Topological isomorphisms for some universal operator algebras

Let I⊂C[z1,…,zd] be a radical homogeneous ideal, and let AI be the norm-closed non-selfadjoint algebra generated by the compressions of the d-shift on Drury–Arveson space to the co-invariant subspace . Then AI 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 AI and AJ? We provide a positive answer to a conjecture of Davidson, Ramsey and Shalit: AI and AJ 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 AI and AJ.