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 $H_d^2$ to the co-invariant subspace $H_d^2\ominus I$. 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 ${\mathcal{A}}_J$.

20 pages; two references added; to appear in the Journal of Functional Analysis

References in corpus (2)

• The isomorphism problem for some universal operator algebras
• Stable polynomial division and essential normality of graded Hilbert modules

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