98 research outputs found
Progress in noncommutative function theory
In this expository paper we describe the study of certain non-self-adjoint
operator algebras, the Hardy algebras, and their representation theory. We view
these algebras as algebras of (operator valued) functions on their spaces of
representations. We will show that these spaces of representations can be
parameterized as unit balls of certain -correspondences and the
functions can be viewed as Schur class operator functions on these balls. We
will provide evidence to show that the elements in these (non commutative)
Hardy algebras behave very much like bounded analytic functions and the study
of these algebras should be viewed as noncommutative function theory
A Simple Proof of the Fundamental Theorem about Arveson Systems
With every Eo-semigroup (acting on the algebra of of bounded operators on a
separable infinite-dimensional Hilbert space) there is an associated Arveson
system. One of the most important results about Arveson systems is that every
Arveson system is the one associated with an Eo-semigroup. In these notes we
give a new proof of this result that is considerably simpler than the existing
ones and allows for a generalization to product systems of Hilbert module (to
be published elsewhere).Comment: Publication data added, acknowledgements and a note after acceptance
added, corrects a number of inconveniences that have been produced in the
published version during the publication proces
The Index of (White) Noises and their Product Systems
(See detailed abstract in the article.) We single out the correct class of
spatial product systems (and the spatial endomorphism semigroups with which the
product systems are associated) that allows the most far reaching analogy in
their classifiaction when compared with Arveson systems. The main differences
are that mere existence of a unit is not it sufficient: The unit must be
CENTRAL. And the tensor product under which the index is additive is not
available for product systems of Hilbert modules. It must be replaced by a new
product that even for Arveson systems need not coincide with the tensor
product
Composition Operators and Endomorphisms
If is an inner function, then composition with induces an
endomorphism, , of that leaves
invariant. We investigate the structure of the
endomorphisms of and that implement
through the representations of and
in terms of multiplication operators on
and . Our analysis, which is based on work
of R. Rochberg and J. McDonald, will wind its way through the theory of
composition operators on spaces of analytic functions to recent work on Cuntz
families of isometries and Hilbert -modules
Cartan subalgebras in C*-algebras of Hausdorff etale groupoids
The reduced -algebra of the interior of the isotropy in any Hausdorff
\'etale groupoid embeds as a -subalgebra of the reduced
-algebra of . We prove that the set of pure states of with unique
extension is dense, and deduce that any representation of the reduced
-algebra of that is injective on is faithful. We prove that there
is a conditional expectation from the reduced -algebra of onto if
and only if the interior of the isotropy in is closed. Using this, we prove
that when the interior of the isotropy is abelian and closed, is a Cartan
subalgebra. We prove that for a large class of groupoids with abelian
isotropy---including all Deaconu--Renault groupoids associated to discrete
abelian groups--- is a maximal abelian subalgebra. In the specific case of
-graph groupoids, we deduce that is always maximal abelian, but show by
example that it is not always Cartan.Comment: 14 pages. v2: Theorem 3.1 in v1 incorrect (thanks to A. Kumjain for
pointing out the error); v2 shows there is a conditional expectation onto
iff the interior of the isotropy is closed. v3: Material (including some
theorem statements) rearranged and shortened. Lemma~3.5 of v2 removed. This
version published in Integral Equations and Operator Theor
Operator theory and function theory in Drury-Arveson space and its quotients
The Drury-Arveson space , also known as symmetric Fock space or the
-shift space, is a Hilbert function space that has a natural -tuple of
operators acting on it, which gives it the structure of a Hilbert module. This
survey aims to introduce the Drury-Arveson space, to give a panoramic view of
the main operator theoretic and function theoretic aspects of this space, and
to describe the universal role that it plays in multivariable operator theory
and in Pick interpolation theory.Comment: Final version (to appear in Handbook of Operator Theory); 42 page
Quantized reduction as a tensor product
Symplectic reduction is reinterpreted as the composition of arrows in the
category of integrable Poisson manifolds, whose arrows are isomorphism classes
of dual pairs, with symplectic groupoids as units. Morita equivalence of
Poisson manifolds amounts to isomorphism of objects in this category.
This description paves the way for the quantization of the classical
reduction procedure, which is based on the formal analogy between dual pairs of
Poisson manifolds and Hilbert bimodules over C*-algebras, as well as with
correspondences between von Neumann algebras. Further analogies are drawn with
categories of groupoids (of algebraic, measured, Lie, and symplectic type). In
all cases, the arrows are isomorphism classes of appropriate bimodules, and
their composition may be seen as a tensor product. Hence in suitable categories
reduction is simply composition of arrows, and Morita equivalence is
isomorphism of objects.Comment: 44 pages, categorical interpretation adde
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