TY - JOUR

T1 - Operator algebras for analytic varieties

AU - Davidson, Kenneth R.

AU - Ramsey, Christopher

AU - Shalit, Orr Moshe

N1 - Publisher Copyright:
© 2014 American Mathematical Society.

PY - 2015/1/1

Y1 - 2015/1/1

N2 - 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.

AB - 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.

KW - Non-Selfadjoint operator algebras

KW - Reproducing kernel Hilbert spaces

UR - http://www.scopus.com/inward/record.url?scp=84928241089&partnerID=8YFLogxK

U2 - 10.1090/s0002-9947-2014-05888-1

DO - 10.1090/s0002-9947-2014-05888-1

M3 - Article

AN - SCOPUS:84928241089

VL - 367

SP - 1121

EP - 1150

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 2

ER -