Multipliers of Embedded Discs

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 ℓ2 which are multiplier biholomorphic but have non-isomorphic multiplier algebras. We also show that there are closed discs in the ball of ℓ2 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 B2 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.