Multipliers of embedded discs

TL;DR

This paper explores the structure of multiplier algebras associated with embedded discs in complex balls, revealing diverse isomorphism classes and counterexamples to existing hypotheses in the context of Nevanlinna-Pick spaces.

Contribution

It demonstrates the existence of uncountably many multiplier biholomorphic discs with non-isomorphic algebras and provides counterexamples to prior assumptions about multiplier algebras on varieties.

Findings

Uncountably many discs are multiplier biholomorphic but have non-isomorphic algebras.

Existence of closed discs in the ball of ℓ² that are varieties with specific multiplier algebra properties.

Counterexamples to previous results regarding biholomorphisms and multiplier algebras in finite-dimensional balls.

Abstract

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 $\ell^2$ which are multiplier biholomorphic but have non-isomorphic multiplier algebras. We also show that there are closed discs in the ball of $\ell^2$ 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 $\mathbb B_2$ such that the multiplier algebra is not all of $H^\infty(V)$. We also show that the transversality property, which is one of their hypotheses, is a consequence of the smoothness that they require.