Operator algebras for analytic varieties
By K. Davidson, Christopher Ramsey, Orr Shalit. 2012
58 citations recorded by the source corpus.
We study the isomorphism problem for the multiplier algebras of irreducible complete Pick kernels. These are precisely the restrictionsMV of the multiplier algebraM of Drury-Arveson space to a holomorphic subvariety V of the unit ball Bd. We nd that MV is completely isometrically isomorphic toMW if and only if W is the image of V under a biholomorphic auto- morphism of the ball. In this case, the isomorphism is unitarily implemented. This is then strengthend to show that, when d <1, every isometric isomorphism is completely isometric. The problem of characterizing when two such algebras are (al- gebraically) isomorphic is also studied. When V and W are each a nite union of irreducible varieties and a discrete variety in Bd with d <1, then an isomorphism betweenMV andMW deter- mines 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.
Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/9a24b7382e9b20e0ee324c23c4ee2fdf562546bb
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In Algebra and Number Theory
In Applied Mathematics
In Mathematical Physics
Citations in the corpus
- Operator Theory and Function Theory in Drury–Arveson Space and Its Quotients (2013, unknown): “ it induce an algebraic isomorphism MV →MW? If f ∈MV and φ∈W →V is holomorphic then evidently f◦φ∈H∞(W); the crux of the matter is whether or not it is a multiplier. The answer is negative in general [45, 52]. The first positive result in this direction was obtained by D. Alpay, M. Putinar and V. Vin”
- Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball (2017, methodology): “In [24], AV was denoted by AI , to highlight the role of the ideal I.”