A Connes correspondence approach to the dilation theory

2020 ◽  
Vol 31 (05) ◽  
pp. 2050040
Author(s):  
Yusuke Sawada
Keyword(s):  
Von Neumann Algebra ◽  
Classification Theory ◽  
Type I ◽  
View Point ◽  
Product System ◽  
Von Neumann ◽  
Product Systems ◽  
Dilation Theory ◽  
One To One

Product systems have been originally introduced to classify E0-semigroups on type I factors by Arveson. The idea of product systems is influenced the constructions of dilations of CP0-semigroups. In this paper, we will develop the dilation theory and the classification theory of E0-semigroups on a general von Neumann algebra in the view point of Connes correspondences. For this, we will provide a concept of product system whose components are Connes correspondences. There exists a one-to-one correspondence between a CP0-semigroup and a unit of a product system of Connes correspondences. The correspondence enables us to construct dilations and classify E0-semigroups by product systems of Connes correspondences up to cocycle equivalence.

scholarly journals ON PRODUCT SYSTEMS ARISING FROM SUM SYSTEMS

2005 ◽  
Vol 08 (01) ◽  
pp. 1-31 ◽  
Author(s):  
B. V. RAJARAMA BHAT ◽  
R. SRINIVASAN
Keyword(s):  
Hilbert Spaces ◽  
Von Neumann Algebra ◽  
Type I ◽  
Product System ◽  
Von Neumann ◽  
Type Iii ◽  
Product Systems ◽  
Weyl Representation ◽  
Uncountable Family ◽  
Type Spaces

B. Tsirelson constructed an uncountable family of type III product systems of Hilbert spaces through the theory of Gaussian spaces, measure type spaces and "slightly colored noises", using techniques from probability theory. Here we take a purely functional analytic approach and try to have a better understanding of Tsireleson's construction and his examples. We prove an extension of Shale's theorem connecting symplectic group and Weyl representation. We show that the "Shale map" respects compositions (this settles an old conjecture of K. R. Parthasarathy8). Using this we associate a product system to a sum system. This construction includes the exponential product system of Arveson, as a trivial case, and the type III examples of Tsirelson. By associating a von Neumann algebra to every "elementary set" in [0, 1], in a much simpler and direct way, we arrive at the invariants of the product system introduced by Tsirelson, given in terms of the sum system. Then we introduce a notion of divisibility for a sum system, and prove that the examples of Tsirelson are divisible. It is shown that only type I and type III product systems arise out of divisible sum systems. Finally, we give a sufficient condition for a divisible sum system to give rise to a unitless (type III) product system.

On the Unitary Equivalence of Certain Classes of Non-Normal Operators. I

1971 ◽  
Vol 23 (5) ◽  
pp. 849-856 ◽  
Author(s):  
P. K. Tam
Keyword(s):  
Von Neumann Algebra ◽  
Unitary Operator ◽  
Equivalence Problem ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Type I ◽  
Unitary Equivalence ◽  
Bounded Linear ◽  
Von Neumann ◽  
Normal Operators

The following (so-called unitary equivalence) problem is of paramount importance in the theory of operators: given two (bounded linear) operators A1, A2 on a (complex) Hilbert space , determine whether or not they are unitarily equivalent, i.e., whether or not there is a unitary operator U on such that U*A1U = A2. For normal operators this question is completely answered by the classical multiplicity theory [7; 11]. Many authors, in particular, Brown [3], Pearcy [9], Deckard [5], Radjavi [10], and Arveson [1; 2], considered the problem for non-normal operators and have obtained various significant results. However, most of their results (cf. [13]) deal only with operators which are of type I in the following sense [12]: an operator, A, is of type I (respectively, II1, II∞, III) if the von Neumann algebra generated by A is of type I (respectively, II1, II∞, III).

scholarly journals LOCAL DERIVATIONS ON ALGEBRAS OF MEASURABLE OPERATORS

2011 ◽  
Vol 13 (04) ◽  
pp. 643-657 ◽  
Author(s):  
S. ALBEVERIO ◽  
SH. A. AYUPOV ◽  
K. K. KUDAYBERGENOV ◽  
B. O. NURJANOV
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Sufficient Conditions ◽  
Continuity Condition ◽  
Type I ◽  
Atomic Lattice ◽  
Von Neumann ◽  
Measurable Operators ◽  
Finite Trace ◽  
Necessary And Sufficient

The paper is devoted to local derivations on the algebra [Formula: see text] of τ-measurable operators affiliated with a von Neumann algebra [Formula: see text] and a faithful normal semi-finite trace τ. We prove that every local derivation on [Formula: see text] which is continuous in the measure topology, is in fact a derivation. In the particular case of type I von Neumann algebras, they all are inner derivations. It is proved that for type I finite von Neumann algebras without an abelian direct summand, and also for von Neumann algebras with the atomic lattice of projections, the continuity condition on local derivations in the above results is redundant. Finally we give necessary and sufficient conditions on a commutative von Neumann algebra [Formula: see text] for the algebra [Formula: see text] to admit local derivations which are not derivations.

scholarly journals Discrete product systems with twisted units

1995 ◽  
Vol 52 (2) ◽  
pp. 317-326 ◽  
Author(s):  
Marcelo Laca
Keyword(s):  
Dynamical System ◽  
Crossed Product ◽  
Type I ◽  
Crossed Products ◽  
Product System ◽  
C Algebra ◽  
Covariant Representations ◽  
Product Systems ◽  
Recent Theorem ◽  
I Factor

The spectral C*-algebra of the discrete product systems of H.T. Dinh is shown to be a twisted semigroup crossed product whenever the product system has a twisted unit. The covariant representations of the corresponding dynamical system are always faithful, implying the simplicity of these crossed products; an application of a recent theorem of G.J. Murphy gives their nuclearity. Furthermore, a semigroup of endomorphisms of B(H) having an intertwining projective semigroup of isometries can be extended to a group of automorphisms of a larger Type I factor.

APPROXIMATELY INNER FLOWS ON SEPARABLE C*-ALGEBRAS

2002 ◽  
Vol 14 (07n08) ◽  
pp. 649-673 ◽  
Author(s):  
AKITAKA KISHIMOTO
Keyword(s):  
Irreducible Representation ◽  
Von Neumann Algebra ◽  
Automorphism Groups ◽  
Unitary Groups ◽  
Type I ◽  
Von Neumann ◽  
C Algebra ◽  
C Algebras ◽  
Inner Automorphism ◽  
Uniformly Continuous

We present two types of result for approximately inner one-parameter automorphism groups (referred to as AI flows hereafter) of separable C*-algebras. First, if there is an irreducible representation π of a separable C*-algebra A such that π(A) does not contain non-zero compact operators, then there is an AI flow α such that π is α-covariant and α is far from uniformly continuous in the sense that α induces a flow on π(A) which has full Connes spectrum. Second, if α is an AI flow on a separable C*-algebra A and π is an α-covariant irreducible representation, then we can choose a sequence (hn) of self-adjoint elements in A such that αt is the limit of inner flows Ad eithn and the sequence π(eithn) of one-parameter unitary groups (referred to as unitary flows hereafter) converges to a unitary flow which implements α in π. This latter result will be extended to cover the case of weakly inner type I representations. In passing we shall also show that if two representations of a separable simple C*-algebra on a separable Hilbert space generate the same von Neumann algebra of type I, then there is an approximately inner automorphism which sends one into the other up to equivalence.

scholarly journals ON HAAG DUALITY FOR PURE STATES OF QUANTUM SPIN CHAINS

2008 ◽  
Vol 20 (06) ◽  
pp. 707-724 ◽  
Author(s):  
M. KEYL ◽  
TAKU MATSUI ◽  
D. SCHLINGEMANN ◽  
R. F. WERNER
Keyword(s):  
Von Neumann Algebra ◽  
Quantum Spin ◽  
Spin Chains ◽  
Type I ◽  
Quantum Spin Chains ◽  
Haag Duality ◽  
Von Neumann ◽  
Infinite Intervals ◽  
Pure States

In this note, we consider quantum spin chains and their translationally invariant pure states. We prove Haag duality for quasilocal observables localized in semi-infinite intervals (-∞ , 0] and [1, ∞) when the von Neumann algebra generated by observables localized in [0, ∞) is non-type I.

scholarly journals Derivations on the algebra of measurable operators affiliated with a type I von Neumann algebra

2008 ◽  
Vol 18 (2) ◽  
pp. 86-94 ◽  
Author(s):  
S. Albeverio ◽  
Sh. A. Ayupov ◽  
K. K. Kudaybergenov
Keyword(s):  
Von Neumann Algebra ◽  
Type I ◽  
Von Neumann ◽  
Measurable Operators

scholarly journals A REMARK ON THE STRUCTURE OF SYMMETRIC QUANTUM DYNAMICAL SEMIGROUPS ON VON NEUMANN ALGEBRAS

2002 ◽  
Vol 05 (04) ◽  
pp. 571-579 ◽  
Author(s):  
SERGIO ALBEVERIO ◽  
DEBASHISH GOSWAMI
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Type I ◽  
Quantum Dynamical Semigroup ◽  
Von Neumann ◽  
Matrix Algebras ◽  
Quantum Dynamical ◽  
Dynamical Semigroup ◽  
Quantum Dynamical Semigroups ◽  
Symmetric Quantum

We study the structure of the generator of a symmetric, conservative quantum dynamical semigroup with norm-bounded generator on a von Neumann algebra equipped with a faithful semifinite trace. For von Neumann algebras with Abelian commutant (i.e. type I von Neumann algebras), we give a necessary and sufficient algebraic condition for the generator of such a semigroup to be written as a sum of square of self-adjoint derivations of the von Neumann algebra. This generalizes some of the results obtained by Albeverio, Høegh-Krohn and Olsen1 for the special case of the finite-dimensional matrix algebras. We also study similar questions for a class of quantum dynamical semigroups with unbounded generators.

scholarly journals Structure of block quantum dynamical semigroups and their product systems

2020 ◽  
Vol 23 (01) ◽  
pp. 2050001
Author(s):  
B. V. Rajarama Bhat ◽  
U. Vijaya Kumar
Keyword(s):  
Von Neumann Algebra ◽  
Markov Semigroup ◽  
Map Building ◽  
Quantum Dynamical Semigroup ◽  
Von Neumann ◽  
Quantum Dynamical ◽  
Product Systems ◽  
Dynamical Semigroup ◽  
Quantum Markov Semigroup ◽  
Unit Formula

Paschke’s version of Stinespring’s theorem associates a Hilbert [Formula: see text]-module along with a generating vector to every completely positive map. Building on this, to every quantum dynamical semigroup (QDS) on a [Formula: see text]-algebra [Formula: see text] one may associate an inclusion system [Formula: see text] of Hilbert [Formula: see text]-[Formula: see text]-modules with a generating unit [Formula: see text]. Suppose [Formula: see text] is a von Neumann algebra, consider [Formula: see text], the von Neumann algebra of [Formula: see text] matrices with entries from [Formula: see text]. Suppose [Formula: see text] with [Formula: see text] is a QDS on [Formula: see text] which acts block-wise and let [Formula: see text] be the inclusion system associated to the diagonal QDS [Formula: see text] with the generating unit [Formula: see text] It is shown that there is a contractive (bilinear) morphism [Formula: see text] from [Formula: see text] to [Formula: see text] such that [Formula: see text] for all [Formula: see text] We also prove that any contractive morphism between inclusion systems of von Neumann [Formula: see text]-[Formula: see text]-modules can be lifted as a morphism between the product systems generated by them. We observe that the [Formula: see text]-dilation of a block quantum Markov semigroup (QMS) on a unital [Formula: see text]-algebra is again a semigroup of block maps.

scholarly journals Cartan Triples

2019 ◽  
Author(s):  
Allan P Donsig ◽  
Adam H Fuller ◽  
David R Pitts
Keyword(s):  
Inverse Semigroup ◽  
Von Neumann Algebra ◽  
Inverse Semigroups ◽  
Spectral Theorem ◽  
Von Neumann ◽  
Support Sets ◽  
One To One

Abstract We introduce the class of Cartan triples as a generalization of the notion of a Cartan MASA in a von Neumann algebra. We obtain a one-to-one correspondence between Cartan triples and certain Clifford extensions of inverse semigroups. Moreover, there is a spectral theorem describing bimodules in terms of their support sets in the fundamental inverse semigroup and, as a corollary, an extension of Aoi’s theorem to this setting. This context contains that of Fulman’s generalization of Cartan MASAs and we discuss his generalization in an appendix.

Sign in / Sign up

Export Citation Format

Share Document