Topological isomorphisms for some universal operator algebras

TL;DR

This paper characterizes when two universal operator algebras, generated by compressions of the d-shift related to radical homogeneous ideals, are topologically isomorphic, linking algebraic isomorphisms to geometric isometries of vanishing loci.

Contribution

It proves that topological isomorphisms between these algebras occur if and only if their vanishing loci are related by an invertible linear isometry, confirming a conjecture of Davidson, Ramsey, and Shalit.

Findings

Topological isomorphisms correspond to isometric linear maps between vanishing loci.

Finite algebraic sums of full Fock spaces over subspaces are closed.

The isomorphism induces a completely bounded algebra isomorphism.

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 $\mathcal A_J$.