Alexandr Usachev

On the distinction between the classes of Dixmier and Connes-Dixmier traces

In the present paper we prove that the classes of Dixmier and Connes-Dixmier traces differ even on the Dixmier ideal

DIXMIER TRACES GENERATED BY EXPONENTIATION INVARIANT GENERALISED LIMITS

\mathcal M_{1,\infty}

The space of almost convergent sequences

. We construct a Marcinkiewicz space

Banach limits: Invariance and functional characteristics

\mathcal M_\psi

Characterization of singular traces on the weak trace class ideal generated by exponentiation invariant extended limits

and a positive operator

Banach limits and traces on L1

T\in \mathcal M_\psi

Geometric properties of the set of Banach limits

which is Connes-Dixmier measurable but which is not Dixmier measurable.

Generalized limits with additional invariance properties and their applications to noncommutative geometry

We define a new class of singular positive traces on the ideal M1,1 of B(H) generated by exponentiation invariant generalized limits. We prove that this new class is strictly contained in the class of all Dixmier traces. We also prove a Lidskii-type formula for this class of traces.

Fourier-Haar Coefficients and Banach Limits

, then(5)This bound is sharp in the sense that in (5) cannot bereplaced by a smaller number. In [8], the existence ofBanach limits invariant w ith respect to the Cesaro trans-form was proved. For such functionals, estimate (5) canbe sharpened.Obviously,The constant 2 in this formula is exact; i.e., for any λ < 2,there exists an x

Order and geometric properties of the set of Banach limits

Banach limits invariant with respect to the Cesàro transform are studied. New functional characteristics of Banach limits are introduced and studied.