A Lidskii type formula for Dixmier traces

Lidskii-Type Formulae for Dixmier Traces

“…Finally, various formulae of noncommutative geometry (in particular, those involving heat kernel estimates and generalised ζ-function) were established in [3,5,7] for yet a smaller subset of Connes-Dixmier traces, when the functional ω was assumed to be M -invariant. This class (and its further modifications) was first introduced in [3] (see also [10]) and further studied and used in [1,2,5]. For brevity we refer to the latter class (a proper subclass of Connes-Dixmier traces) as a class of M -invariant Dixmier traces.…”

“…In this section, we extend results of [1] (and, partially, those of [2]) to a wider class of Marcinkiewicz spaces and Connes-Dixmier traces. To this end, we need some extra assumptions on ψ ∈ Ψ.…”

Dixmier traces and some applications in non-commutative geometry

“…Finally, following [8], we show how Lidskii's formula may be extended to Dixmier traces in the von Neumann setting in Section 18. In a series of corollaries we explain its relevance to the question of measurability.…”

“…In [31], except for the case of pseudodifferential operators, measurability results are proved for positive operators. The approach of [8] allows one to address the problem of removing the positivity assumption.…”

“…As a result of the extra effort needed to handle this level of generality it is possible to find improvements even in the standard type I theory of [31]. A number of authors have contributed to the development of this framework and new applications of Dixmier and more general singular traces [2], [3], [4], [8], [10], [32] [19], [20], [21], [22], [23], [24], [25], [47], [48], [58], [59], [68], [83], [61], [60], [91].…”