Category Archives: Open problem

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)

The remarkable Hilbert space H^2 (part III – three open problems)

This is the last in the series of three posts on the d–shift space, which accompany/replace the colloquium talk I was supposed to give. The first two parts are available here and here. In this post I will discuss three open problems that I have been thinking about, which are formulated within the setting of $latex H^2_d$.

Continue reading The remarkable Hilbert space H^2 (part III – three open problems)