Essential normality, essential norms and hyperrigidity

Let S=(S₁, S_d) denote the compression of the d-shift to the complement of a homogeneous ideal I of C[z1,. . .,zd]. 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.The unital norm closed algebra BI generated by S₁, . . ., S_d modulo the compact operators is shown to be completely isometrically isomorphic to the uniform algebra generated by polynomials on V-:=Z(I)∩Bd-, where Z(I) is the variety corresponding to I. Consequently, the essential norm of an element in BI is equal to the sup norm of its Gelfand transform, and the C*-envelope of BI is identified as the algebra of continuous functions on V-∩∂Bd, 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 BI, 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 Arveson's 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.