L 2-topological invariants of 3-manifolds

Summary We give results on theL ²-Betti numbers and Novikov-Shubin invariants of compact manifolds, especially 3-manifolds. We first study the Betti numbers and Novikov-Shubin invariants of a chain complex of Hilbert modules over a finite von Neumann algebra. We establish inequalities among the Novikov-Shubin invariants of the terms in a short exact sequence of chain complexes. Our algebraic results, along with some analytic results on geometric 3-manifolds, are used to compute theL ²-Betti numbers of compact 3-manifolds which satisfy a weak form of the geometrization conjecture, and to compute or estimate their Novikov-Shubin invariants.Oblatum 6-V-1993 & 20-VI-1994Partially supported by NSF-grant DMS 9101920Partially supported by NSF-grant DMS 9103327     

Amenable groups, topological entropy and Betti numbers

We investigate an analogue of theL ²-Betti numbers for amenable linear subshifts. The role of the von Neumann dimension shall be played by the topological entropy. Partially supported by OTKA grant T 25004 and the Bolyai Fellowship.     

Generalized notion of a trace for von Neumann algebras

On a von Neumann algebra M, we consider traces with values in the algebra L ⁰ of measurable complex-valued functions. We show that every faithful normal L ⁰-valued trace on M generates an L ⁰-valued metric on the algebra of measurable operators that are affiliated with M. Moreover, convergence in this metric coincides with local convergence in measure.     

Group cocycles and the ring of affiliated operators

In this article we study cocycles of discrete countable groups with values in ℓ ² G and the ring of affiliated operators UG\mathcal{U}G. We clarify properties of the first cohomology of a group G with coefficients in ℓ ² G and answer several questions from De Cornulier et al. (Transform. Groups 13(1):125–147, 2008). Moreover, we obtain strong results about the existence of free subgroups and the subgroup structure, provided the group has a positive first ℓ ²-Betti number. We give numerous applications and examples of groups which satisfy our assumptions.     

Schatten–von Neumann properties for Fourier integral operators with non-smooth symbols,I

We consider Fourier integral operators with symbols in modulation spaces and non-smooth phase functions whose second orders of derivatives belong to certain types of modulation space. We prove continuity and Schatten–von Neumann properties of such operators when acting on L ².     

Hyperinvariant subspaces for some <Emphasis Type="Italic">B</Emphasis>–circular operators

We show that if A is a Hilbert–space operator, then the set of all projections onto hyperinvariant subspaces of A, which is contained in the von Neumann algebra υN(A) that is generated by A, is independent of the representation of υ N(A), thought of as an abstract W^*–algebra. We modify a technique of Foias, Ko, Jung and Pearcy to get a method for finding nontrivial hyperinvariant subspaces of certain operators in finite von Neumann algebras. We introduce the B–circular operators as a special case of Speicher's B–Gaussian operators in free probability theory, and we prove several results about a B–circular operator z, including formulas for the B–valued Cauchy– and R–transforms of z^*z. We show that a large class of L^∞([0,1])–circular operators in finite von Neumann algebras have nontrivial hyperinvariant subspaces, and that another large class of them can be embedded in the free group factor L(F₃). These results generalize some of what is known about the quasinilpotent DT–operator. Supported in part by NSF Grant DMS-0300336. with an Appendix by Gabriel Tucci     

On the K – and L – Theory of the Algebra of Operators Affiliated to a Finite von Neumann Algebra

We construct a real valued dimension for arbitrary modules over the algebra of operators affiliated to a finite von Neumann algebra. Moreover we determine the algebraic K ₀- and K ₁-group and the L-groups of such an algebra. Mathematics Subject Classifications (2000): Primary 18F25; Secondary 46L10.     

Derivations on the algebra of measurable operators affiliated with a type I von Neumann algebra

Let M be a type I von Neumann algebra with the center Z and let LS(M) be the algebra of all locally measurable operators affiliated with M. We prove that every Z-linear derivation on LS(M) is inner. In particular, all Z-linear derivations on the algebras of measurable and respectively totally measurable operators are spatial and implemented by elements of LS(M). The text was submitted by the authors in English.     

The first <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-cohomology of some groups with one end

Let p be a real number greater than one. In this paper we study the vanishing and nonvanishing of the first L ^p -cohomology space of some groups that have one end. We also make a connection between the first L ^p -cohomolgy space and the Floyd boundary of the Cayley graph of a group. We apply the result about Floyd boundaries to show that there exists a real number p such that the first L ^p -cohomology space of a nonelementary hyperbolic group does not vanish. Received: 4 August 2006 Revised: 2 November 2006     

Amenability and vanishing of L2-Betti numbers: An operator algebraic approach

We introduce a Følner condition for dense subalgebras in finite von Neumann algebras and prove that it implies dimension flatness of the inclusion in question. It is furthermore proved that the Følner condition naturally generalizes the existing notions of amenability and that the ambient von Neumann algebra of a Følner algebra is automatically injective. As an application, we show how our techniques unify previously known results concerning vanishing of $L^2$-Betti numbers for amenable groups, quantum groups and groupoids and moreover provide a large class of new examples of algebras with vanishing $L^2$-Betti numbers.     

The Spectral Measure of Certain Elements of the Complex Group Ring of a Wreath Product

We use elementary methods to compute the L ²-dimension of the eigenspaces of the Markov operator on the lamplighter group and of generalizations of this operator on other groups. In particular, we give a transparent explanation of the spectral measure of the Markov operator on the lamplighter group found by Grigorchuk and Zuk, and later used by them, together with Linnell and Schick, to produce a counterexample to a strong version of the Atiyah conjecture about the range of L ²-Betti numbers. We use our results to construct manifolds with certain L ²-Betti numbers (given as convergent infinite sums of rational numbers) which are not obviously rational, but we have been unable to determine whether any of them are irrational.     

Local and 2-local derivations on noncommutative Arens algebras

The paper is devoted to so-called local and 2-local derivations on the noncommutative Arens algebra L ^ω(M,τ) associated with a von Neumann algebra M and a faithful normal semi-finite trace τ. We prove that every 2-local derivation on L ^ω(M,τ) is a spatial derivation, and if M is a finite von Neumann algebra, then each local derivation on L ^ω(M,τ) is also a spatial derivation and every 2-local derivation on M is in fact an inner derivation.     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-Betti numbers of one-relator groups

We determine the L ²-Betti numbers of all one-relator groups and all surface-plus-one-relation groups. We also obtain some information about the L ²-cohomology of left-orderable groups, and deduce the non-L ² result that, in any left-orderable group of homological dimension one, all two-generator subgroups are free. Warren Dicks was Funded by the DGI (Spain) through Project BFM2003-06613.     

A cohomological description of property (T) for quantum groups

We prove a Delorme-Guichardet type theorem for discrete quantum groups expressing property (T) of the quantum group in question in terms of its first cohomology groups. As an application, we show that the first L²-Betti number of a discrete property (T) quantum group vanishes.     

Quantum extensions of semigroups generated by Bessel processes

We construct a quantum extension of the Markov semigroup of the classical Bessel process of orderv≥1 to the noncommutative von Neumann algebra ß(L ²(0, +∞)) of bounded operators onL ²(0, +∞).     

To the theory of operator monotone and operator convex functions

We prove that a real function is operator monotone (operator convex) if the corresponding monotonicity (convexity) inequalities are valid for some normal state on the algebra of all bounded operators in an infinite-dimensional Hilbert space. We describe the class of convex operator functions with respect to a given von Neumann algebra in dependence of types of direct summands in this algebra. We prove that if a function from ℝ⁺ into ℝ⁺ is monotone with respect to a von Neumann algebra, then it is also operator monotone in the sense of the natural order on the set of positive self-adjoint operators affiliated with this algebra.     

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.     

On the A. M. Bikchentaev conjecture

In 1998 A. M. Bikchentaev conjectured that for positive ??-measurable operators a and b affiliated with a semifinite von Neumann algebra, the operator b ^1/2 ab ^1/2 is submajorized by the operator ab in the sense of Hardy-Littlewood. We prove this conjecture in its full generality and obtain a number of consequences for operator ideals, Golden-Thompson inequalities, and singular traces.