scholarly journals Subalgebras of C*-algebras and von Neumann algebras

1984 ◽  
Vol 25 (1) ◽  
pp. 19-25 ◽  
Author(s):  
Charles A. Akemann
Keyword(s):  
Recent Work ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Weierstrass Theorem ◽  
Von Neumann ◽  
Matrix Algebras ◽  
C Algebras ◽  
Weakly Closed

Recent work [2, 6] on subalgebras of matrix algebras leads naturally to the following situation. Let A be a C*-subalgebra of the C*-algebra B andM be a weakly closed *-subalgebra of the von Neumann algebra N. Consider the following Conditions.Condition 1. For every b≠ 0 in B there exists a ∈ A such that O≠ab ∈ A.Condition 2. For every b∈B there exists a ≠ 0 in A such that ab ∈ A.If we replace A by M and B by N in Conditions 1 and 2 we get von Neumann algebra versions which we shall call Condition 1'and Condition 2'. Clearly Condition 1 implies Condition 2, and both conditions suggest that A is some kind of weak ideal of B. This paper explores the extent to which this is true. The paper grew out of the author's attempts [1, 3] to generalize the Stone-Weierstrass theorem to C*-algebras.

scholarly journals A REMARK ON THE STRUCTURE OF SYMMETRIC QUANTUM DYNAMICAL SEMIGROUPS ON VON NEUMANN ALGEBRAS

2002 ◽  
Vol 05 (04) ◽  
pp. 571-579 ◽  
Author(s):  
SERGIO ALBEVERIO ◽  
DEBASHISH GOSWAMI
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Type I ◽  
Quantum Dynamical Semigroup ◽  
Von Neumann ◽  
Matrix Algebras ◽  
Quantum Dynamical ◽  
Dynamical Semigroup ◽  
Quantum Dynamical Semigroups ◽  
Symmetric Quantum

We study the structure of the generator of a symmetric, conservative quantum dynamical semigroup with norm-bounded generator on a von Neumann algebra equipped with a faithful semifinite trace. For von Neumann algebras with Abelian commutant (i.e. type I von Neumann algebras), we give a necessary and sufficient algebraic condition for the generator of such a semigroup to be written as a sum of square of self-adjoint derivations of the von Neumann algebra. This generalizes some of the results obtained by Albeverio, Høegh-Krohn and Olsen1 for the special case of the finite-dimensional matrix algebras. We also study similar questions for a class of quantum dynamical semigroups with unbounded generators.

On Approximately Finite-Dimensional Von Neuman Algebras, II

1978 ◽  
Vol 21 (4) ◽  
pp. 415-418 ◽  
Author(s):  
George A. Elliott
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Von Neumann ◽  
Intrinsic Characterization ◽  
C Algebras ◽  
Finite Dimensional ◽  
Positive Maps ◽  
Injective Von Neumann Algebra

AbstractAn intrinsic characterization is given of those von Neumann algebras which are injective objects in the category of C*-algebras with completely positive maps. For countably generated von Neumann algebras several such characterizations have been given, so it is in fact enough to observe that an injective von Neumann algebra is generated by an upward directed collection of injective countably generated sub von Neumann algebras. The present work also shows that three of the intrinsic characterizations known in the countably generated case hold in general.

Approximately Periodic Functionals on C*-Algebras and von Neumann Algebras

1985 ◽  
Vol 37 (5) ◽  
pp. 769-784 ◽  
Author(s):  
John C. Quigg
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Related Question ◽  
Compact Groups ◽  
Von Neumann ◽  
Linear Map ◽  
C Algebras ◽  
Preceding Question ◽  
Special Case ◽  
Algebraic Tensor Product

In the duality for locally compact groups, much use is made of a version of the Hopf algebra technique in the context of von Neumann algebras, culminating in the theory of Kac algebras [6], [14]. It seems natural to ask whether something like a Hopf algebraic structure can be defined on the pre-dual of a Kac algebra. This leads to the question of whether the multiplication on a von Neumann algebra M, viewed as a linear map m from M ⊙ M (the algebraic tensor product) to M, can be pre-transposed to give a co-multiplication on the pre-dual M*, i.e., a linear map m* from M* to the completion of M* ⊙ M* with respect to some cross-norm. A related question is whether the multiplication on a C*-algebra A can be transposed to give a co-multiplication on the dual A*. Of course, this can be regarded as a special case of the preceding question by taking M = A**, where the double dual A** is identified with the enveloping von Neumann algebra of A.

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.

Analytic Subalgebras of Von Neumann Algebras

1987 ◽  
Vol 39 (1) ◽  
pp. 74-99 ◽  
Author(s):  
Paul S. Muhly ◽  
Kichi-Suke Saito
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Identity Operator ◽  
Boundary Values ◽  
Von Neumann ◽  
Weakly Continuous ◽  
Closed Subalgebra ◽  
Weakly Closed ◽  
Upper Half Plane ◽  
Image Position

Let M be a von Neumann algebra and let {αt}t∊R be a σ-weakly continuous flow on M; i.e., suppose that {αt}t∊R is a one-parameter group of *-automorphisms of M such that for each ρ in the predual, M∗, of M and for each x ∊ M, the function of t, ρ(αt(x)), is continuous on R. In recent years, considerable attention has been focused on the subspace of M, H∞(α), which is defined to bewhere H∞(R) is the classical Hardy space consisting of the boundary values of functions bounded analytic in the upper half-plane. In Theorem 3.15 of [8] it is proved that in fact H∞(α) is a σ-weakly closed subalgebra of M containing the identity operator such thatis σ-weakly dense in M, and such that

scholarly journals Free Product C* -algebras Associated with Graphs, Free Differentials, and Laws of Loops

2017 ◽  
Vol 69 (3) ◽  
pp. 548-578 ◽  
Author(s):  
Michael Hartglass
Keyword(s):  
Differential Calculus ◽  
Von Neumann Algebras ◽  
Sufficient Conditions ◽  
Weighted Graph ◽  
Positive Cone ◽  
Von Neumann ◽  
Planar Algebras ◽  
C Algebra ◽  
C Algebras ◽  
Necessary And Sufficient

AbstractWe study a canonical C* -algebra, 𝒮(Г,μ), that arises from a weighted graph (Г,μ), speci fic cases of which were previously studied in the context of planar algebras. We discuss necessary and sufficient conditions of the weighting that ensure simplicity and uniqueness of trace of 𝒮(Г,μ), and study the structure of its positive cone. We then study the *-algebra,𝒜, generated by the generators of 𝒮(Г,μ), and use a free differential calculus and techniques of Charlesworth and Shlyakhtenko as well as Mai, Speicher, and Weber to show that certain “loop” elements have no atoms in their spectral measure. After modifying techniques of Shlyakhtenko and Skoufranis to show that self adjoint elements x ∊ Mn(𝒜) have algebraic Cauchy transform, we explore some applications to eigenvalues of polynomials inWishart matrices and to diagrammatic elements in von Neumann algebras initially considered by Guionnet, Jones, and Shlyakhtenko.

scholarly journals ON EQUALITY CONDITION FOR TRACE JENSEN INEQUALITY IN SEMI-FINITE VON NEUMANN ALGEBRAS

2008 ◽  
Vol 19 (04) ◽  
pp. 481-501 ◽  
Author(s):  
TETSUO HARADA ◽  
HIDEKI KOSAKI
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Adjoint Operator ◽  
Strict Convexity ◽  
Jensen Inequality ◽  
Von Neumann ◽  
Convexity Assumption ◽  
Finite Von Neumann Algebra ◽  
Equality Condition ◽  
Finite Von Neumann Algebras

Let τ be a faithful semi-finite normal trace on a semi-finite von Neumann algebra, and f(t) be a convex function with f(0) = 0. The trace Jensen inequality states τ(f(a* xa)) ≤ τ(a* f(x)a) for a contraction a and a self-adjoint operator x. Under certain strict convexity assumption on f(t), we will study when this inequality reduces to the equality.

Decomposability of von Neumann Algebras and the Mazur Property of Higher Level

2006 ◽  
Vol 58 (4) ◽  
pp. 768-795 ◽  
Author(s):  
Zhiguo Hu ◽  
Matthias Neufang
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Group Structure ◽  
Cardinal Number ◽  
Banach Algebras ◽  
Locally Compact Group ◽  
Measure Algebra ◽  
Locally Compact ◽  
Von Neumann ◽  
Topological Centre

AbstractThe decomposability number of a von Neumann algebra ℳ (denoted by dec(ℳ)) is the greatest cardinality of a family of pairwise orthogonal non-zero projections in ℳ. In this paper, we explore the close connection between dec(ℳ) and the cardinal level of the Mazur property for the predual ℳ* of ℳ, the study of which was initiated by the second author. Here, our main focus is on those von Neumann algebras whose preduals constitute such important Banach algebras on a locally compact group G as the group algebra L1(G), the Fourier algebra A(G), the measure algebra M(G), the algebra LUC(G)*, etc. We show that for any of these von Neumann algebras, say ℳ, the cardinal number dec(ℳ) and a certain cardinal level of the Mazur property of ℳ* are completely encoded in the underlying group structure. In fact, they can be expressed precisely by two dual cardinal invariants of G: the compact covering number κ(G) of G and the least cardinality ᙭(G) of an open basis at the identity of G. We also present an application of the Mazur property of higher level to the topological centre problem for the Banach algebra A(G)**.

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.

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