arXiv subjects

Fedor A. Sukochev

Publications and source records attributed to Fedor A. Sukochev.

3 recordsLinked to original sources

arXiv2021

A multidimensional solution to additive homological equations

In this paper we prove that for a finite-dimensional real normed space $V$, every bounded mean zero function $f\in L_\infty([0,1];V)$ can be written in the form $f = g\circ T - g$ for some $g\in L_\infty([0,1];V)$ and some ergodic invertible measure preserving transformation $T$ of $[0,1]$. Our method moreover allows us to choose $g$, for any given $\varepsilon>0$, to be such that $\|g\|_\infty\leq (S_V+\varepsilon)\|f\|_\infty$, where $S_V$ is the Steinitz constant corresponding to $V$.

math.DS↗

arXiv2019

Full proof of Kwapień's theorem on representing bounded mean zero functions on $[0,1]$

In [7], Kwapień announced that every mean zero function $f\in L_\infty[0,1]$ can be written as a coboundary $f = g\circ T -g$ for some $g\in L_\infty[0,1]$ and some measure preserving transformation $T$ of $[0,1]$. Whereas the original proof in [7] holds for continuous functions, there is a serious gap in the proof for functions with discontinuities. In this article we fill in this gap and establish Kwapień's result in full generality. Our method also allows to improve the original result by showing that for any given $ε>0$ the function $g$ can be chosen to satisfy a bound $\|g\|_\infty\leq (1+ε)\|f\|_\infty$.

math.DS↗

arXiv2009

Noncommutative Residues and a Characterisation of the Noncommutative Integral

We continue the study of the relationship between Dixmier traces and noncommutative residues initiated by A. Connes. The utility of the residue approach to Dixmier traces is shown by a characterisation of the noncommutative integral in Connes' noncommutative geometry (for a wide class of Dixmier traces) as a generalised limit of vector states associated to the eigenvectors of a compact operator (or an unbounded operator with compact resolvent), i.e. as a generalised quantum limit. Using the characterisation, a criteria involving the eigenvectors of a compact operator and the projections of a von Neumann subalgebra of bounded operators is given so that the noncommutative integral associated to the compact operator is normal, i.e. satisfies a monotone convergence theorem, for the von Neumann subalgebra.

math.OA↗