Dixmier traces of operators on Banach and Hilbert spaces

Abstract

We present a simple approach to the theory of traces on the Marcinkiewicz operator ideal Particular attention is paid to the Dixmier traces, which may be obtained from both dilation‐invariant functionals and shift‐invariant functionals on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathfrak l_\infty$\end{document}. Since the concept of positivity does not make sense for traces defined on operator ideals over Banach spaces, traces are not assumed to be positive in the Hilbert space setting, as well. This standpoint raises some new problems. The present studies will be continued in 35, 36.

Banach limits and traces on L1,∞

“…It is a strict subset since it is known that the set of Dixmier and Connes-Dixmier traces are distinct [42,Theorem 6.1]. This distinction was studied by A. Pietsch in a series of three papers [43,44,42], where the deep techniques were developed, which are of a wider interest in the theory of singular traces. The inclusion C D was also proved in [57, Theorem 2.2] using a different approach.…”

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

“…A priori, C ⊆ D and the question about precise relationship between these two classes arises naturally. Recently the distinction between C and D was studied by A. Pietsch in terms of density characters (see [11]- [13]). For the discussion of various classes of singular traces we refer to [1,2,10].…”

Shift-invariant functionals on Banach sequence spaces

“…By [23,Prop. 2.3], it suffices to verify the condition above either for S − or S + ; the other one follows automatically.…”

“…We stress that X//S is just the usual quotient X/R(I − S), where R(I − S) denotes the range of I − S. The quotient map from X onto X//S is denoted by Q X S . Note that the dual (X//S) * can be identified with the space of all S-invariant functionals on X; see [23,Prop. 9.9].…”