On the Tensor Products of JW-Algebras

1995 ◽  
Vol 47 (4) ◽  
pp. 786-800 ◽  
Author(s):  
Fatmah B. Jamjoom
Keyword(s):  
Tensor Product ◽  
Von Neumann Algebra ◽  
Tensor Products ◽  
Von Neumann ◽  
The Relationship ◽  
Type Decomposition

AbstractIn this article we introduce and develop a theory of tensor products of JW-algebras. Since JW-algebras are so close to W*-algebras, one can expect that the W*-algebra tensor product theory will be actively involved. It is shown that if Mand N are JW-algebras with centres Z1 and Z2 respectively, then Z1 ⊗ Z2 is not the centre of the JW-tensor product JW() (see below for notation) ofMand N, in general. Also, the type decomposition of JW() has been determined in terms of the type decomposition of the JW-algebras M and N which, essentially, rely on the relationship between the types of the JW-algebra and the types of its universal enveloping Von Neumann algebra.

A CONTINUUM OF C*-NORMS ON (H) ⊗ (H) AND RELATED TENSOR PRODUCTS

2015 ◽  
Vol 58 (2) ◽  
pp. 433-443 ◽  
Author(s):  
NARUTAKA OZAWA ◽  
GILLES PISIER
Keyword(s):  
Tensor Product ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Tensor Products ◽  
Von Neumann ◽  
C Algebras ◽  
Injective Tensor Product ◽  
Algebraic Tensor Product

AbstractFor any pair M, N of von Neumann algebras such that the algebraic tensor product M ⊗ N admits more than one C*-norm, the cardinal of the set of C*-norms is at least 2ℵ0. Moreover, there is a family with cardinality 2ℵ0 of injective tensor product functors for C*-algebras in Kirchberg's sense. Let ${\mathbb B}$=∏nMn. We also show that, for any non-nuclear von Neumann algebra M⊂ ${\mathbb B}$(ℓ2), the set of C*-norms on ${\mathbb B}$ ⊗ M has cardinality equal to 22ℵ0.

scholarly journals On the Tensor Products of Maximal Abelian -Algebras

2009 ◽  
Vol 2009 ◽  
pp. 1-7
Author(s):  
F. B. H. Jamjoom
Keyword(s):  
Tensor Product ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Tensor Products ◽  
Von Neumann ◽  
Close Relationship

It is well known in the work of Kadison and Ringrose (1983)that if and are maximal abelian von Neumann subalgebras of von Neumann algebras and , respectively, then is a maximal abelian von Neumann subalgebra of . It is then natural to ask whether a similar result holds in the context of -algebras and the -tensor product. Guided to some extent by the close relationship between a -algebra M and its universal enveloping von Neumann algebra , we seek in this article to investigate the answer to this question.

Extension of normal functionals on W*-tensor products

1975 ◽  
Vol 78 (2) ◽  
pp. 301-307 ◽  
Author(s):  
Simon Wassermann
Keyword(s):  
Representation Theory ◽  
Tensor Product ◽  
Hilbert Spaces ◽  
General Result ◽  
Von Neumann Algebras ◽  
Tensor Products ◽  
Von Neumann ◽  
Hilbert Algebras ◽  
Special Cases

A deep result in the theory of W*-tensor products, the Commutation theorem, states that if M and N are W*-algebras faithfully represented as von Neumann algebras on the Hilbert spaces H and K, respectively, then the commutant in L(H ⊗ K) of the W*-tensor product of M and N coincides with the W*-tensor product of M′ and N′. Although special cases of this theorem were established successively by Misonou (2) and Sakai (3), the validity of the general result remained conjectural until the advent of the Tomita-Takesaki theory of Modular Hilbert algebras (6). As formulated, the Commutation theorem is a spatial result; that is, the W*-algebras in its statement are taken to act on specific Hilbert spaces. Not surprisingly, therefore, known proofs rely heavily on techniques of representation theory.

scholarly journals TYPE DECOMPOSITION OF A PSEUDOEFFECT ALGEBRA

2010 ◽  
Vol 89 (3) ◽  
pp. 335-358 ◽  
Author(s):  
DAVID J. FOULIS ◽  
SYLVIA PULMANNOVÁ ◽  
ELENA VINCEKOVÁ
Keyword(s):  
Effect Algebra ◽  
Von Neumann Algebra ◽  
Von Neumann ◽  
Pseudoeffect Algebras ◽  
Noncommutative Version ◽  
Pseudoeffect Algebra ◽  
Determining Set ◽  
Direct Decompositions ◽  
Type Decomposition ◽  
Unsharp Observables

AbstractEffect algebras, which generalize the lattice of projections in a von Neumann algebra, serve as a basis for the study of unsharp observables in quantum mechanics. The direct decomposition of a von Neumann algebra into types I, II, and III is reflected by a corresponding decomposition of its lattice of projections, and vice versa. More generally, in a centrally orthocomplete effect algebra, the so-called type-determining sets induce direct decompositions into various types. In this paper, we extend the theory of type decomposition to a (possibly) noncommutative version of an effect algebra called a pseudoeffect algebra. It has been argued that pseudoeffect algebras constitute a natural structure for the study of noncommuting unsharp or fuzzy observables. We develop the basic theory of centrally orthocomplete pseudoeffect algebras, generalize the notion of a type-determining set to pseudoeffect algebras, and show how type-determining sets induce direct decompositions of centrally orthocomplete pseudoeffect algebras.

The connection between the universal enveloping C*-algebra and the universal enveloping von Neumann algebra of a JW-algebra

1992 ◽  
Vol 112 (3) ◽  
pp. 575-579 ◽  
Author(s):  
Fatmah B. Jamjoom
Keyword(s):  
Von Neumann Algebra ◽  
Main Result Theorem ◽  
Von Neumann ◽  
C Algebra ◽  
Result Theorem ◽  
The Relationship

AbstractThis article aims to study the relationship between the universal enveloping C*-algebra C*(M) and the universal enveloping von Neumann algebra W*(M), when M is a JW-algebra. In our main result (Theorem 2·7) we show that C*(M) can be realized as the C*-subalgebra of W*(M) generated by M.

scholarly journals VON NEUMANN ENTROPY AND RELATIVE POSITION BETWEEN SUBALGEBRAS

2013 ◽  
Vol 24 (08) ◽  
pp. 1350066 ◽  
Author(s):  
MARIE CHODA
Keyword(s):  
Tensor Product ◽  
Relative Position ◽  
Von Neumann Algebra ◽  
Positive Definite ◽  
The Other ◽  
Von Neumann Entropy ◽  
Product Decomposition ◽  
Positive Definite Matrices ◽  
Von Neumann ◽  
Finite Von Neumann Algebra

In order to give numerical characterizations of the notion of "mutual orthogonality", we introduce two kinds of family of positive definite matrices for a unitary u in a finite von Neumann algebra M. They are arising from u naturally depending on the decompositions of M. One corresponds to the tensor product decomposition and the other does to the crossed product decomposition. By using the von Neumann entropy for these positive definite matrices, we characterize the notion of mutual orthogonality between subalgebras.

scholarly journals Schur Multipliers of Cartan Pairs

2016 ◽  
Vol 60 (2) ◽  
pp. 413-440
Author(s):  
R. H. Levene ◽  
N. Spronk ◽  
I. G. Todorov ◽  
L. Turowska
Keyword(s):  
Tensor Product ◽  
Direct Summand ◽  
Von Neumann Algebra ◽  
Von Neumann ◽  
Schur Multipliers

AbstractWe define the Schur multipliers of a separable von Neumann algebrawith Cartan maximal abelian self-adjoint algebra, generalizing the classical Schur multipliers of(ℓ2). We characterize these as the normal-bimodule maps on. Ifcontains a direct summand isomorphic to the hyperfinite II1factor, then we show that the Schur multipliers arising from the extended Haagerup tensor product⊗ehare strictly contained in the algebra of all Schur multipliers.

Type Decomposition and the Rectangular AFD Property for W*-TRO’s

2004 ◽  
Vol 56 (4) ◽  
pp. 843-870 ◽  
Author(s):  
Zhong-Jin Ruan
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Operator Space ◽  
Dual Operator ◽  
Von Neumann ◽  
Direct Sum ◽  
Cardinal Numbers ◽  
Finite Dimensional ◽  
Dimensional Property ◽  
Type Decomposition

AbstractWe study the type decomposition and the rectangular AFD property for W*-TRO’s. Like von Neumann algebras, every W*-TRO can be uniquely decomposed into the direct sum of W*- TRO's of type I, type II, and type III. We may further considerW*-TRO's of type Im,n with cardinal numbers m and n, and considerW*-TRO's of type IIλ,μ with λ, μ = 1 or ∞. It is shown that every separable stable W*-TRO (which includes type I∞, ∞, type II∞, ∞ and type III) is TRO-isomorphic to a von Neumann algebra. We also introduce the rectangular version of the approximately finite dimensional property for W*-TRO’s. One of our major results is to show that a separable W*-TRO is injective if and only if it is rectangularly approximately finite dimensional. As a consequence of this result, we show that a dual operator space is injective if and only if its operator predual is a rigid rectangular space (equivalently, a rectangular space).

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.

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