Operator algebras for analytic varieties

Kenneth R. Davidson, Christopher Ramsey, Orr Moshe Shalit

Research output: Contribution to journalArticlepeer-review

30 Scopus citations

Abstract

We study the isomorphism problem for the multiplier algebras of irreducible complete Pick kernels. These are precisely the restrictions MV of the multiplier algebra M of Drury-Arveson space to a holomorphic subvariety V of the unit ball Bd. We find that MV is completely isometrically isomorphic to MW if and only if W is the image of V under a biholomorphic automorphism of the ball. In this case, the isomorphism is unitarily implemented. This is then strengthened to show that when d < ∞ every isometric isomorphism is completely isometric. The problem of characterizing when two such algebras are (algebraically) isomorphic is also studied. When V andW are each a finite union of irreducible varieties and a discrete variety, when d < ∞, an isomorphism between MV and MW determines a biholomorphism (with multiplier coordinates) between the varieties; and the isomorphism is composition with this function. These maps are automatically weak-∗ continuous. We present a number of examples showing that the converse fails in several ways. We discuss several special cases in which the converse does hold— particularly, smooth curves and Blaschke sequences. We also discuss the norm closed algebras associated to a variety, and point out some of the differences.

Original languageEnglish
Pages (from-to)1121-1150
Number of pages30
JournalTransactions of the American Mathematical Society
Volume367
Issue number2
DOIs
StatePublished - 1 Jan 2015

Keywords

  • Non-Selfadjoint operator algebras
  • Reproducing kernel Hilbert spaces

ASJC Scopus subject areas

  • Mathematics (all)
  • Applied Mathematics

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