Relative commutant of a von Neumann algebra in its crossed product by a group action

1978 ◽  
Vol 163 (1) ◽  
pp. 5-13 ◽  
Author(s):  
William L. Paschke
Keyword(s):  
Group Action ◽  
Von Neumann Algebra ◽  
Crossed Product ◽  
Von Neumann ◽  
Relative Commutant

Relative commutant of an unbounded operator affiliated with a finite von Neumann algebra

2016 ◽  
Vol 75 (1) ◽  
pp. 209-223 ◽  
Author(s):  
Don Hadwin ◽  
Junhao Shen ◽  
Wenming Wu ◽  
Wei Yuan
Keyword(s):  
Unbounded Operator ◽  
Von Neumann Algebra ◽  
Von Neumann ◽  
Finite Von Neumann Algebra ◽  
Relative Commutant

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.

A mean ergodic theorem of an amenable group action

2014 ◽  
Vol 17 (01) ◽  
pp. 1450003 ◽  
Author(s):  
Anilesh Mohari
Keyword(s):  
Ergodic Theorem ◽  
Group Action ◽  
Von Neumann Algebra ◽  
Amenable Group ◽  
Ergodic Theorems ◽  
Von Neumann ◽  
Mean Ergodic Theorem ◽  
Faithful Normal State ◽  
Contractive Maps ◽  
Dual Banach Space

We consider a sequence of weak Kadison–Schwarz maps τn on a von-Neumann algebra ℳ with a faithful normal state ϕ sub-invariant for each (τn, n ≥ 1) and use a duality argument to prove strong convergence of their pre-dual maps when their induced contractive maps (Tn, n ≥ 1) on the GNS space of (ℳ, ϕ) are strongly convergent. The result is applied to deduce improvements of some known ergodic theorems and Birkhoff's mean ergodic theorem for any locally compact second countable amenable group action on the pre-dual Banach space ℳ*.

Non-Isomorphic Non-Hyperfinite Factors

1969 ◽  
Vol 21 ◽  
pp. 1293-1308 ◽  
Author(s):  
Wai-Mee Ching
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Crossed Product ◽  
Type Ii ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Type Iii ◽  
Infinite Dimensional Hilbert Space ◽  
Finite Dimensional ◽  
Dimensional Hilbert Space

A von Neumann algebra is called hyperfinite if it is the weak closure of an increasing sequence of finite-dimensional von Neumann subalgebras. For a separable infinite-dimensional Hilbert space the following is known: there exist hyperfinite and non-hyperfinite factors of type II1 (4, Theorem 16’), and of type III (8, Theorem 1); all hyperfinite factors of type Hi are isomorphic (4, Theorem 14); there exist uncountably many non-isomorphic hyperfinite factors of type III (7, Theorem 4.8); there exist two nonisomorphic non-hyperfinite factors of type II1 (10), and of type III (11). In this paper we will show that on a separable infinite-dimensional Hilbert space there exist three non-isomorphic non-hyperfinite factors of type II1 (Theorem 2), and of type III (Theorem 3).Section 1 contains an exposition of crossed product, which is developed mainly for the construction of factors of type III in § 3.

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 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}}.

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