Operator algebras for analytic varieties

Davidson, Kenneth R.
Ramsey, Christopher
Shalit, Orr Moshe
Faculty Advisor
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
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