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.

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.

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]).

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.

scholarly journals On the central Haagerup tensor product

1994 ◽  
Vol 37 (1) ◽  
pp. 161-174 ◽  
Author(s):  
Pere Ara ◽  
Martin Mathieu
Keyword(s):  
Tensor Product ◽  
Large Class ◽  
Von Neumann Algebras ◽  
Bounded Operators ◽  
Multiplier Algebra ◽  
Von Neumann ◽  
Isometric Representation

For a large class of C*-algebras including all von Neumann algebras, the central Haagerup tensor product of the multiplier algebra with itself has an isometric representation as completely bounded operators.

Entropy for extensions of Bernoulli shifts

1996 ◽  
Vol 16 (6) ◽  
pp. 1197-1206 ◽  
Author(s):  
Marie Choda
Keyword(s):  
Tensor Product ◽  
Von Neumann Algebras ◽  
Bernoulli Shift ◽  
Product Formula ◽  
Von Neumann ◽  
Bernoulli Shifts ◽  
Outer Automorphisms ◽  
Finite Von Neumann Algebras

AbstractWe give a condition for automorphisms α and β on finite von Neumann algebras which induces the tensor product formula for entropies: H(α ⊗ β) = H(α) + H(β). As an application, the Bernoulli shift (1/n, 1/n, …, 1/n) has extensions to ergodic outer automorphisms {αk; k = 1,2, …} on the hyperfinite II1 factor R with the entropies H(αk) = (1/2)kn log n.

The Krull and global dimension of the tensor product of quantum tori

2016 ◽  
Vol 15 (09) ◽  
pp. 1650174
Author(s):  
Ashish Gupta
Keyword(s):  
Tensor Product ◽  
Upper Bound ◽  
Tensor Products ◽  
Global Dimension ◽  
Base Field ◽  
Quantum Torus ◽  
Common Value ◽  
Quantum Tori ◽  
Special Cases ◽  
The Common

An [Formula: see text]-dimensional quantum torus is defined as the [Formula: see text]-algebra generated by variables [Formula: see text] together with their inverses satisfying the relations [Formula: see text], where [Formula: see text]. The Krull and global dimensions of this algebra are known to coincide and the common value is equal to the supremum of the rank of certain subgroups of [Formula: see text] that can be associated with this algebra. In this paper we study how these dimensions behave with respect to taking tensor products of quantum tori over the base field. We derive a best possible upper bound for the dimension of such a tensor product and from this special cases in which the dimension is additive with respect to tensoring.

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