On the distinction between the classes of Dixmier and Connes-Dixmier traces (Q2838975)

For a nonnegative self-adjoint operator \(T\) its Dixmier trace can be defined as \(\mathrm{Tr}_{\omega}(T)=\lim_{N\to\infty} {1\over N} \sum_{i=0}^N \mu_i\), where \(\mu_i\) are the eigenvalues of \(T\) and \(\lim\) is a generalized limit depending on a parameter \(\omega\). A Connes-Dixmier trace has been introduced by \textit{S. Lord, A. Sedaev} and \textit{F. Sukochev} in [J. Funct. Anal. 224, No. 1, 72--106 (2005; Zbl 1081.46042)]. The main result of the paper says that in general the Dixmier and the Connes-Dixmier traces do not coincide with each other; concrete examples of this phenomenon are given. A draft of the paper is available at \url{arXiv:1210.3397}.