Fedor Sukochev

Symmetric norms and spaces of operators

Abstract We show that if (E, ∥ · ∥ E ) is a symmetric Banach sequence space then the corresponding space of operators on a separable Hilbert space, defined by if and only if , is a Banach space under the norm . Although this was proved for finite-dimensional spaces by von Neumann in 1937, it has never been established in complete generality in infinite-dimensional spaces; previous proofs have used the stronger hypothesis of full symmetry on E. The proof that is a norm requires the apparently new concept of uniform Hardy-Littlewood majorization; completeness also requires a new proof. We also give the analogous results for operator spaces modelled on a semifinite von Neumann algebra with a normal faithful semi-finite trace.

Dixmier traces and some applications in non-commutative geometry

This is a discussion of recent progress in the theory of singular traces on ideals of compact operators, with emphasis on Dixmier traces and their applications in non-commutative geometry. The starting point is the book Non-commutative geometry by Alain Connes, which contains several open problems and motivations for their solutions. A distinctive feature of the exposition is a treatment of operator ideals in general semifinite von Neumann algebras. Although many of the results presented here have already appeared in the literature, new and improved proofs are given in some cases. The reader is referred to the table of contents below for an overview of the topics considered.

Operator integrals, spectral shift and spectral flow

We present a new and simple approach to the theory of multiple operator integrals that ap- plies to unbounded operators affiliated with general von Neumann algebras. For semifinite von Neu- mann algebras we give applications to the Frechet differentiation of operator functions that sharpen existing results, and establish the Birman-Solomyak representation of the spectral shift function of M.G.Krein in terms of an average of spectral measuresin the type II setting. We also exhibit a surpris- ing connection between the spectral shift function and spectral flow.

Series of independent random variables in rearrangement invariant spaces: An operator approach

This paper studies series of independent random variables in rearrangement invariant spacesX on [0, 1]. Principal results of the paper concern such series in Orlicz spaces exp(Lp), 1≤p≤∞ and Lorentz spacesAΨ. One by-product of our methods is a new (and simpler) proof of a result due to W. B. Johnson and G. Schechtman that the assumptionLp ⊂X, p<∞ is sufficient to guarantee that convergence of such series inX (under the side condition that the sum of the measures of the supports of all individual terms does not exceed 1) is equivalent to convergence inX of the series of disjoint copies of individual terms. Furthermore, we prove the converse (in a certain sense) to that result.

Symmetric Functionals and Singular Traces

We study the construction and properties of positive linear functionals on symmetric spaces of measurable functions which are monotone with respect to submajorization. The construction of such functionals may be lifted to yield the existence of singular traces on certain non-commutative Marcinkiewicz spaces which generalize the notion of Dixmier trace.

Index theory for locally compact noncommutative geometries

Introduction Pseudodifferential calculus and summability Index pairings for semifinite spectral triples The local index formula for semifinite spectral triples Applications to index theorems on open manifolds Noncommutative examples Appendix A. Estimates and technical lemmas Bibliography Index

A uniform Kadec-Klee property for symmetric operator spaces

We show that if a rearrangement invariant Banach function space E on the positive semi-axis satisfies a non-trivial lower q -estimate with constant 1 then the corresponding space E (M) of τ-measurable operators, affiliated with an arbitrary semi-finite von Neumann algebra M equipped with a distinguished faithful, normal, semi-finite trace τ, has the uniform Kadec-Klee property for the topology of local convergence in measure. In particular, the Lorentz function spaces L q, p and the Lorentz-Schatten classes C g, p have the UKK property for convergence locally in measure and for the weak-operator topology, respectively. As a partial converse, we show that if E has the UKK property with respect to local convergence in measure then E must satisfy some non-trivial lower q -estimate. We also prove a uniform Kadec-Klee result for local convergence in any Banach lattice satisfying a lower q -estimate.

The chern character of semifinite spectral triples

In previous work we generalised both the odd and 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. Our proofs are novel even in the setting of the original theorem and rely on the introduction of a function valued cocycle (called the resolvent cocycle) which is ‘almost’ a .b; B/-cocycle in the cyclic cohomology of A. In this paper we show that this resolvent cocycle ‘almost’ represents the Chern character and assuming analytic continuation properties for zeta functions we show that the associated residue cocycle, which appears in our statement of the local index theorem does represent the Chern character. Mathematics Subject Classification (2000). Primary: 19K56, 46L80; Secondary: 58B30, 46L87.

On the Witten index in terms of spectral shift functions

AbstractWe study the model operator DA = (d/dt) + A in L2(R;H) associated with the operator path {A(t)}t=−∞∞, where (Af)(t) = A(t)f(t) for a.e. t ∈ R, and appropriate f ∈ L2(R;H) (with H a separable, complex Hilbert space). Denoting by A± the norm resolvent limits of A(t) as t → ±∞, our setup permits A(t) in H to be an unbounded, relatively trace class perturbation of the unbounded self-adjoint operator A-, and no discrete spectrum assumptions are made on A±. Introducing H1 = DA*DA, H2 = DADA*, the resolvent and semigroup regularized Witten indices of DA, denoted by Wr(DA) and Ws(DA), are defined by