Operator algebras for analytic varieties DAVIDSON, KENNETH R.; RAMSEY, CHRISTOPHER; SHALIT, ORR MOSHE
Transactions of the American Mathematical Society,
02/2015, Volume:
367, Issue:
2
Journal Article
Peer reviewed
Open access
We find that \mathcal {M}_V is completely isometrically isomorphic to \mathcal {M}_W 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>< 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, when d>< 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 study non-selfadjoint operator algebras that can be entirely understood via their finite-dimensional representations. In contrast with the elementary matricial description of finite-dimensional ...C⁎-algebras, in the non-selfadjoint setting we show that an additional level of flexibility must be allowed. Motivated by this peculiarity, we consider a natural non-selfadjoint notion of residual finite-dimensionality. We identify sufficient conditions for the tensor algebra of a C⁎-correspondence to enjoy this property. To clarify the connection with the usual self-adjoint notion, we investigate the residual finite-dimensionality of the minimal and maximal C⁎-covers associated to an operator algebra.
We continue the investigation of the isomorphism problem for multiplier algebras of reproducing kernel Hilbert spaces with the complete Nevanlinna-Pick property. In contrast to previous work in this ...area, we do not study these spaces by identifying them with the restrictions of a universal space, namely the Drury-Arveson space. Instead, we work directly with the Hilbert spaces and their reproducing kernels. In particular, we show that two multiplier algebras of Nevanlinna-Pick spaces on the same set are equal if and only if the Hilbert spaces are equal. Most of the article is devoted to the study of a special class of complete Nevanlinna-Pick spaces on homogeneous varieties. We provide a complete answer to the question of when two multiplier algebras of spaces of this type are algebraically or isometrically isomorphic.This generalizes results of Davidson, Ramsey,Shalit, and the author.
We consider the closed algebra Ad generated by the polynomial multipliers on the Drury–Arveson space. We identify Ad⁎ as a direct sum of the preduals of the full multiplier algebra and of a ...commutative von Neumann algebra, and establish analogues of many classical results concerning the dual space of the ball algebra. These developments are deeply intertwined with the problem of peak interpolation for multipliers, and we generalize a theorem of Bishop–Carleson–Rudin to this setting by means of Choquet type integral representations. As a byproduct we shed some light on the nature of the extreme points of the unit ball of Ad⁎.
Let I⊂Cz1,…,zd be a radical homogeneous ideal, and let AI be the norm-closed non-selfadjoint algebra generated by the compressions of the d-shift on Drury–Arveson space Hd2 to the co-invariant ...subspace Hd2⊖I. Then AI is the universal operator algebra for commuting row contractions subject to the relations in I. We ask under which conditions are there topological isomorphisms between two such algebras AI and AJ? We provide a positive answer to a conjecture of Davidson, Ramsey and Shalit: AI and AJ are topologically isomorphic if and only if there is an invertible linear map A on Cd which maps the vanishing locus of J isometrically onto the vanishing locus of I. Most of the proof is devoted to showing that finite algebraic sums of full Fock spaces over subspaces of Cd are closed. This allows us to show that the map A induces a completely bounded isomorphism between AI and AJ.
Written by a distinguished specialist in functional analysis, this book presents a comprehensive treatment of the history of Banach spaces and (abstract bounded) linear operators. Banach space theory ...is presented as a part of a broad mathematics context, using tools from such areas as set theory, topology, algebra, combinatorics, probability theory, logic, etc. Equal emphasis is given to both spaces and operators. The book may serve as a reference for researchers and as an introduction for graduate students who want to learn Banach space theory with some historical flavor.
We consider a number of examples of multiplier algebras on Hilbert spaces associated to discs embedded into a complex ball in order to examine the isomorphism problem for multiplier algebras on ...complete Nevanlinna–Pick reproducing kernel Hilbert spaces. In particular, we exhibit uncountably many discs in the ball of
ℓ
2
which are multiplier biholomorphic but have non-isomorphic multiplier algebras. We also show that there are
closed
discs in the ball of
ℓ
2
which are varieties, and examine their multiplier algebras. In finite balls, we provide a counterpoint to a result of Alpay, Putinar and Vinnikov by providing a proper rational biholomorphism of the disc onto a variety
V
in
B
2
such that the multiplier algebra is not all of
H
∞
(
V
)
. We also show that the transversality property, which is one of their hypotheses, is a consequence of the smoothness that they require.
We study the triangular subalgebras of UHF algebras which provide new examples of algebras with the Dirichlet property and the Ando property. This in turn allows us to describe the semicrossed ...product by an isometric automorphism. We also study the isometric automorphism group of these algebras and prove that it decomposes into the semidirect product of an abelian group by a torsion free group. Various other structure results are proven as well.
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