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.

scholarly journals Derivations on commutative operator algebras

1985 ◽  
Vol 32 (3) ◽  
pp. 415-418
Author(s):  
Mark Spivack
Keyword(s):  
Hilbert Space ◽  
Banach Spaces ◽  
Von Neumann Algebra ◽  
Operator Algebras ◽  
Bounded Operator ◽  
Alternative Proof ◽  
Reflexive Banach Spaces ◽  
Von Neumann ◽  
Simple Alternative

It is well-known that any derivation on a commutative von Neumann algebra is implemented by a bounded operator. In this note we present a simple alternative proof, which generalizes the result further within Hilbert space, and to reflexive Banach spaces.

Nonlinear ∗-Jordan triple derivations on von Neumann algebras

2018 ◽  
Vol 68 (1) ◽  
pp. 163-170 ◽  
Author(s):  
Fangfang Zhao ◽  
Changjing Li
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
New Product ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Von Neumann

AbstractLetB(H) be the algebra of all bounded linear operators on a complex Hilbert spaceHand 𝓐 ⊆B(H) be a von Neumann algebra with no central summands of typeI1. ForA,B∈ 𝓐, define byA∙B=AB+BA∗a new product ofAandB. In this article, it is proved that a map Φ: 𝓐 →B(H) satisfies Φ(A∙B∙C) = Φ(A) ∙B∙C+A∙ Φ(B) ∙C+A∙B∙Φ(C) for allA,B,C∈ 𝓐 if and only if Φ is an additive *-derivation.

Ergodic theorems for semifinite von Neumann algebras: II

1980 ◽  
Vol 88 (1) ◽  
pp. 135-147 ◽  
Author(s):  
F. J. Yeadon
Keyword(s):  
Banach Spaces ◽  
Ergodic Theorem ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Operator Norm ◽  
Ergodic Theorems ◽  
Unbounded Operators ◽  
Von Neumann ◽  
The Mean ◽  
Image Position

In (7) we proved maximal and pointwise ergodic theorems for transformations a of a von Neumann algebra which are linear positive and norm-reducing for both the operator norm ‖ ‖∞ and the integral norm ‖ ‖1 associated with a normal trace ρ on . Here we introduce a class of Banach spaces of unbounded operators, including the Lp spaces defined in (6), in which the transformations α reduce the norm, and in which the mean ergodic theorem holds; that is the averagesconverge in norm.

On the von Neumann algebras associated to Yang–Baxter operators

2020 ◽  
pp. 1-24
Author(s):  
Panchugopal Bikram ◽  
Rahul Kumar ◽  
Rajeeb Mohanta ◽  
Kunal Mukherjee ◽  
Diptesh Saha
Keyword(s):  
Hilbert Space ◽  
Special Form ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Adjoint Operator ◽  
Complex Hilbert Space ◽  
Von Neumann ◽  
Finite Von Neumann Algebra

Bożejko and Speicher associated a finite von Neumann algebra M T to a self-adjoint operator T on a complex Hilbert space of the form $\mathcal {H}\otimes \mathcal {H}$ which satisfies the Yang–Baxter relation and $ \left\| T \right\| < 1$ . We show that if dim $(\mathcal {H})$ ⩾ 2, then M T is a factor when T admits an eigenvector of some special form.

scholarly journals Remarks on complemented subspaces of von Neumann algebras

1992 ◽  
Vol 121 (1-2) ◽  
pp. 1-4 ◽  
Author(s):  
G. Pisier
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Von Neumann ◽  
C Algebra ◽  
Know How ◽  
Complemented Subspaces ◽  
Reflexive Subspace

SynopsisIn this note we include two remarks about bounded (not necessarily contractive) linear projections on a von Neumann algebra. We show that if M is a von Neumann subalgebra of B(H) which is complemented in B(H) and isomorphic to M⊗M, then M is injective (or equivalently M is contractively complemented). We do not know how to get rid of the second assumption on M. In the second part, we show that any complemented reflexive subspace of a C*-algebra is necessarily linearly isomorphic to a Hilbert space.

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 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.

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.

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.

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