspired by a semi-plenary talk by Jim Agler, and also by two interesting talks by Joe Ball and Spela Spenko. I also attended nice talks related to this subject by Victor Vinnikov, Dmitry Kalyuhzni-Verbovetskyi, Baruch Solel, Igor Klep and Bill Helton. This topic has attracted the attention of many operator theorists, for its applications as well as for its inherent beauty, and seems to be accelerating in the last several years; I will only try to give a taste of some neat things that are going on, by telling you about Agler’s talk. What I will not be able to do is to convey Agler’s intense and unique mathematical charisma.

Here is the program of the conference, so you can check out other things that were going on there.

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