Topological isomorphisms for some universal operator algebras

Abstract

Let $I \subset \mathbb C[z_1,...,z_d]$ be a radical homogeneous ideal, and let $\mathcal A_I$ be the norm-closed non-selfadjoint algebra generated by the compressions of the $d$-shift on Drury-Arveson space $H^2_d$ to the co-invariant subspace $H^2_d \ominus I$. Then $\mathcal A_I$ is the universal operator algebra for commuting row contractions subject to the relations in $I$. We ask under which conditions are there topological isomorphisms between two such algebras $\mathcal A_I$ and $\mathcal A_J$? We provide a positive answer to a conjecture of Davidson, Ramsey and Shalit: $\mathcal A_I$ and $\mathcal A_J$ are topologically isomorphic if and only if there is an invertible linear map $A$ on $\mathbb C^d$ which maps the vanishing locus of $J$ isometrically onto the vanishing locus of $I$. Most of the proof is devoted to showing that finite algebraic sums of full Fock spaces over subspaces of $\mathbb C^d$ are closed. This allows us to show that the map $A$ induces a completely bounded isomorphism between $\mathcal A_I$ and $\mathcal A_J$.