Last week I attended a workshop titled “Hilbert modules and complex geometry” in MFO (Oberwolfach). In this post I wish to tell about some interesting things that I have learned. There were many great talks to choose from. Below is a sample, in short form, with links.
Category Archives: Operator algebras
K-spectral sets and the holomorphic functional calculus
In two previous posts I discussed the holomorphic functional calculus as part of a standard course in functional analysis (lectures notes 18 and 19). In this post I wish to discuss a slightly different approach, which relies also on the notion of K-spectral sets, and relies a little less on contour integration of Banach-space valued functions.
In my very personal opinion this approach is a little more natural then the standard one, and it would be even more natural if one was able to altogether remove the dependence on Banach-space valued integrals (unfortunately, right now I don’t know how to do this completely).
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Major advances in the operator amenability problem
Laurent Marcoux and Alexey Popov recently published a preprint, whose title speaks for itself :”Abelian, amenable operator algebras are similar to C*-algebras“. This complements another recent contribution, by Yemon Choi, Ilijas Farah and Narutaka Ozawa, “A nonseparable amenable operator algebra which is not isomorphic to a C*-algebra“.
The open problem that these two papers address is whether every amenable Banach algebra, which is a subalgebra of $latex B(H)$, is similar to a (nuclear) C*-algebra. As the titles clearly indicate (good titling!), we now know that an abelian amenable operator algebra is similar to a C*-algebra, and on the other hand, that a non-separable, non-abelian operator algebra is not necessarily similar to a C*-algebra.
I recommend reading the introduction to the Marcoux-Popov paper (which is very friendly to non-experts too) to get a picture of this problem, its history, and an outline of the solution.
Essential normality, essential norms and hyper rigidity
Matt Kennedy and I recently posted on the arxiv a our paper “Essential normality, essential norms and hyper rigidity“. This paper treats Arveson’s conjecture on essential normality (see the first open problem in this previous post). From the abstract:
Let $latex S = (S_1, ldots, S_d)$ denote the compression of the $latex d$-shift to the complement of a homogeneous ideal $latex I$ of $latex mathbb{C}[z_1, ldots, z_d]$. Arveson conjectured that $latex S$ is essentially normal. In this paper, we establish new results supporting this conjecture, and connect the notion of essential normality to the theory of the C*-envelope and the noncommutative Choquet boundary.
Previous works on the conjecture verified it for certain classes of ideals, for example ideals generated by monomials, principal ideals, or ideals of “low dimension”. In this paper we find results that hold for all ideals, but – alas! – these are only partial results.
Denote by $latex Z = (Z_1, ldots, Z_d)$ the image of $latex S$ in the Calkin algebra (here as in the above paragraph, $latex S$ is the compression of the $latex d$-shift to the complement of an ideal $latex I$ in $latex H^2_d$). Another way of stating Arveson’s conjecture is that the C*-algebra generated by $latex Z$ is commutative. This would have implied that the norm closed (non-selfadjoint) algebra generated by $latex Z$ is equal to the sup-norm closure of polynomials on the zero variety of the ideal $latex I$. One of our main results is that we are able to show that the non-selfadjoint algebra is indeed as the conjecture predicts, and this gives some evidence for the conjecture. This is also enough to obtain a von Neumann inequality on subvarieties of the ball, what would have been a consequence of the conjecture being true.
Another main objective is to connect between essential normality and the noncommutative Choquet boundary (see this and this previous posts). A main result here is we have is that the tuple $latex S$ is essentially normal if and only if it is hyperrigid (meaning in particular that all irreducible representations of $latex C^*(S)$ are boundary representations).
Another one bites the dust (actually many of them)
Boom. In the arxiv mailing list of a few days ago appeared the following paper: “Interlacing Families II: Mixed Characteristic Polynomials and The Kadison-Singer Problem” (Markus, Spielman and Srivastava). The abstract says:
We use the method of interlacing families of polynomials to prove Weaver’s conjecture KS2, which is known to imply a positive solution to the Kadison-Singer problem via a projection paving conjecture of Akemann and Anderson. Our proof goes through an analysis of the largest roots of a family of polynomials that we call the “mixed characteristic polynomials” of a collection of matrices.
From the abstract it might not be immediately clear that this paper claims to solve the Kadison-Singer problem, because it says that their result implies KS via another conjecture; what they mean, however, is that the conjecture they prove was proven to be equivalent to another conjecture which has already been shown in the past to be equivalent to a positive solution to the Kadison-Singer problem.
Blog posts on the solution appeared here and here, with links to excellent references. I will add here a few remarks of my own.
Continue reading Another one bites the dust (actually many of them)
Forty five years later, a major open problem in operator algebras is solved
A couple of days ago, Ken Davidson and Matt Kennedy posted a preprint on the arxiv, “The Choquet boundary of an operator system“. In this paper they solve a major open problem in operator algebras, showing that every operator system has sufficiently many boundary representations.
In 1969, William Arveson published the seminal paper, [“Subalgebras of C*-algebras”, Acta Math. 123, 1969], which is one of the cornerstones, (if not the cornerstone) of the theory of operator spaces and nonself-adjoint operator algebras. In that paper, among other things, Arveson introduced and put to good use the notion of a boundary representation. I wrote on “Subalgebras of C*-algebras” in a previous post dedicated to Arveson, and for some background material the reader is invited to look into that old post. I did not, however, write much about boundary representations (because I was emphasizing his contributions rather what he has left open). Below I wish to explain what are boundary representations, what does it mean that there are sufficiently many of these, and where Davidson and Kennedy’s new results fits in the chain of results leading to the solution of the problem. The paper itself is accessible to anyone who understands the problem, and the main ideas are clearly presented in its introduction.
Continue reading Forty five years later, a major open problem in operator algebras is solved
Souvenirs from Bangalore
I recently returned from the two week long workshop and conference Recent Advances in Operator Theory and Operator Algebras which took place in ISI Bangalore. As I promised myself before going, I was on the look-out for something new to be excited about and to learn. The event (beautifully organized and run) was made of two parts: a workshop, which was a one week mini-school on several topics (see here for topics) and a one week conference. It was very very broad, and there were several talks (or informal discussions) which I plan to pursue further.
In this post and also perhaps in a future one I will try to work out (for my own benefit, mostly) some details of a small part of the research presented in two of the talks. The first part is the Superproduct Systems which arise in the theory of E_0-semigroups on type II_1 factors (following the talk of R. Srinivasan). The second (which I will not discuss here, but perhpas in the future) is the equivalence between the Baby Corona Theorem and the Full Corona Theorem (following the mini-course given by B. Wick). In neither case will I describe the most important aspect of the work, but something that I felt was urgent for me to learn.
Advanced Analysis, Notes 15: C*-algebras (square root)
This post contains some make–up material for the course Advanced Analysis. It is a theorem about the positive square root of a positive element in a C*-algebra which does not appear in the text book we are using. My improvisation for this in class came out kakha–kakha, so here is the clarification.
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