We show that whenever m≥1 and M1,…,Mm are nonamenable factors in a large class of von Neumann algebras that we call C(AO) and which contains all free Araki–Woods factors, the tensor product factor ...M1⊗‾⋯⊗‾Mm retains the integer m and each factor Mi up to stable isomorphism, after permutation of the indices. Our approach unifies the Unique Prime Factorization (UPF) results from 33,25 and moreover provides new UPF results in the case when M1,…,Mm are free Araki–Woods factors. In order to obtain the aforementioned UPF results, we show that Connes's bicentralizer problem has a positive solution for all type III1 factors in the class C(AO).
We show that every ℓ 2 \ell _2 -strictly singular operator on the predual of any atomless hyperfinite finite von Neumann algebra M \mathcal {M} is Dunford–Pettis, which extends a Rosenthal’s theorem ...for the case of commutative algebra M = L ∞ ( 0 , 1 ) \mathcal {M}=L_\infty (0,1) . We also apply our result to the study of noncommutative symmetric spaces X = E ( M , τ ) X=E(\mathcal {M},\tau ) for which every ℓ 2 \ell _2 -strictly singular operator from L p ( 0 , 1 ) L_p(0,1) into X X is narrow.
We study which von Neumann algebras can be embedded into uniform Roe algebras and quasi-local algebras associated to a uniformly locally finite metric space X. Under weak assumptions, these ...C⁎-algebras contain embedded copies of ∏kMnk(C) for any bounded countable (possibly finite) collection (nk)k of natural numbers; we aim to show that they cannot contain any other von Neumann algebras.
One of our main results shows that L∞0,1 does not embed into any of those algebras, even by a not-necessarily-normal ⁎-homomorphism. In particular, it follows from the structure theory of von Neumann algebras that any von Neumann algebra which embeds into such algebra must be of the form ∏kMnk(C) for some countable (possibly finite) collection (nk)k of natural numbers. Under additional assumptions, we also show that the sequence (nk)k has to be bounded: in other words, the only embedded von Neumann algebras are the “obvious” ones.
We study the relationship between the dynamics of the action α of a discrete group G on a von Neumann algebra M, and structural properties of the associated crossed product inclusion L(G)⊆M⋊αG, and ...its intermediate subalgebras. This continues a thread of research originating in classical structural results for ergodic actions of discrete, abelian groups on probability spaces. A key tool in the setting of a noncommutative dynamical system is the set of quasinormalizers for an inclusion of von Neumann algebras. We show that the von Neumann algebra generated by the quasinormalizers captures analytical properties of the inclusion L(G)⊆M⋊αG such as the Haagerup Approximation Property, and is essential to capturing “almost periodic” behavior in the underlying dynamical system. Our von Neumann algebraic point of view yields a new description of the Furstenberg-Zimmer distal tower for an ergodic action on a probability space, and we establish new versions of the Furstenberg-Zimmer structure theorem for general, tracial W⁎-dynamical systems. We present a number of examples contrasting the noncommutative and classical settings which also build on previous work concerning singular inclusions of finite von Neumann algebras.
We consider the tracial crossed product algebra M=A⋊Λ arising from a trace preserving action σ:Λ↷A of a discrete group Λ on a tracial von Neumann algebra A. For a unitary subgroup G⊂U(M), we study ...when this G can be conjugated into U(A)⋅Λ in M. We provide a general sufficient condition for this to happen. Our result generalizes 14, Theorem 3.1 which treats the case when M is the group von Neumann algebra L(Λ).
A countable discrete group G is called W⁎-superrigid (resp. C⁎-superrigid) if it is completely recognizable from its von Neumann algebra L(G) (resp. reduced C⁎-algebra Cr⁎(G)). Developing new ...technical aspects in Popa's deformation/rigidity theory we introduce several new classes of W⁎-superrigid groups which appear as direct products, semidirect products with non-amenable core and iterations of amalgamated free products and HNN-extensions. As a byproduct we obtain new rigidity results in C⁎-algebra theory including additional examples of C⁎-superrigid groups and explicit computations of symmetries of reduced group C⁎-algebras.
We introduce a new equivalence relation on groups, which we call von Neumann equivalence, that is coarser than both measure equivalence and W⁎-equivalence. We introduce a general procedure for ...inducing actions in this setting and use this to show that many analytic properties, such as amenability, property (T), and the Haagerup property, are preserved under von Neumann equivalence. We also show that proper proximality, which was defined recently in 9 using dynamics, is also preserved under von Neumann equivalence. In particular, proper proximality is preserved under both measure equivalence and W⁎-equivalence, and from this we obtain examples of non-inner amenable groups that are not properly proximal.
We provide an example of two elementarily equivalent countable ICC groups G and H such that G is amenable and H is not inner amenable. As a result, we provide the first example of elementarily ...equivalent groups whose group von Neumann algebras are not elementarily equivalent, answering a question asked by many researchers.
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