Abstract. We continue the investigation of the isomorphism problem for multiplier algebras of reproducing kernel Hilbert spaces with the complete Nevanlinna-Pick property. In contrast to previous work in this area, we do not study these spaces by identifying them with restrictions of a universal space, namely the Drury-Arveson space. Instead, we work directly with the Hilbert spaces and their reproducing kernels. In particular, we show that two multiplier algebras of Nevanlinna-Pick spaces on the same set are equal if and only if the Hilbert spaces are equal. Most of the article is devoted to the study of a special class of complete Nevanlinna-Pick spaces on homogeneous varieties. We provide a complete answer to the question of when two multiplier algebras of spaces of this type are algebraically or isometrically isomorphic.is generalizes results of Davidson, Ramsey, Shalit, and the author.

We show that every function in a reproducing kernel Hilbert space with a normalized complete Pick kernel is the quotient of a multiplier and a cyclic multiplier. This extends a theorem of Alpay, Bolotnikov and Kaptano\u{g}lu. We explore various consequences of this result regarding zero sets, spaces on compact sets and Gleason parts. In particular, using a construction of Salas, we exhibit a rotationally invariant complete Pick space of analytic functions on the unit disc for which the corona theorem fails.Comment: 18 pages; minor change

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 $\mathcal A_J$.Comment: 20 pages; two references added; to appear in the Journal of Functional Analysi

We consider a number of examples of multiplier algebras on Hilbert spaces associated to discs embedded into a complex ball in order to examine the isomorphism problem for multiplier algebras on complete Nevanlinna-Pick reproducing kernel Hilbert spaces. In particular, we exhibit uncountably many discs in the ball of ℓ 2 which are multiplier biholomorphic but have non-isomorphic multiplier algebras. We also show that there are closed discs in the ball of ℓ 2 which are varieties, and examine their multiplier algebras. In finite balls, we provide a counterpoint to a result of Alpay, Putinar and Vinnikov by providing a proper rational biholomorphism of the disc onto a variety V in B 2 such that the multiplier algebra is not all of H ∞ (V ). We also show that the transversality property, which is one of their hypotheses, is a consequence of the smoothness that they require.H V = span{k y : y ∈ V }.

For s ∈ R the weighted Besov space on the unit ballHere R s is a power of the radial derivative operator R = d i=1 z i ∂ ∂zi , V denotes Lebesgue measure, and ω is a radial weight function not supported on any ball of radius < 1.Our results imply that for all such weights ω and ν, every bounded column multiplication operator B s ω → B t ν ⊗ ℓ 2 induces a bounded row multiplier B s ω ⊗ℓ 2 → B t ν . Furthermore we show that if a weight ω satisfies that for some α > −1 the ratio ω(z)/(1 − |z| 2 ) α is nondecreasing for t 0 < |z| < 1, then B s ω is a complete Pick space, whenever s ≥ (α + d)/2.