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