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: Research
Spectral sets and distinguished varieties in the symmetrized bidisc
In this post I will write about a new paper, “Spectral sets and distinguished varieties in the symmetrized bidisc“, that Sourav Pal and I posted on the arxiv, and give the background to understand what we do in that paper.
Continue reading Spectral sets and distinguished varieties in the symmetrized bidisc
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).
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.
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)
On the isomorphism question for complete Pick multiplier algebras
In this post I want to tell you about our new preprint, “On the isomorphism question for complete Pick multiplier algebras“, which Matt Kerr, John McCarthy and myself just uploaded to the arXiv. Very broadly speaking, the motif of this paper is the connection between algebra and geometry; to be a little bit more precise, it is the connection between complex geometry and nonself-adjoint operator algebras. Continue reading On the isomorphism question for complete Pick multiplier algebras
Essential normality and the decomposability of algebraic varieties
I am very proud, because few days ago Matt Kennedy and I have had our new paper, Essential normality and the decomposability of algebraic varieties, published in the New York Journal of Mathematics.
In this paper we treat a strong version of a conjecture of Arveson, which we call the Arveson-Douglas conjecture, and we obtain some new results in particular cases (the conjecture is still far from being settled). I think that we do a good job in the introduction of the paper explaining what this conjecture is and what we do, so I invite you to take a look.
Now I want to say a few words about the New York Journal of Mathematics. I’ll say it in a different post.