The local index formula in semifinite von Neumann algebras II: The even case

We generalise the even local index formula of Connes and Moscovici to the case of spectral triples for a *-subalgebra A of a general semifinite von Neumann algebra. The proof is a variant of that for the odd case which appears in Part I. To allow for algebras with a non-trivial centre we have to establish a theory of unbounded Fredholm operators in a general semifinite von Neumann algebra and in particular prove a generalised McKean-Singer formula.     

Maximal Abelian von Neumann Algebras and Toeplitz Operators with Separately Radial Symbols

This paper mainly concerns abelian von Neumann algebras generated by Toeplitz operators on weighted Bergman spaces. Recently a family of abelian w^*-closed Toeplitz algebras has been obtained (see [5,6,7,8]). We show that this algebra is maximal abelian and is singly generated by a Toeplitz operator with a “common” symbol. A characterization for Toeplitz operators with radial symbols is obtained and generalized to the high dimensional case. We give several examples for abelian von Neumann algebras in the case of high dimensional weighted Bergman spaces, which are different from the one dimensional case.     

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.     

Structure and Classification of Superconformal Nets

We study the general structure of Fermi conformal nets of von Neumann algebras on S ¹ and consider a class of topological representations, the general representations, that we characterize as Neveu–Schwarz or Ramond representations, in particular a Jones index can be associated with each of them. We then consider a supersymmetric general representation associated with a Fermi modular net and give a formula involving the Fredholm index of the supercharge operator and the Jones index. We then consider the net associated with the super-Virasoro algebra and discuss its structure. If the central charge c belongs to the discrete series, this net is modular by the work of F. Xu and we get an example where our setting is verified by taking the Ramond irreducible representation with lowest weight c/24. We classify all the irreducible Fermi extensions of any super-Virasoro net in the discrete series, thus providing a classification of all superconformal nets with central charge less than 3/2. Sebastino Carpi: Supported by MIUR, GNAMPA-INDAM and EU network “Noncommutative Geometry” MRTN-CT-2006-0031962 Yasuyuki Kawahigashi: Supported in part by the Grants-in-Aid for Scientific Research, JSPS. Submitted: March 3, 2008. Accepted: May 5, 2008.     

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.     

Operator Algebraic Quotient Structures Induced by Direct Graphs II

The main purpose of this paper is to investigate the operator algebraic quotient structures induced by directed graphs. We enlarge our study of Cho (Compl Anal Oper Theory, 2008) to the general case. This can be done by constructing new graphs from given graphs called the pull-back graphs. We consider the corresponding groupoids, and von Neumann algebras of pull-back graphs.     

The Daugavet Property of C*-Algebras and Non-commutative Lp-Spaces

We prove that a C ^*-algebra A or a predual N _* of a von Neumann algebra N has the Daugavet property if and only if A (or N) is non-atomic. We also prove a similar (although somewhat weaker) result for non-commutative L _p-spaces corresponding to non-atomic von Neumann algebras.     

Ergodic actions of compact groups on operator algebras

Summary We prove that the only ergodic actions ofSU(2) are on Type I von Neumann algebras, using the theory developed in [8].     

Lifting endomorphisms to automorphisms

Normal endomorphisms of von Neumann algebras need not be extendable to automorphisms of a larger von Neumann algebra, but they always have asymptotic lifts. We describe the structure of endomorphisms and their asymptotic lifts in some detail, and apply those results to complete the identification of asymptotic lifts of unital completely positive linear maps on von Neumann algebras in terms of their minimal dilations to endomorphisms.

     

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.     

Vertex-Compressed Subalgebras of a Graph von Neumann Algebra

In Cho (Acta Appl. Math. 95:95–134, 2007 and Complex Anal. Oper. Theory 1:367–398, 2007), we introduced Graph von Neumann Algebras which are the (groupoid) crossed product algebras of von Neumann algebras and graph groupoids via graph-representations, which are groupoid actions. In Cho (Acta Appl. Math. 95:95–134, 2007), we showed that such crossed product algebras have the amalgamated reduced free probabilistic properties, where the reduction is totally depending on given directed graphs. Moreover, in Cho (Complex Anal. Oper. Theory 1:367–398, 2007), we characterize each amalgamated free blocks of graph von Neumann algebras: we showed that they are characterized by the well-known von Neumann algebras: Classical group crossed product algebras and (operator-valued) matricial algebras. This shows that we can provide a nicer way to investigate such groupoid crossed product algebras, since we only need to concentrate on studying graph groupoids and characterized algebras. How about the compressed subalgebras of them? i.e., how about the inner (cornered) structures of a graph von Neumann algebra? In this paper, we will provides the answer of this question. Consequently, we show that vertex-compressed subalgebras of a graph von Neumann algebra are characterized by other graph von Neumann algebras. This gives the full characterization of the vertex-compressed subalgebras of a graph von Neumann algebra, by other graph von Neumann algebras.     

VITALI-HAHN-SAKS THEOREM FOR VECTOR MEASURES ON OPERATOR ALGEBRAS

We show that the direct generalization of the Vitali–Hahn–Sakstheorem is not valid for all measures on von Neumann algebras.By applying a general equicontinuity argument, we prove a directextension of the Vitali–Hahn–Saks theorem for awide range of vector measures on von Neumann algebra s and JBWalgebras. We also characterize relatively compact sets of vectormeasures on operator algebras.     

Strong clustering in type III entropic K-systems

It is shown that Complete Memory Loss (CML) formulated in terms of the Quantum Dynamical Entropy of Connes, Narnhofer and Thirring implies Strong Clustering for some typeIII von Neumann algebras including infinite quantum systems with quasi-free states. This generalizes analogous conclusions on Abelian and typeII ₁ von Neumann algebras. The result is based on the fact that optimal decompositions that obtain the so-called Entropy of a Subalgebra are under control in the two-dimensional case.     

Affiliated subspaces and infiniteness of von Neumann algebras

We show that the structural properties of von Neumann algebra s are connected with the metric and order theoretic properties of various classes of affiliated subspaces. Among others we show that properly infinite von Neumann algebra s always admit an affiliated subspace for which (1) closed and orthogonally closed affiliated subspaces are different; (2) splitting and quasi‐splitting affiliated subspaces do not coincide. We provide an involved construction showing that concepts of splitting and quasi‐splitting subspaces are non‐equivalent in any GNS representation space arising from a faithful normal state on a Type I factor. We are putting together the theory of quasi‐splitting subspaces developed for inner product spaces on one side and the modular theory of von Neumann algebra s on the other side.     

COMPLEX INTERPOLATION OF NONCOMMUTATIVE HARDY SPACES ASSOCIATED WITH SEMIFINITE VON NEUMANN ALGEBRAS

We proved a complex interpolation theorem of noncommutative Hardy spaces associated with semi-finite von Neumann algebras and extend the Riesz type factorization to the semi-finite case.     

Commutativity of projections and characterization of traces on Von Neumann algebras

We find new necessary and sufficient conditions for the commutativity of projections in terms of operator inequalities. We apply these inequalities to characterize a trace on von Neumann algebras in the class of all positive normal functionals. We obtain some characterization of a trace on von Neumann algebras in terms of the commutativity of products of projections under a weight.     

Commutativity of projectors and trace characterization on von Neumann algebras. I

We obtain the necessary and sufficient conditions for commutativity of projectors in terms of operator inequalities. We apply these conditions for the trace characterization on von Neumann algebras in the class of all positive normal functionals. We also propose a trace characterization on von Neumann algebras in terms of the commutation of products of projectors under the weight sign.     

von Neumann代数中套子代数的摄动

本文主要讨论ｖｏｎ　Ｎｅｕｍａｎｎ代数中套子代数的摄动．给出了因子ｖｏｎ　Ｎｅｕｍａｎｎ代数中套相似的一个充分条件．证明了任何因子ｖｏｎ　Ｎｅｕｍａｎｎ代数中相邻的套子代数经由一个邻近于单位元的可逆算子是相似的．     

Operator K-Theory and Functor N 0. Some Applications

In this paper, we present a general introduction to the K-theory of C ^*-algebras and survey of our previous papers, where the functor N ₀ from the category of von Neumann algebras to the category of Abelian groups was defined. We investigate the properties of this functor (in particular, its interrelation with the functor K ₀) and point out some applications of the functor N ₀ in noncommutative geometry. In addition, we recall facts of theory of C ^*-algebras, von Neumann algebras, and Hilbert C ^*-modules.