Topological isomorphisms for some universal operator algebras

Details Topological isomorphisms for some universal operator algebras Topological isomorphisms for some universal operator algebras

2012-04-09

arxiv.org/abs/1204.1940v1

Mathematics

Operator Algebras

20 pages

Scientific paper

Let $I$ be a radical homogeneous ideal of complex polynomials in $d$ variables, 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 $I$. In this note, we study the question, under which conditions there are topological isomorphisms between two such algebras $\mathcal A_I$ and $\mathcal A_J$. We provide a positive answer to a conjecture of Davdison, Ramsey and Shalit: that $\mathcal A_I$ 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 $\mathcal A_J$.

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