192 research outputs found
Three Ways to Representations of B^a(E)
We describe three methods to determine the structure of (sufficiently
continuous) representations of the algebra B^a(E) of all adjointable operators
on a Hilbert B-module E by operators on a Hilbert C-module. While the last and
latest proof is simple and direct and new even for normal representations of
B(H) (H some Hilbert space), the other ones are direct generalizations of the
representation theory of B(H) (based on Arveson's and on Bhat's approaches to
product systems of Hilbert spaces) and depend on technical conditions (for
instance, existence of a unit vector or restriction to von Neumann algebras and
von Neumann modules). We explain why for certain problems the more specific
information available in the older approaches is more useful for the solution
of the problem
Commutants of von Neumann Correspondences and Duality of Eilenberg-Watts Theorems by Rieffel and by Blecher
The category of von Neumann correspondences from B to C (or von Neumann
B-C-modules) is dual to the category of von Neumann correspondences from C' to
B' via a functor that generalizes naturally the functor that sends a von
Neumann algebra to its commutant and back. We show that under this duality,
called commutant, Rieffel's Eilenberg-Watts theorem (on functors between the
categories of representations of two von Neumann algebras) switches into
Blecher's Eilenberg-Watts theorem (on functors between the categories of von
Neumann modules over two von Neumann algebras) and back.Comment: 20 page
Generalized Unitaries and the Picard Group
After discussing some basic facts about generalized module maps, we use the
representation theory of the algebra of adjointable operators on a Hilbert
B-module E to show that the quotient of the group of generalized unitaries on E
and its normal subgroup of unitaries on E is a subgroup of the group of
automorphisms of the range ideal of E in B. We determine the kernel of the
canonical mapping into the Picard group of the range ideal in terms of the
group of its quasi inner automorphisms. As a by-product we identify the group
of bistrict automorphisms of the algebra of adjointable operators on E modulo
inner automorphisms as a subgroup of the (opposite of the) Picard group.Comment: minor corrections, some parts extended, this version is to appear in
Proceedings of the Indian Academy of Science
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
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
Representations of B^a(E)
We present a short and elegant proof of the complete theory of strict
representations of the algebra B^a(E) of all adjointable operators on a Hilbert
B-module E by operators on a Hilbert C-module F. Aanalogue for W*-modules and
normal representations is also proved. As an application we furnish a new proof
of Blecher's Eilenberg-Watts theorem.Comment: Publication data added, corrected an ambiguity in the formulation of
Corollary 1.20 (present also in the published version) and adjusted the proof
appropriatel
Maximal Commutative Subalgebras Invariant for CP-Maps: (Counter-)Examples
We solve, mainly by counterexamples, many natural questions regarding maximal
commutative subalgebras invariant under CP-maps or semigroups of CP-maps on a
von Neumann algebra. In particular, we discuss the structure of the generators
of norm continuous semigroups on B(G) leaving a maximal commutative subalgebra
invariant and show that there exists Markov CP-semigroups on M_d without
invariant maximal commutative subalgebras for any d>2.Comment: After the elemenitation in Version 2 of a false class of examples in
Version 1, we now provide also correct examples for unital CP-maps and Markov
semigroups on M_d for d>2 without invariant masa
Hilbert module realization of the square of white noise and the finite difference algebra
We develop an approach to the representations theory of the algebra of the square of white noise based on the construction of Hilbert modules. We find the unique Fock representation and show that the representation space is the usual symmetric Fock space. Although we started with one degree of freedom we end up with countably many degrees of freedom. Surprisingly, our representation turns out to have a close relation to Feinsilver's finite difference algebra. In fact, there exists a holomorphic image of the finite difference algebra in the algebra of square of white noise. Our representation restricted to this image is the Boukas representation on the finite difference Fock space. Thus we extend the Boukas representation to a bigger algebra, which is generated by creators, annihilators, and number operators
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