We show that any free product of finite-dimensional von Neumann algebras equipped with non-tracial states is isomorphic to a free Araki–Woods factor with its free quasi-free state possibly direct sum ...a finite-dimensional von Neumann algebra. This gives a complete answer to questions posed by Dykema in 5 and Shlyakhtenko in 10, which had been partially answered by Houdayer in 7 and Ueda in 16. We also extend this to suitable infinite-dimensional von Neumann algebras with almost periodic states.
Operator algebras of free wreath products Fima, Pierre; Troupel, Arthur
Advances in mathematics (New York. 1965),
April 2024, 2024-04-00, 2024-04, Volume:
441
Journal Article
Peer reviewed
Open access
We give a description of operator algebras of free wreath products in terms of fundamental algebras of graphs of operator algebras as well as an explicit formula for the Haar state. This allows us to ...deduce stability properties for certain approximation properties such as exactness, Haagerup property, hyperlinearity and K-amenability. We study qualitative properties of the associated von Neumann algebra: factoriality, fullness, primeness and absence of Cartan subalgebra and we give a formula for Connes' T-invariant and τ-invariant. We also study maximal amenable von Neumann subalgebras. Finally, we give some explicit computations of K-theory groups for C*-algebras of free wreath products. As an application we show that the reduced C*-algebras of quantum reflection groups are pairwise non-isomorphic.
Abstract
We 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.
Let M be a finite von Neumann algebra and u1,…,uN be unitaries in M. We show that u1,…,uN freely generate L(FN) if and only if‖∑i=1Nui⊗(uiop)⁎+ui⁎⊗uiop‖M⊗‾Mop=22N−1.
Let
$I$
be any nonempty set and let
$(M_{i},\unicodeSTIX{x1D711}_{i})_{i\in I}$
be any family of nonamenable factors, endowed with arbitrary faithful normal states, that belong to a large class
...${\mathcal{C}}_{\text{anti}\text{-}\text{free}}$
of (possibly type
$\text{III}$
) von Neumann algebras including all nonprime factors, all nonfull factors and all factors possessing Cartan subalgebras. For the free product
$(M,\unicodeSTIX{x1D711})=\ast _{i\in I}(M_{i},\unicodeSTIX{x1D711}_{i})$
, we show that the free product von Neumann algebra
$M$
retains the cardinality
$|I|$
and each nonamenable factor
$M_{i}$
up to stably inner conjugacy, after permutation of the indices. Our main theorem unifies all previous Kurosh-type rigidity results for free product type
$\text{II}_{1}$
factors and is new for free product type
$\text{III}$
factors. It moreover provides new rigidity phenomena for type
$\text{III}$
factors.
To any strongly continuous orthogonal representation of R on a real Hilbert space HR, Hiai constructed q-deformed Araki–Woods von Neumann algebras for −1<q<1, which are W⁎-algebras arising from ...non-tracial representations of the q-commutation relations, the latter yielding an interpolation between the Bosonic and Fermionic statistics. We prove that if the orthogonal representation is not ergodic then these von Neumann algebras are factors whenever dim(HR)≥2 and q∈(−1,1). In such case, the centralizer of the q-quasi free state has trivial relative commutant. In the process, we study ‘generator MASAs’ in these factors and establish that they are strongly mixing.
We prove that any weak* continuous semigroup (Tt)t⩾0 of factorizable Markov maps acting on a von Neumann algebra M equipped with a normal faithful state can be dilated by a group of Markov ...⁎-automorphisms analogous to the case of a single factorizable Markov operator, which is an optimal result. We also give a version of this result for strongly continuous semigroups of operators acting on noncommutative Lp-spaces and examples of semigroups to which the results of this paper can be applied. Our results imply the boundedness of the McIntosh's H∞ functional calculus of the generators of these semigroups on the associated noncommutative Lp-spaces generalising some previous work from Junge, Le Merdy and Xu. Finally, we also give concrete dilations for Poisson semigroups which are even new in the case of Rn.
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