Operator algebras for analytic varieties
Faculty Advisor
Date
2015
Keywords
non-selfadjoint operator algebras, reproducing kernel Hilbert spaces
Abstract (summary)
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 and W are each a finite union of irreducible varieties and a discrete variety in Bd with d<∞, then 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.
Publication Information
Davidson, K., Ramsey, C. & Shalit, O. (2015). Operator algebras for analytic varieties. Transactions of the American Mathematical Society 367(2), 1121-1150. https://doi.org/10.1090/S0002-9947-2014-05888-1
Notes
Item Type
Language
English
Rights
All Rights Reserved