Locally von Neumann algebras II

In this paper, we will prove some properties of locally von Neumann algebras. In particular, we will show that every locally von Neumann algebra is the dual of a certain locally convex space and also, we will show the existence of a polar decomposition for every element in a locally von Neumann algebra.     

Nest-subalgebras of von Neumann algebras

The structure of a certain class of separably acting reflexive operator algebras is investigated for which the nest algebras of J. Ringrose can be considered prototypes. To a fixed von Neumann algebra and a complete nest of projections contained therein one associates the algebra of all operators in the von Neumann algebra which leave every member of the nest invariant. A generalization of the Ringrose criterion for inclusion in the Jacobson radical of a nest algebra is given for this more general class of algebras. Further properties of the radical are studied.     

Stochastic coupling for von Neumann algebras

We consider even and odd stochastic transitions of von Neumann algebras when dual mappings intertwine (couple) modular groups of the corresponding states (with the occurrence of a sign exchange for the odd case). We show that one can define modular objects and cones associated to linear combinations of von Neumann algebras, which generalize objects and cones in the standard modular theory. In the odd case, we find sufficient conditions for the intertwining property and consider several applications to noncommutative Markov processes. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 760–774, May, 1999.     

Countable T-groups and von Neumann algebras

Factors with countable fundamental groups are investigated in connection with representations and actions of discrete with the property T. An example of an ICC-group is given, whose group factor has a countable fundamental group and a continuous locally compact group of outer automorphisms. In addition, for this factor all the tensor powers are mutually nonisomorphic. It is proved that there exists a continuum of nonisomorphic full factors of type III₁ with a fixed Connes invariant Sd.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 45, pp. 5–16, 1986.     

On weakly injective von Neumann algebras

A von Neumann algebra \({M\subset B(H)}\) is called weakly injective if there exist an ultraweakly dense unital C*-subalgebra \({A\subset M}\) and a unital completely positive map φ : B(H) → M such that φ(a) = a for all \({a\in A}\). In this note we present several properties of weakly injective von Neumann algebras and highlight the role these algebras play in relation to the QWEP conjecture.     

Approximate equivalence in von Neumann algebras

One formulation of D. Voiculescu's theorem on approximate unitary equivalence is that two unital representations π and ρ of a separable C*-algebra are approximately unitarily equivalent if and only if rank οπ = rank ορ. We study the analog when the ranges of π and ρ are contained in a von Neumann algebra R, the unitaries inducing the approximate equivalence must come from R, and "rank" is replaced with "R -rank" (defined as the Murray-von Neumann equivalence of the range projection).     

Asymptotic properties of von Neumann algebras

One gives a survey of the results regarding asymptotic commutativity in von Neumann algebras.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki (Noveishie Dostizheniya), Vol. 27, pp. 129–166, 1985.     

Commutator estimates in von Neumann algebras

Let M be a von Neumann algebra. For every self-adjoint locally measurable operator a, there exists a central self-adjoint locally measurable operator c ₀ such that, given any ? > 0, |[a, u _ε ]| ? (1 ? ε)|aε ? c ₀| for some unitary operator u _ε ∈ M. In particular, every derivation δ: M → I (where I is an ideal in M) is inner, and δ = δ _a for a ∈ I.     

HNN extensions of von Neumann algebras

Reduced HNN extensions of von Neumann algebras (as well as C^*-algebras) will be introduced, and their modular theory, factoriality and ultraproducts will be discussed. In several concrete settings, detailed analysis on them will be also carried out.     

Measurable vectors for von Neumann algebras

Let A be a semifinite von Neumann algebra, with countably decomposable center, on the Hilbert space H. A measurable vector is a linear functional on H whose domain contains a strongly dense domain and which satisfies certain continuity conditions. H can be embedded as a dense subspace of the topological vector space of measurable vectors. The measurable vectors are a module over the measurable operators, and the action of measurable operators on measurable vectors is jointly continuous with respect to suitable topologies. If A is standard, then the measurable operators and measurable vectors are isomorphic as topological vector spaces. If the center of A is not countably decomposable, the results hold with minor changes.     

Topologies on central extensions of von Neumann algebras

Given a von Neumann algebra M, we consider the central extension E(M) of M. We introduce the topology t _c(M) on E(M) generated by a center-valued norm and prove that it coincides with the topology of local convergence in measure on E(M) if and only if M does not have direct summands of type II. We also show that t _c(M) restricted to the set E(M) _h of self-adjoint elements of E(M) coincides with the order topology on E(M) _h if and only if M is a σ-finite type I_fin von Neumann algebra.     

The range of operators on von Neumann algebras

We prove that for every bounded linear operator , where is a non-reflexive quotient of a von Neumann algebra, the point spectrum of is non-empty (i.e., for some the operator fails to have dense range). In particular, and as an application, we obtain that such a space cannot support a topologically transitive operator.

     

Local 3-cocycles of von Neumann algebras

We show that every local 3-cocycle of a von Neumann algebra R into an arbitrary unital dual R-bimodule S is a 3-cocycle.