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.

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.

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.

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.

The Haagerup invariant for tensor products of operator spaces

1996 ◽  
Vol 120 (1) ◽  
pp. 147-153 ◽  
Author(s):  
Allan M. Sinclair ◽  
Roger R. Smith
Keyword(s):  
Tensor Product ◽  
Approximation Property ◽  
Von Neumann Algebras ◽  
Tensor Products ◽  
Dual Operator ◽  
Operator Spaces ◽  
Von Neumann

In [7, 8] Haagerup introduced two isomorphism invariants and for C*-algebras and von Neumann algebras , based on appropriate forms of the completely bounded approximation property defined below. These definitions have obvious extensions to operator spaces and dual operator spaces respectively, and in [16] we established the multiplicativity of A on the ultraweakly closed spatial tensor product of two dual operator spaces and :

scholarly journals Tensor Products of Noncommutative Lp-Spaces

ISRN Algebra ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Somlak Utudee
Keyword(s):  
Tensor Product ◽  
Von Neumann Algebras ◽  
Tensor Products ◽  
Von Neumann ◽  
Lp Spaces ◽  
Finite Von Neumann Algebras

We consider the notion of tensor product of noncommutative Lp spaces associated with finite von Neumann algebras and define the notion of tensor product of Haagerup noncommutative Lp spaces associated with σ-finite von Neumann algebras.

scholarly journals Matrix models for $$\varvec{\varepsilon }$$-free independence

2021 ◽  
Author(s):  
Ian Charlesworth ◽  
Benoît Collins
Keyword(s):  
Tensor Product ◽  
Matrix Models ◽  
Random Matrices ◽  
Von Neumann Algebras ◽  
Tensor Products ◽  
Product Structure ◽  
Graph Products ◽  
Products Of Random Matrices ◽  
Von Neumann ◽  
Free Independence

AbstractWe investigate tensor products of random matrices, and show that independence of entries leads asymptotically to $$\varepsilon $$ ε -free independence, a mixture of classical and free independence studied by Młotkowski and by Speicher and Wysoczański. The particular $$\varepsilon $$ ε arising is prescribed by the tensor product structure chosen, and conversely, we show that with suitable choices an arbitrary $$\varepsilon $$ ε may be realized in this way. As a result, we obtain a new proof that $$\mathcal {R}^\omega $$ R ω -embeddability is preserved under graph products of von Neumann algebras, along with an explicit recipe for constructing matrix models.

STRUCTURE RESULTS FOR NORMALIZERS OF II1 FACTORS

2011 ◽  
Vol 22 (07) ◽  
pp. 947-979 ◽  
Author(s):  
JAN M. CAMERON
Keyword(s):  
Tensor Product ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Type Theorem ◽  
Discrete Structure ◽  
Von Neumann ◽  
Product Group ◽  
Countable Subgroup ◽  
Finite Von Neumann Algebras ◽  
Group Von Neumann Algebra

For an inclusion N ⊆ M of II1 factors, we study the group of normalizers [Formula: see text] and the von Neumann algebra it generates. We first show that [Formula: see text] imposes a certain "discrete" structure on the generated von Neumann algebra. By analyzing the bimodule structure of certain subalgebras of [Formula: see text], this leads to a "Galois-type" theorem for normalizers, in which we find a description of the subalgebras of [Formula: see text] in terms of a unique countable subgroup of [Formula: see text]. Implications for inclusions B ⊆ M arising from the crossed product, group von Neumann algebra, and tensor product constructions will also be addressed. Our work leads to a construction of new examples of norming subalgebras in finite von Neumann algebras: If N ⊆ M is a regular inclusion of II1 factors, then N norms M.

Non-Isomorphic Tensor Products of Von Neumann Algebras

1974 ◽  
Vol 26 (02) ◽  
pp. 492-512 ◽  
Author(s):  
J. J. Williams
Keyword(s):  
Tensor Product ◽  
Von Neumann Algebras ◽  
Tensor Products ◽  
Von Neumann ◽  
Algebraic Invariants

This paper investigates special conditions under which the tensor product of two von Neumann algebras will be non-isomorphic to the tensor product of two others. The main tools are the algebraic invariants property Λ x (x ≧ 0) (first defined by Powers [18]) and the r ∞ and ρ sets (defined by Araki and Woods [3]).

scholarly journals CO-REPRESENTATIONS OF HOPF–VON NEUMANN ALGEBRAS ON OPERATOR SPACES OTHER THAN COLUMN HILBERT SPACE

2010 ◽  
Vol 82 (2) ◽  
pp. 205-210 ◽  
Author(s):  
VOLKER RUNDE
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Banach Algebra ◽  
Banach Spaces ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Operator Spaces ◽  
Reflexive Banach Spaces ◽  
Von Neumann ◽  
Reflexive Operator

AbstractRecently, Daws introduced a notion of co-representation of abelian Hopf–von Neumann algebras on general reflexive Banach spaces. In this note, we show that this notion cannot be extended beyond subhomogeneous Hopf–von Neumann algebras. The key is our observation that, for a von Neumann algebra 𝔐 and a reflexive operator space E, the normal spatial tensor product $\M \btensor \CB (E)$ is a Banach algebra if and only if 𝔐 is subhomogeneous or E is completely isomorphic to column Hilbert space.

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