Category Archives: d shift space

Essential normality, essential norms and hyper rigidity

Arveson, d shift space, Open problem, Operator algebras, Operator theory, Research

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).

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

d shift space, Expository, Open problem, Operator theory, Research, Talks, ,

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 $latex H^2_d$.

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The remarkable Hilbert space H^2 (Part II – multivariable operator theory and model theory)

d shift space, Expository, Operator theory, Talks,

This post is the second post in the series of posts on the d–shift space, a.k.a. the Drury–Arveson space, a.k.a. $latex H^2_d$ (see this previous post about the space $latex H^2$).

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The remarkable Hilbert space H^2 (Part I – definition and interpolation theory)

d shift space, Expository, Operator theory, Talks,

This series of posts is based on the colloquium talk that I was supposed to give on November 20, at our department. As fate had it, that week studies were cancelled.

Several people in our department thought that it would be a nice idea if alongside the usual colloquium talks given by invited speakers which highlight their recent achievements, we would also have some talks by department members that will be more of an exposition to the fields they work in. So my talk was supposed to be an exposition to the setting in which much of the research I do goes on.

The topic of the “talk”  is the Hilbert space $latex H^2_d$. There will be three parts to this series:

  1. Definition and interpolation theory. 
  2. Multivariate operator theory and model theory
  3. Current research problems

Continue reading The remarkable Hilbert space H^2 (Part I – definition and interpolation theory) →