Fyodor Sukochev

Spectral flow and Dixmier traces

Abstract We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the heat operator in a general semi-finite von Neumann algebra. Our results have several applications. We deduce a formula for the Chern character of an odd L (1,∞) -summable Breuer–Fredholm module in terms of a Hochschild 1-cycle. We explain how to derive a Wodzicki residue for pseudo-differential operators along the orbits of an ergodic R n action on a compact space X. Finally, we give a short proof of an index theorem of Lesch for generalised Toeplitz operators.

Dixmier traces as singular symmetric functionals and applications to measurable operators

Abstract We unify various constructions and contribute to the theory of singular symmetric functionals on Marcinkiewicz function/operator spaces. This affords a new approach to the non-normal Dixmier and Connes–Dixmier traces (introduced by Dixmier and adapted to non-commutative geometry by Connes) living on a general Marcinkiewicz space associated with an arbitrary semifinite von Neumann algebra. The corollaries to our approach, stated in terms of the operator ideal L ( 1 , ∞ ) (which is a special example of an operator Marcinkiewicz space), are: (i) a new characterization of the set of all positive measurable operators from L ( 1 , ∞ ) , i.e. those on which an arbitrary Connes–Dixmier trace yields the same value. In the special case, when the operator ideal L ( 1 , ∞ ) is considered on a type I infinite factor, a bounded operator x belongs to L ( 1 , ∞ ) if and only if the sequence of singular numbers { s n ( x ) } n ⩾ 1 (in the descending order and counting the multiplicities) satisfies ∥ x ∥ ( 1 , ∞ ) ≔ sup N ⩾ 1 1 Log ( 1 + N ) ∑ n = 1 N s n ( x ) ∞ . In this case, our characterization amounts to saying that a positive element x ∈ L ( 1 , ∞ ) is measurable if and only if lim N → ∞ 1 Log N ∑ n = 1 N s n ( x ) exists; (ii) the set of Dixmier traces and the set of Connes–Dixmier traces are norming sets (up to equivalence) for the space L ( 1 , ∞ ) / L 0 ( 1 ∞ ) , where the space L 0 ( 1 , ∞ ) is the closure of all finite rank operators in L ( 1 , ∞ ) in the norm ∥ . ∥ ( 1 , ∞ ) .

CHARACTERIZATIONS OF KADEC-KLEE PROPERTIES IN SYMMETRIC SPACES OF MEASURABLE FUNCTIONS

We present several characterizations of Kadec-Klee properties in symmetric function spaces on the half-line, based on the K-functional of J. Peetre. In addition to the usual Kadec-Klee property, we study those symmetric spaces for which sequential convergence in measure (respectively, local convergence in measure) on the unit sphere coincides with norm convergence.

Unbounded Fredholm modules and double operator integrals

Abstract In noncommutative geometry one is interested in invariants such as the Fredholm index or spectral flow and their calculation using cyclic cocycles. A variety of formulae have been established under side conditions called summability constraints. These can be formulated in two ways, either for spectral triples or for bounded Fredholm modules. We study the relationship between these by proving various properties of the map on unbounded self adjoint operators D given by ƒ(D) = D(1 + D 2)−1/2. In particular we prove commutator estimates which are needed for the bounded case. In fact our methods work in the setting of semifinite noncommutative geometry where one has D as an unbounded self adjoint linear operator affiliated with a semi-finite von Neumann algebra ℳ. More precisely we show that for a pair D, D 0 of such operators with D – D 0 a bounded self-adjoint linear operator from ℳ and , where 𝓔 is a noncommutative symmetric space associated with ℳ, then . This result is further used to show continuous differentiability of the mapping between an odd 𝓔-summable spectral triple and its bounded counterpart.

Weak compactness criteria in symmetric spaces of measurable operators

The principal result of the paper reduces the study of certain weakly compact sets in Banach spaces of measurable operators to that of the corresponding sets of generalized singular value functions. In particular, under natural conditions, it is shown that the orbit of a relatively weakly compact subset of the Kothe dual of a symmetric space of measurable operators affiliated with some semi-finite von Neumann algebra is again relatively weakly compact.

Spectral averaging for trace compatible operators

In this note the notions of trace compatible operators and infin- itesimal spectral flow are introduced. We define the spectral shift function as the integral of infinitesimal spectral flow. It is proved that the spectral shift function thus defined is absolutely continuous and Kreins formula is es- tablished. Some examples of trace compatible affine spaces ofoperators are given.

RUC-Bases in Orlicz and Lorentz Operator Spaces

AbstractIf n

COMMUTATOR ESTIMATES AND

{ x_n } _{n = 1}^infty