A Galois correspondence for compact group actions on C-algebras

In this paper, we prove a Galois correspondence for compact group actions on C^?-algebras in the presence of a commuting minimal action. Namely, we show that there is a one-to-one correspondence between the C^?-subalgebras that are globally invariant under the compact action and the commuting minimal action, that in addition contain the fixed point algebra of the compact action and the closed, normal subgroups of the compact group.     

Haagerup property for C-algebras

We define the Haagerup property for C^?-algebras A and extend this to a notion of relative Haagerup property for the inclusion B⊆A, where B is a C^?-subalgebra of A. Let Γ be a discrete group and Λ a normal subgroup of Γ, we show that the inclusion A_?α,rΛ⊆A_?α,rΓ has the relative Haagerup property if and only if the quotient group Γ/Λ has the Haagerup property. In particular, the inclusion has the relative Haagerup property if and only if Γ/Λ has the Haagerup property; has the Haagerup property if and only if Γ has the Haagerup property. We also characterize the Haagerup property for Γ in terms of its Fourier algebra A(Γ).     

Nonlinear maps preserving Lie products on factor von Neumann algebras

In this paper, we prove that every bijective map preserving Lie products from a factor von Neumann algebra into another factor von Neumann algebra is of the form A→ψ(A)+ξ(A), where is an additive isomorphism or the negative of an additive anti-isomorphism and is a map with ξ(AB-BA)=0 for all .     

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.     

Amenable representations and finite injective von Neumann algebras

Let be the unitary group of a finite, injective von Neumann algebra . We observe that any subrepresentation of a group representation into is amenable in the sense of Bekka; this yields short proofs of two known results-one by Robertson, one by Haagerup-concerning group representations into .

     

Isometries between projection lattices of von Neumann algebras

We investigate surjective isometries between projection lattices of two von Neumann algebras. We show that such a mapping is characterized by means of Jordan ^?-isomorphisms. In particular, we prove that two von Neumann algebras without type ${\mathrm{I}}_1$ direct summands are Jordan ^?-isomorphic if and only if their projection lattices are isometric. Our theorem extends the recent result for type I factors by G.P. Gehér and P. ?emrl, which is a generalization of Wigner's theorem.     

Abelian subalgebras of von Neumann algebras from flat tori in locally symmetric spaces

Consider a compact locally symmetric space M of rank r, with fundamental group Γ. The von Neumann algebra VN(Γ) is the convolution algebra of functions fℓ₂(Γ) which act by left convolution on ℓ₂(Γ). Let T^r be a totally geodesic flat torus of dimension r in M and let be the image of the fundamental group of T^r in Γ. Then VN(Γ₀) is a maximal abelian -subalgebra of VN(Γ) and its unitary normalizer is as small as possible. If M has constant negative curvature then the Pukánszky invariant of VN(Γ₀) is ∞.     

A weak* similarity degree characterization for injective von Neumann algebras

In this note, we show that a von Neumann algebra M is injective if and only if the weak*similarity degree d*(M) ≤ 2.     

Simple flat Leavitt path algebras are von Neumann regular

For a unital ring, it is an open question whether flatness of simple modules implies all modules are flat and thus the ring is von Neumann regular. The question was raised by Ramamurthi over 40?years ago who called such rings SF-rings (i.e. simple modules are flat). In this note we show that an SF Steinberg algebra of an ample Hausdorff groupoid, graded by an ordered group, has an aperiodic unit space. For graph groupoids, this implies that the graphs are acyclic. Combining with the Abrams–Rangaswamy Theorem, it follows that SF Leavitt path algebras are regular, answering Ramamurthi’s question in positive for the class of Leavitt path algebras.     

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.     

Reidemeister torsion and the K-theory of von Neumann algebras

We introduce a topological-type invariant for a cocompact properly discontinuous action of a discrete group on a Riemannian manifold generalizing classical notions of Reidemeister torsion. It takes values in the weak algebraic K-theory of the von Neumann algebra of . We give basic tools for its computation like sum and product formulas and calculate it in several cases. It encompasses, for instance, the Alexander polynomial and is related to analytic torsion.     

Products of projections in von Neumann algebras

We describe the elements of von Neumann algebras which can be represented as products of orthogonal projections and idempotents, and estimate the minimal number of terms in the product.     

Von Neumann algebras and linear independence of translates

For and , define and if , define . It has been conjectured that if , then is linearly independent over ; one motivation for this problem comes from Gabor analysis. We shall prove that is linearly independent if and is contained in a discrete subgroup of , and as a byproduct we shall obtain some results on the group von Neumann algebra generated by the operators . Also, we shall prove these results for the obvious generalization to .

     

von Neumann代数的T-局部导子

In this paper, we introduce the concept of T-local derivations and obtain the main result: each T-local derivation of a von Neumann algebra A into a dual A-bimodule M is a T-derivation, where T is an endomorphism of A to A.     

Similarity degrees for the crossed product of von Neumann algebras

In this paper, we will estimate an upper bound for the similarity degree of the crossed product of a hyperfinite finite von Neumann algebra by weakly compact action of an infinite discrete group. We will also improve some upper bounds for similarity degrees of some finite von Neumann algebras.     

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.

     

AlgebraicK-theory of von Neumann algebras

To every von Neumann algebra, one can associate a (multiplicative) determinant defined on the invertible elements of the algebra with range a subgroup of the Abelian group of the invertible elements of the center of the von Neumann algebra. This determinant is a normalization of the usual determinant for finite von Neumann algebras of type I, for the type II₁-case it is the Fuglede-Kadison determinant, and for properly infinite von Neumann algebras the determinant is constant equal to 1. It is proved that every invertible element of determinant 1 is a product of a finite number of commutators. This extends a result of T. Fack and P. de la Harpe for II₁-factors. As a corollary, it follows that the determinant induces an injection from the algebraicK ₁-group of the von Neumann algebra into the Abelian group of the invertible elements of the center. Its image is described. Another group,K ₁ ^w (A), which is generated by elements in matrix algebras overA that induce injective right multiplication maps, is also computed. We use the Fuglede-Kadison determinant to detect elements in the Whitehead group Wh(G).Partially supported by NSF Grant DMS-9103327.     

Type II non-commutative geometry. I. Dixmier trace in von Neumann algebras

We define the notion of Connes-von Neumann spectral triple and consider the associated index problem. We compute the analytic Chern-Connes character of such a generalized spectral triple and prove the corresponding local formula for its Hochschild class. This formula involves the Dixmier trace for II_∞ von Neumann algebras. In the case of foliations, we identify this Dixmier trace with the corresponding measured Wodzicki residue.