Essential normality, essential norms and hyper rigidity

by Orr Shalit

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 S = (S_1, \ldots, S_d) denote the compression of the d-shift to the complement of a homogeneous ideal I of \mathbb{C}[z_1, \ldots, z_d]. Arveson conjectured that 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 Z = (Z_1, \ldots, Z_d) the image of S in the Calkin algebra (here as in the above paragraph, S is the compression of the d-shift to the complement of an ideal I in H^2_d). Another way of stating Arveson’s conjecture is that the C*-algebra generated by Z is commutative. This would have implied that the norm closed (non-selfadjoint) algebra generated by Z is equal to the sup-norm closure of polynomials on the zero variety of the ideal 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 S is essentially normal if and only if it is hyperrigid  (meaning in particular that all irreducible representations of C^*(S) are boundary representations).

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