arXiv subjects
Publications and source records attributed to Evgenii Semenov.
The aim of this work is to describe subsets of Banach limits in terms of a certain functional characteristic. We compute radii and cardinalities for some of these subsets.
We study two subclasses of Banach limits: the one consisting of Banach limits which are invariant with respect of the Cesàro operator and another one consists of Banach limits which are invariant with respect to all dilations. We prove that the first is a proper subset of the second. We also show that these classes are at the maximal distance from the set ${\rm ext} \mathfrak B$ of all extreme points of the set of all Banach limits.
We introduce a new approach to traces on the principal ideal $\mathcal L_{1,\infty}$ generated by any positive compact operator whose singular value sequence is the harmonic sequence. Distinct from the well-known construction of J.~Dixmier, the new approach provides the explicit construction of every trace of every operator in $\mathcal L_{1,\infty}$ in terms of translation invariant functionals applied to a sequence of restricted sums of eigenvalues. The approach is based on a remarkable bijection between the set of all traces on $\mathcal L_{1,\infty}$ and the set of all translation invariant functionals on $l_\infty$. This bijection allows us to identify all known and commonly used subsets of traces (Dixmier traces, Connes-Dixmier traces, etc.) in terms of invariance properties of linear functionals on $l_\infty$, and definitively classify the measurability of operators in $\mathcal L_{1,\infty}$ in terms of qualified convergence of sums of eigenvalues. This classification has led us to a resolution of several open problems (for the class $\mathcal L_{1,\infty}$) from~\cite{CS}. As an application we extend Connes' classical trace theorem to positive normalised traces.