Dixmier traces and some applications in non-commutative geometry

Invariant Banach limits and applications

“…In Section 3 we present a detailed study of the case when H = C, where C is the classical Cesàro operator. In particular, we study the behavior of the iterations C m , m ∈ N. In Section 4 we apply the results of Section 3 to the study of singular traces appearing in noncommutative geometry [4,3]. The results obtained in Section 4 partially resolve open questions raised in [3] and complement earlier results of [9] describing the "measurable operators" of noncommutative geometry.…”

The Dixmier trace of Hankel operators on the Bergman space

“…Although this definition does depend on ω the operators A we consider are measurable, that is, the value of Tr ω (A) is independent of the particular instance of Tr ω considered. We refer to [9] and [8] for details and for discussion of the role of these functionals.…”

Lidskii-Type Formulae for Dixmier Traces

“…It is well known (see e.g. § 5 in [6] and additional references therein) that τ ω is an additive functional on the positive part of M 1,∞ . Thus, τ ω admits a linear extension to a unitarily invariant functional (trace) on M 1,∞ .…”

“…Dixmier traces τ ω defined such ω's are termed Connes-Dixmier traces. We refer to [6,7,15] for discussion of their properties. 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.…”