Tensor product of quaternion hilbert modules

1991 ◽  
Vol 24 (2) ◽  
pp. 141-178 ◽  
Author(s):  
Aharon Razon ◽  
L. P. Horwitz
Keyword(s):  
Tensor Product ◽  
Hilbert Modules

scholarly journals Tensor product of quotient Hilbert modules

2015 ◽  
Vol 424 (1) ◽  
pp. 727-747 ◽  
Author(s):  
Arup Chattopadhyay ◽  
B. Krishna Das ◽  
Jaydeb Sarkar
Keyword(s):  
Tensor Product ◽  
Hilbert Modules

Projection operators and states in the tensor product of quaternion hilbert modules

1991 ◽  
Vol 24 (2) ◽  
pp. 179-194 ◽  
Author(s):  
Aharon Razon ◽  
L. P. Horwitz
Keyword(s):  
Tensor Product ◽  
Projection Operators ◽  
Hilbert Modules

scholarly journals THE INDEX OF (WHITE) NOISES AND THEIR PRODUCT SYSTEMS

2006 ◽  
Vol 09 (04) ◽  
pp. 617-655 ◽  
Author(s):  
MICHAEL SKEIDE
Keyword(s):  
Tensor Product ◽  
Hilbert Spaces ◽  
Hilbert Modules ◽  
Product System ◽  
Invariant Vector ◽  
Product Systems ◽  
System A ◽  
Definition Of ◽  
General Product ◽  
White Noises

Almost every paper about Arveson systems (i.e. product systems of Hilbert spaces) starts by recalling their basic classification assigning to every Arveson system a type and an index. So it is natural to ask in how far an analogue classification can also be proposed for product systems of Hilbert modules. However, while the definition of type is plain, there are obstacles for the definition of index. But all obstacles can be removed when restricting to the category which we introduce here as spatial product systems and that matches the usual definition of spatial in the case of Arveson systems. This is not really a loss because the definition of index for nonspatial Arveson systems is rather formal and does not reflect the information the index carries for spatial Arveson systems.E0-semigroups give rise to product systems. Our definition of spatial product system, namely, existence of a unital unit that is central, matches Powers' definition of spatial in the sense that the E0-semigroup from which the product system is derived admits a semigroup of intertwining isometries. We show that every spatial product system contains a unique maximal completely spatial subsystem (generated by all units) that is isomorphic to a product system of time ordered Fock modules. (There exist nonspatial product systems that are generated by their units. Consequently, these cannot be Fock modules.) The index of a spatial product system we define as the (unique) Hilbert bimodule that determines the Fock module. In order to show that the index merits the name index we provide a product of product systems under which the index is additive (direct sum). While for Arveson systems there is the tensor product, for general product systems the tensor product does not make sense as a product system. Even for Arveson systems our product is, in general, only a subsystem of the tensor product. Moreover, its construction depends explicitly on the choice of the central reference units of its factors.Spatiality of a product system means that it may be derived from an E0-semigroup with an invariant vector expectation, i.e. from a noise. We extend our product of spatial product systems to a product of noises and study its properties.Finally, we apply our techniques to show the module analogue of Fowler's result that free flows are comletely spatial, and we compute their indices.

FRAMES IN HILBERT C*-MODULES AND MORITA EQUIVALENT C*-ALGEBRAS

2016 ◽  
Vol 59 (1) ◽  
pp. 1-10 ◽  
Author(s):  
MASSOUD AMINI ◽  
MOHAMMAD B. ASADI ◽  
GEORGE A. ELLIOTT ◽  
FATEMEH KHOSRAVI
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Discrete Spectrum ◽  
Compact Operators ◽  
Morita Equivalence ◽  
Hilbert Modules ◽  
C Algebra ◽  
C Algebras ◽  
Morita Equivalent ◽  
Continuous Trace

AbstractWe show that the property of a C*-algebra that all its Hilbert modules have a frame, in the case of σ-unital C*-algebras, is preserved under Rieffel–Morita equivalence. In particular, we show that a σ-unital continuous-trace C*-algebra with trivial Dixmier–Douady class, all of whose Hilbert modules admit a frame, has discrete spectrum. We also show this for the tensor product of any commutative C*-algebra with the C*-algebra of compact operators on any Hilbert space.

TENSOR PRODUCT SYSTEMS OF HILBERT MODULES AND DILATIONS OF COMPLETELY POSITIVE SEMIGROUPS

2000 ◽  
Vol 03 (04) ◽  
pp. 519-575 ◽  
Author(s):  
B. V. RAJARAMA BHAT ◽  
MICHAEL SKEIDE
Keyword(s):  
Tensor Product ◽  
Hilbert Spaces ◽  
Hilbert Modules ◽  
Completely Positive ◽  
Positive Semigroups ◽  
C Algebras ◽  
Product Systems ◽  
Dilation Theory ◽  
Cocycle Conjugacy ◽  
The Right

In this paper we study the problem of dilating unital completely positive (CP) semigroups (quantum dynamical semigroups) to weak Markov flows and then to semigroups of endomorphisms (E0-semigroups) using the language of Hilbert modules. This is a very effective, representation free approach to dilation. In this way we are able to identify the right algebra (maximal in some sense) for endomorphisms to act. We are led inevitably to the notion of tensor product systems of Hilbert modules and units for them, generalizing Arveson's notions for Hilbert spaces. In the course of our investigations we are not only able to give new natural and transparent proofs of well-known facts for semigroups on [Formula: see text], but also extend the results immediately to much more general setups. For instance, Arveson classifies E0-semigroups on [Formula: see text] up to cocycle conjugacy by product systems of Hilbert spaces.5 We find that conservative CP-semigroups on arbitrary unital C*-algebras are classified up to cocycle conjugacy by product systems of Hilbert modules. Looking at other generalizations, it turns out that the role played by E0-semigroups on [Formula: see text] in dilation theory for CP-semigroups on [Formula: see text] is now played by E0-semigroups on [Formula: see text], the full algebra of adjointable operators on a Hilbert module E. We have CP-semigroup versions of many results proved by Paschke27 for CP maps.

scholarly journals On the index of product systems of Hilbert modules

Filomat ◽  
2015 ◽  
Vol 29 (5) ◽  
pp. 1093-1111
Author(s):  
Dragoljub Keckic ◽  
Biljana Vujosevic
Keyword(s):  
Tensor Product ◽  
Equivalence Relation ◽  
Covariant Functor ◽  
Hilbert Modules ◽  
Product System ◽  
Initial Product ◽  
C Algebra ◽  
Product Systems ◽  
Uniformly Continuous ◽  
Continuous Product

In this note we prove that the set of all uniformly continuous units on a product system over a C* algebra B can be endowed with a structure of left-right B-B Hilbert module after identifying similar units by the suitable equivalence relation. We use this construction to define the index of the initial product system, and prove that it is a generalization of earlier defined indices by Arveson (in the case B=C) and Skeide (in the case of spatial product system). We prove that such defined index is a covariant functor from the category of continuous product systems to the category of B bimodules. We also prove that the index is subadditive with respect to the outer tensor product of product systems, and prove additional properties of the index of product systems that can be embedded into a spatial one.

Uniqueness of the scalar product in the tensor product of quaternion Hilbert modules

1992 ◽  
Vol 33 (9) ◽  
pp. 3098-3104 ◽  
Author(s):  
A. Razon ◽  
L. P. Horwitz
Keyword(s):  
Tensor Product ◽  
Scalar Product ◽  
Hilbert Modules
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