Intermediate subalgebras and bimodules for general crossed products of von Neumann algebras

2016 ◽  
Vol 27 (11) ◽  
pp. 1650091 ◽  
Author(s):  
Jan M. Cameron ◽  
Roger R. Smith
Keyword(s):  
Discrete Group ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Crossed Product ◽  
Crossed Products ◽  
General Version ◽  
Von Neumann ◽  
Outer Automorphisms ◽  
Continuous Maps ◽  
Mercer’S Theorem

Let [Formula: see text] be a discrete group acting on a von Neumann algebra [Formula: see text] by properly outer ∗-automorphisms. In this paper, we study the containment [Formula: see text] of [Formula: see text] inside the crossed product. We characterize the intermediate von Neumann algebras, extending earlier work of other authors in the factor case. We also determine the [Formula: see text]-bimodules that are closed in the Bures topology and which coincide with the [Formula: see text]-closed ones under a mild hypothesis on [Formula: see text]. We use these results to obtain a general version of Mercer’s theorem concerning the extension of certain isometric [Formula: see text]-continuous maps on [Formula: see text]-bimodules to ∗-automorphisms of the containing von Neumann algebras.

scholarly journals On the entropy in II1 von Neumann algebras

1981 ◽  
Vol 1 (4) ◽  
pp. 419-429 ◽  
Author(s):  
O. Besson
Keyword(s):  
Lebesgue Space ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Point Spectrum ◽  
Crossed Product ◽  
Pure Point ◽  
Pure Point Spectrum ◽  
Von Neumann ◽  
Finite Von Neumann Algebra ◽  
Spectrum Transformation

AbstractLet α be an automorphism of a finite von Neumann algebra and let H(α) be its Connes-Størmer's entropy. We show that H(α) = 0 if α is the induced automorphism on the crossed product of a Lebesgue space by a pure point spectrum transformation. We also show that H is not continuous in α and we compute H(α) for some α.

BOUNDED COCYCLES ON FINITE VON NEUMANN ALGEBRAS

2001 ◽  
Vol 12 (06) ◽  
pp. 743-750 ◽  
Author(s):  
TERESA BATES ◽  
THIERRY GIORDANO
Keyword(s):  
Discrete Group ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Von Neumann ◽  
Countable Discrete Group ◽  
Finite Von Neumann Algebra ◽  
Finite Von Neumann Algebras ◽  
Uniformly Bounded

In this note we prove that if G is a countable discrete group, then every uniformly bounded cocycle from a standard Borel G-space into a finite Von Neumann algebra is cohomologous to a unitary cocycle. This generalizes results of both F. H. Vasilescu and L. Zsidó and R. J. Zimmer.

A Galois Correspondence for Reduced Crossed Products of Simple -algebras by Discrete Groups

2019 ◽  
Vol 71 (5) ◽  
pp. 1103-1125 ◽  
Author(s):  
Jan Cameron ◽  
Roger R. Smith
Keyword(s):  
Dynamical System ◽  
Discrete Group ◽  
Crossed Product ◽  
Discrete Groups ◽  
Crossed Products ◽  
Outer Automorphisms ◽  
Galois Correspondence ◽  
Simple Algebras ◽  
Unital Simple

AbstractLet a discrete group $G$ act on a unital simple $\text{C}^{\ast }$-algebra $A$ by outer automorphisms. We establish a Galois correspondence $H\mapsto A\rtimes _{\unicode[STIX]{x1D6FC},r}H$ between subgroups of $G$ and $\text{C}^{\ast }$-algebras $B$ satisfying $A\subseteq B\subseteq A\rtimes _{\unicode[STIX]{x1D6FC},r}G$, where $A\rtimes _{\unicode[STIX]{x1D6FC},r}G$ denotes the reduced crossed product. For a twisted dynamical system $(A,G,\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D70E})$, we also prove the corresponding result for the reduced twisted crossed product $A\rtimes _{\unicode[STIX]{x1D6FC},r}^{\unicode[STIX]{x1D70E}}G$.

scholarly journals Spectral gap characterization of full type III factors

2019 ◽  
Vol 2019 (753) ◽  
pp. 193-210 ◽  
Author(s):  
Amine Marrakchi
Keyword(s):  
Discrete Group ◽  
Von Neumann Algebra ◽  
Spectral Gap ◽  
Crossed Product ◽  
Von Neumann ◽  
Type Iii ◽  
Outer Action ◽  
Usual Topology ◽  
Sigma G

AbstractWe give a spectral gap characterization of fullness for type {\mathrm{III}} factors which is the analog of a theorem of Connes in the tracial case. Using this criterion, we generalize a theorem of Jones by proving that if M is a full factor and {\sigma:G\rightarrow\mathrm{Aut}(M)} is an outer action of a discrete group G whose image in {\mathrm{Out}(M)} is discrete, then the crossed product von Neumann algebra {M\rtimes_{\sigma}G} is also a full factor. We apply this result to prove the following conjecture of Tomatsu–Ueda: the continuous core of a type {\mathrm{III}_{1}} factor M is full if and only if M is full and its τ invariant is the usual topology on {\mathbb{R}}.

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.

The Asymptotic Ratio Set and Direct Integral Decompositions of a Von Neumann Algebra

1971 ◽  
Vol 23 (4) ◽  
pp. 598-607 ◽  
Author(s):  
Ole A. Nielsen
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Direct Integral ◽  
Von Neumann ◽  
Ratio Set ◽  
Almost All ◽  
Direct Integral Decomposition ◽  
Integral Decomposition ◽  
Remarkable Progress

The fact that any von Neumann algebra on a separable Hilbert space has an essentially unique direct integral decomposition into factors means that there is a global as well as a local aspect to any partial classification of von Neumann algebras. More precisely, suppose that J is a statement about von Neumann algebras which is either true or false for any given von Neumann algebra. Then a von Neumann algebra is said to satisfy J globally if it satisfies J, and to satsify J locally if almost all the factors appearing in some (and hence in any) central decomposition of it satisfy J . In a recent paper [3], H. Araki and E. J. Woods introduced the notion of the asymptotic ratio set of a factor, and by means of this they made remarkable progress in the classification of factors.

Compact Derivations on Von Neumann Algebras

1981 ◽  
Vol 24 (1) ◽  
pp. 87-90
Author(s):  
Sze-Kai Tsui
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Central Projection ◽  
Von Neumann

AbstractIf is a von Neumann algebra that thas no nonzero finite discrete central projection, then there is no nontrivial compact derivation of into itself.

scholarly journals Gradient forms and strong solidity of free quantum groups

2020 ◽  
Author(s):  
Martijn Caspers
Keyword(s):  
Quantum Groups ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Dirichlet Forms ◽  
Von Neumann ◽  
Finite Von Neumann Algebra ◽  
Crucial Part

AbstractConsider the free orthogonal quantum groups $$O_N^+(F)$$ O N + ( F ) and free unitary quantum groups $$U_N^+(F)$$ U N + ( F ) with $$N \ge 3$$ N ≥ 3 . In the case $$F = \text {id}_N$$ F = id N it was proved both by Isono and Fima-Vergnioux that the associated finite von Neumann algebra $$L_\infty (O_N^+)$$ L ∞ ( O N + ) is strongly solid. Moreover, Isono obtains strong solidity also for $$L_\infty (U_N^+)$$ L ∞ ( U N + ) . In this paper we prove for general $$F \in GL_N(\mathbb {C})$$ F ∈ G L N ( C ) that the von Neumann algebras $$L_\infty (O_N^+(F))$$ L ∞ ( O N + ( F ) ) and $$L_\infty (U_N^+(F))$$ L ∞ ( U N + ( F ) ) are strongly solid. A crucial part in our proof is the study of coarse properties of gradient bimodules associated with Dirichlet forms on these algebras and constructions of derivations due to Cipriani–Sauvageot.

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