Topological isomorphisms for some universal operator algebras

Let I ⊂ C[z1, . . . , zd ] be a radical homogeneous ideal, and let A I be the norm-closed non-selfadjoint algebra generated by the compressions of the d-shift on Drury–Arveson space H2 d to the co-invariant subspace H2 d I. Then 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 A I and A J ? We provide a positive answer to a conjecture of Davidson, Ramsey and Shalit: A I and A J are topologically isomorphic if and only if there is an invertible linear map A on Cd 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 Cd are closed. This allows us to show that the map A induces a completely bounded isomorphism between A I and A J . © 2012 Elsevier Inc. All rights reserved.