Multipliers of embedded discs

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.

34 pages; the earlier version relied on a result of Davidson and Pitts that the fibre of the maximal ideal space of the multiplier algebra over a point in the open ball consists only of point evaluation. This result fails for $d=\infty$, and has necessitated some changes; to appear in Complex Analysis and Operator Theory

References in corpus (2)

• The isomorphism problem for some universal operator algebras
• Topological isomorphisms for some universal operator algebras

Cited by in corpus (11)

• Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball
• On the isomorphism problem for multiplier algebras of Nevanlinna-Pick spaces
• Interpolating sequences in spaces with the complete Pick property
• Algebras of noncommutative functions on subvarieties of the noncommutative ball: the bounded and completely bounded isomorphism problem
• Spaces of Dirichlet series with the complete Pick property
• The classification problem for operator algebraic varieties and their multiplier algebras
• Tensor algebras of subproduct systems and noncommutative function theory
• Three Classification Results In The Theory Of Weighted Hardy Spaces On The Ball
• Traces of analytic uniform algebras on subvarieties and test collections
• Shift-cyclicity in analytic function spaces
• Distance between reproducing kernel Hilbert spaces and geometry of finite sets in the unit ball