Traces of compact operators and the noncommutative residue

Abstract

We extend the noncommutative residue of M. Wodzicki on compactly supported classical pseudo-differential operators of order $-d$ and generalise A. Connes' trace theorem, which states that the residue can be calculated using a singular trace on compact operators. Contrary to the role of the noncommutative residue for the classical pseudo-differential operators, a corollary is that the pseudo-differential operators of order $-d$ do not have a `unique' trace; pseudo-differential operators can be non-measurable in Connes' sense. Other corollaries are given clarifying the role of Dixmier traces in noncommutative geometry \`{a} la Connes, including the definitive statement of Connes' original theorem.Comment: Version change: added information on Nigel Kalton. *Nigel Kalton (1946-2010). The author passed away during production of this pape

Universal measurability and the Hochschild class of the Chern character

“…The statement below appeared first in [12] and for a detailed proof we refer the reader to Theorem 5.7.6 and Theorem 10.1.3 in [14]. λ(k, T ) = z log(n + 1) + O(1), n ≥ 0, for some constant z ∈ C. In this case, ϕ(T ) = z for every normalised trace ϕ.…”

“…Rather than simply produce an erratum we decided to revisit the whole argument in the light of progress made in the last 10 years 1 [5,6,9,12] which provides, amongst other contributions, a more powerful algebraic approach to these issues resulting in two advances. (i) We prove Connes' result for arbitrary traces on L 1,∞ (other proofs hold only for the original trace discovered by Dixmier).…”

Dixmier traces and extrapolation description of noncommutative Lorentz spaces

“…This formula has a great impact on the theory of pseudo-differential operators [35], and especially on Connes' Trace Theorem [14], which connects Dixmier traces and Wodzicki residue [62,63].…”

“…It has then been followed by an important list of improvements and generalizations of many kinds [4,7,8,60]. In the form we phrase it here, this 1 A very short and simple proof of that fact is displayed in [35,Theorem 3]. Note also that a concrete example a of non-measurable operator in the context of pseudo-differential operators on R n with non-homogeneous symbol, is also given there.…”

“…One of the main features of this formula is that it extends to more general operator ideals than M 1,∞ (H), in particular to the Lorentz ideal M ψ (H) (with ψ ∈ Ω satisfying condition (9) -see below). The second alternative approach relies on the results of [35], and allows one to compute Dixmier traces from expectation values [44,Corollary 8…”