scholarly journals Factoriality, Connes' type III invariants and fullness of amalgamated free product von Neumann algebras

2019 ◽  
Vol 150 (3) ◽  
pp. 1495-1532
Author(s):  
Cyril Houdayer ◽  
Yusuke Isono
Keyword(s):  
Free Product ◽  
Von Neumann Algebras ◽  
Rigidity Theory ◽  
Von Neumann ◽  
Type Iii ◽  
Amalgamated Free Product

AbstractWe investigate factoriality, Connes' type III invariants and fullness of arbitrary amalgamated free product von Neumann algebras using Popa's deformation/rigidity theory. Among other things, we generalize many previous structural results on amalgamated free product von Neumann algebras and we obtain new examples of full amalgamated free product factors for which we can explicitely compute Connes' type III invariants.

scholarly journals Radial multipliers on amalgamated free products of II1-factors

2014 ◽  
Vol 25 (03) ◽  
pp. 1450026
Author(s):  
Sören Möller
Keyword(s):  
Free Product ◽  
Von Neumann Algebras ◽  
Hankel Matrix ◽  
Trace Class ◽  
Free Products ◽  
Von Neumann ◽  
Amalgamated Free Products ◽  
Amalgamated Free Product ◽  
Radial Multipliers ◽  
Completely Bounded Map

Let ℳi be a family of II1-factors, containing a common II1-subfactor 𝒩, such that [ℳi : 𝒩] ∈ ℕ0 for all i. Furthermore, let ϕ: ℕ0 → ℂ. We show that if a Hankel matrix related to ϕ is trace-class, then there exists a unique completely bounded map Mϕ on the amalgamated free product of the ℳi with amalgamation over 𝒩, which acts as a radial multiplier. Hereby, we extend a result of Haagerup and the author for radial multipliers on reduced free products of unital C*- and von Neumann algebras.

scholarly journals Amalgamated free product type III factors with at most one Cartan subalgebra

2013 ◽  
Vol 150 (1) ◽  
pp. 143-174 ◽  
Author(s):  
Rémi Boutonnet ◽  
Cyril Houdayer ◽  
Sven Raum
Keyword(s):  
Free Product ◽  
Measure Space ◽  
Cartan Subalgebra ◽  
Von Neumann Algebras ◽  
Finite Subgroup ◽  
Standard Measure ◽  
Von Neumann ◽  
Cartan Subalgebras ◽  
Amalgamated Free Product ◽  
Normal States

AbstractWe investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras ${\mathop{M{}_{1} \ast }\nolimits}_{B} {M}_{2} $ over an amenable von Neumann subalgebra $B$. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free product von Neumann algebras. Namely, we show that any nonamenable free product von Neumann algebra $({M}_{1} , {\varphi }_{1} )\ast ({M}_{2} , {\varphi }_{2} )$ with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established by A. Ioana [Cartan subalgebras of amalgamated free product ${\mathrm{II} }_{1} $factors, arXiv:1207.0054]. Next, we prove that any countable nonsingular ergodic equivalence relation $ \mathcal{R} $ defined on a standard measure space and which splits as the free product $ \mathcal{R} = { \mathcal{R} }_{1} \ast { \mathcal{R} }_{2} $ of recurrent subequivalence relations gives rise to a nonamenable factor $\mathrm{L} ( \mathcal{R} )$ with a unique Cartan subalgebra, up to unitary conjugacy. Finally, we prove unique Cartan decomposition for a class of group measure space factors ${\mathrm{L} }^{\infty } (X)\rtimes \Gamma $ arising from nonsingular free ergodic actions $\Gamma \curvearrowright (X, \mu )$ on standard measure spaces of amalgamated groups $\Gamma = {\mathop{\Gamma {}_{1} \ast }\nolimits}_{\Sigma } {\Gamma }_{2} $ over a finite subgroup $\Sigma $.

COCYCLE CONJUGACY OF ONE PARAMETER AUTOMORPHISM GROUPS OF AFD FACTORS OF TYPE III

2002 ◽  
Vol 13 (06) ◽  
pp. 579-603 ◽  
Author(s):  
UN KIT HUI
Keyword(s):  
Automorphism Group ◽  
Von Neumann Algebras ◽  
Automorphism Groups ◽  
Crossed Product ◽  
Von Neumann ◽  
Type Iii ◽  
Finite Dimensional ◽  
Cocycle Conjugacy

We classify, up to cocycle conjugacy, one-parameter automorphism groups on an approximately finite dimensional (AFD) factor ℳ of type III with trivial Connes spectrum. Our goal is to find the complete cocycle conjugacy invariants for one-parameter automorphism groups on ℳ. We also study the relations between the flow of weights of ℳ and that of the crossed product ℳ ⋊α ℝ of ℳ by a one-parameter automorphism group α with Γ(α) = {0}. Moreover, we also study model realizations. "Model realizations" means that given certain commutative data, they can be realized as the complete cocycle conjugacy invariants of centrally free and centrally ergodic one-parameter automorphism groups on some properly infinite AFD von Neumann algebras.

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