Chapter 7 Normal States and Unitary Equivalence of von Neumann Algebras

1983 ◽  
pp. 312-367
Keyword(s):  
Von Neumann Algebras ◽  
Unitary Equivalence ◽  
Von Neumann ◽  
Normal States

Arens spaces associated with von Neumann algebras and normal states

Positivity ◽  
2009 ◽  
Vol 14 (1) ◽  
pp. 105-121
Author(s):  
S. Albeverio ◽  
Sh. A. Ayupov ◽  
R. Z. Abdullaev
Keyword(s):  
Von Neumann Algebras ◽  
Von Neumann ◽  
Normal States

scholarly journals On Ultraproduct Embeddings and Amenability for Tracial von Neumann Algebras

2020 ◽  
Author(s):  
Scott Atkinson ◽  
Srivatsav Kunnawalkam Elayavalli
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Analogous Result ◽  
Equivalence Classes ◽  
Embedding Problem ◽  
Unitary Equivalence ◽  
Von Neumann ◽  
Sofic Groups

Abstract We define the notion of self-tracial stability for tracial von Neumann algebras and show that a tracial von Neumann algebra satisfying the Connes embedding problem (CEP) is self-tracially stable if and only if it is amenable. We then generalize a result of Jung by showing that a separable tracial von Neumann algebra that satisfies the CEP is amenable if and only if any two embeddings into $R^{\mathcal{U}}$ are ucp-conjugate. Moreover, we show that for a II$_1$ factor $N$ satisfying CEP, the space $\mathbb{H}$om$(N, \prod _{k\to \mathcal{U}}M_k)$ of unitary equivalence classes of embeddings is separable if and only $N$ is hyperfinite. This resolves a question of Popa for Connes embeddable factors. These results hold when we further ask that the pairs of embeddings commute, admitting a nontrivial action of $\textrm{Out}(N\otimes N)$ on ${\mathbb{H}}\textrm{om}(N\otimes N, \prod _{k\to \mathcal{U}}M_k)$ whenever $N$ is non-amenable. We also obtain an analogous result for commuting sofic representations of countable sofic groups.

Unitary equivalence of unbounded *-representations of *-algebras

1997 ◽  
Vol 122 (2) ◽  
pp. 269-279
Author(s):  
I. IKEDA ◽  
A. INOUE ◽  
M. TAKAKURA
Keyword(s):  
Von Neumann Algebras ◽  
Unitary Equivalence ◽  
Density Condition ◽  
Von Neumann ◽  
Representations Of Algebras ◽  
Unbounded Representations

In this paper the unitary equivalence of unbounded *-representations of *-algebras is investigated. It is shown that if closed *-representations π1 and π2 of a *-algebra [Ascr ] satisfy a certain density condition for the intertwining spaces [Jscr ](π1, π2) and [Jscr ](π2, π1), then a *-isomorphism Φ between the O*-algebras π1([Ascr ]) and π2([Ascr ]) is defined by Φ(π1(x))=π2(x), x∈[Ascr ] and it induces a *-isomorphism Φ¯, between the von Neumann algebras (π1([Ascr ])′w)′ and (π2([Ascr ])′w)′, and further if Φ¯, is spatial (that is, it is unitarily implemented), then π1 and π2 are unitarily equivalent.

scholarly journals On the Relationship between Jordan Algebras and Their Universal Enveloping Algebras

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
F. B. H. Jamjoom ◽  
A. H. Al Otaibi
Keyword(s):  
State Space ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
The State ◽  
Jordan Algebras ◽  
Universal Enveloping Algebras ◽  
Von Neumann ◽  
Enveloping Algebras ◽  
Normal States ◽  
The Relationship

The relationship between JW-algebras (resp. JC-algebras) and their universal enveloping von Neumann algebras (resp. C ∗ -algebras) can be described as significant and influential. Examples of numerous relationships have been established. In this article, we established a relationship between the set of split faces of the state space (resp. normal states) of a JC-algebra (resp. a JW-algebra) and the set of split faces of the state space (resp. normal states) of its universal enveloping C ∗ -algebra (resp. von Neumann algebra), and we tied up this relationship with the correspondence between the classes of invariant faces, closed ideals, and central projections of these Jordan algebras and of their universal enveloping algebras.

scholarly journals Rigidity of free product von Neumann algebras

2016 ◽  
Vol 152 (12) ◽  
pp. 2461-2492 ◽  
Author(s):  
Cyril Houdayer ◽  
Yoshimichi Ueda
Keyword(s):  
Large Class ◽  
Free Product ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Product Type ◽  
Von Neumann ◽  
Rigidity Results ◽  
Cartan Subalgebras ◽  
Normal States

Let $I$ be any nonempty set and let $(M_{i},\unicode[STIX]{x1D711}_{i})_{i\in I}$ be any family of nonamenable factors, endowed with arbitrary faithful normal states, that belong to a large class ${\mathcal{C}}_{\text{anti}\text{-}\text{free}}$ of (possibly type $\text{III}$) von Neumann algebras including all nonprime factors, all nonfull factors and all factors possessing Cartan subalgebras. For the free product $(M,\unicode[STIX]{x1D711})=\ast _{i\in I}(M_{i},\unicode[STIX]{x1D711}_{i})$, we show that the free product von Neumann algebra $M$ retains the cardinality $|I|$ and each nonamenable factor $M_{i}$ up to stably inner conjugacy, after permutation of the indices. Our main theorem unifies all previous Kurosh-type rigidity results for free product type $\text{II}_{1}$ factors and is new for free product type $\text{III}$ factors. It moreover provides new rigidity phenomena for type $\text{III}$ factors.

scholarly journals Amalgamated free product type III factors with at most one Cartan subalgebra

2013 ◽  
Vol 150 (1) ◽  
pp. 143-174 ◽  
Author(s):  
Rémi Boutonnet ◽  
Cyril Houdayer ◽  
Sven Raum
Keyword(s):  
Free Product ◽  
Measure Space ◽  
Cartan Subalgebra ◽  
Von Neumann Algebras ◽  
Finite Subgroup ◽  
Standard Measure ◽  
Von Neumann ◽  
Cartan Subalgebras ◽  
Amalgamated Free Product ◽  
Normal States

AbstractWe investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras ${\mathop{M{}_{1} \ast }\nolimits}_{B} {M}_{2} $ over an amenable von Neumann subalgebra $B$. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free product von Neumann algebras. Namely, we show that any nonamenable free product von Neumann algebra $({M}_{1} , {\varphi }_{1} )\ast ({M}_{2} , {\varphi }_{2} )$ with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established by A. Ioana [Cartan subalgebras of amalgamated free product ${\mathrm{II} }_{1} $factors, arXiv:1207.0054]. Next, we prove that any countable nonsingular ergodic equivalence relation $ \mathcal{R} $ defined on a standard measure space and which splits as the free product $ \mathcal{R} = { \mathcal{R} }_{1} \ast { \mathcal{R} }_{2} $ of recurrent subequivalence relations gives rise to a nonamenable factor $\mathrm{L} ( \mathcal{R} )$ with a unique Cartan subalgebra, up to unitary conjugacy. Finally, we prove unique Cartan decomposition for a class of group measure space factors ${\mathrm{L} }^{\infty } (X)\rtimes \Gamma $ arising from nonsingular free ergodic actions $\Gamma \curvearrowright (X, \mu )$ on standard measure spaces of amalgamated groups $\Gamma = {\mathop{\Gamma {}_{1} \ast }\nolimits}_{\Sigma } {\Gamma }_{2} $ over a finite subgroup $\Sigma $.

scholarly journals Some aspects of quantum sufficiency for finite and full von Neumann algebras

2021 ◽  
Vol 111 (4) ◽  
Author(s):  
Andrzej Łuczak
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Linear Operators ◽  
Bounded Linear ◽  
Von Neumann ◽  
Finite Von Neumann Algebra ◽  
Pure States ◽  
Normal States ◽  
Full Algebra

AbstractSome features of the notion of sufficiency in quantum statistics are investigated. Three kinds of this notion are considered: plain sufficiency (called simply: sufficiency), strong sufficiency and Umegaki’s sufficiency. It is shown that for a finite von Neumann algebra with a faithful family of normal states the minimal sufficient von Neumann subalgebra is sufficient in Umegaki’s sense. Moreover, a proper version of the factorization theorem of Jenčová and Petz is obtained. The structure of the minimal sufficient subalgebra is described in the case of pure states on the full algebra of all bounded linear operators on a Hilbert space.

Sign in / Sign up

Export Citation Format

Share Document