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.

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.

FROM QUANTUM FIELDS TO LOCAL VON NEUMANN ALGEBRAS

1992 ◽  
Vol 04 (spec01) ◽  
pp. 15-47 ◽  
Author(s):  
H.J. BORCHERS ◽  
JAKOB YNGVASON
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebras ◽  
Sufficient Conditions ◽  
Bounded Operators ◽  
Unbounded Operators ◽  
Von Neumann ◽  
Energy Bounds ◽  
Local Algebras ◽  
Quantized Fields ◽  
Necessary And Sufficient

The subject of the paper is an old problem of the general theory of quantized fields: When can the unbounded operators of a Wightman field theory be associated with local algebras of bounded operators in the sense of Haag? The paper reviews and extends previous work on this question, stressing its connections with a noncommutive generalization of the classical Hamburger moment problem. Necessary and sufficient conditions for the existence of a local net of von Neumann algebras corresponding to a given Wightman field are formulated in terms of strengthened versions of the usual positivity property of Wightman functionals. The possibility that the local net has to be defined in an enlarged Hilbert space cannot be ruled out in general. Under additional hypotheses, e.g., if the field operators obey certain energy bounds, such an extension of the Hilbert space is not necessary, however. In these cases a fairly simple condition for the existence of a local net can be given involving the concept of “central positivity” introduced by Powers. The analysis presented here applies to translationally covariant fields with an arbitrary number of components, whereas Lorentz covariance is not needed. The paper contains also a brief discussion of an approach to noncommutative moment problems due to Dubois-Violette, and concludes with some remarks on modular theory for algebras of unbounded 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.

On the Duality of Operator Spaces

1995 ◽  
Vol 38 (3) ◽  
pp. 334-346 ◽  
Author(s):  
Christian Le Merdy
Keyword(s):  
Banach Space ◽  
Von Neumann Algebras ◽  
Related Result ◽  
Space Structure ◽  
Operator Space ◽  
Operator Spaces ◽  
Bounded Operators ◽  
Von Neumann ◽  
Operator Space Structure ◽  
Dual Banach Space

AbstractWe prove that given an operator space structure on a dual Banach space Y*, it is not necessarily the dual one of some operator space structure on Y. This allows us to show that Sakai's theorem providing the identification between C*-algebras having a predual and von Neumann algebras does not extend to the category of operator spaces. We also include a related result about completely bounded operators from B(ℓ2)* into the operator Hilbert space OH.

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 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.

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