In our paper Essential normality, essential norms and hyperrigidity we claimed that the restriction of the identity representation of a certain operator system (constructed from a polynomial ideal) has the unique extension property, however the justi
fication we gave was insufficient. In this note we provide the required justification under some additional assumptions. Fortunately, homogeneous ideals that are sufficiently non-trivial are covered by these assumptions. This affects the section of our paper relating essential normality and hyperrigidity. We show here that Proposition 4.11 and Theorem 4.12 hold under the additional assumptions. We do not know if they hold in the generality considered in our paper.
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 t
his conjecture, and connect the notion of essential normality to the theory of the C*-envelope and the noncommutative Choquet boundary. The unital norm closed algebra $mathcal{B}_I$ generated by $S_1,ldots,S_d$ modulo the compact operators is shown to be completely isometrically isomorphic to the uniform algebra generated by polynomials on $overline{V} := overline{mathcal{Z}(I) cap mathbb{B}_d}$, where $mathcal{Z}(I)$ is the variety corresponding to $I$. Consequently, the essential norm of an element in $mathcal{B}_I$ is equal to the sup norm of its Gelfand transform, and the C*-envelope of $mathcal{B}_I$ is identified as the algebra of continuous functions on $overline{V} cap partial mathbb{B}_d$, which means it is a complete invariant of the topology of the variety determined by $I$ in the ball. Motivated by this determination of the C*-envelope of $mathcal{B}_I$, we suggest a new, more qualitative approach to the problem of essential normality. We prove the tuple $S$ is essentially normal if and only if it is hyperrigid as the generating set of a C*-algebra, which is a property closely connected to Arvesons notion of a boundary representation. We show that most of our results hold in a much more general setting. In particular, for most of our results, the ideal $I$ can be replaced by an arbitrary (not necessarily homogeneous) invariant subspace of the $d$-shift.
