Singular Traces and Compact Operators

Abstract

We give a necessary and sufficient condition on a positive compact operator T for the existence of a singular trace (i.e. a trace vanishing on the finite rank operators) which takes a finite non-zero value on T. This generalizes previous results by Dixmier and Varga. We also give an explicit description of these traces and associated ergodic states on l (N) using tools of non standard analysis in an essential way.

Dixmier traces and some applications in non-commutative geometry

“…In many cases, the theory of singular traces runs in parallel with the theory of symmetric functionals. For instance, results of [2,92] (respectively, [58]) concerning necessary and sufficient conditions on a positive compact operator T (respectively, positive τ -measurable operators T ∈ L 1 (N , τ ) + N ) for the existence of a singular trace which takes a finite non-zero value on T are in fact very close to the result of Theorem 2.2. It should be noted that the results [58] are stated for general τ -measurable operators and not just for the operators from L 1 (N , τ ) + N , however not all the results there treating this general case are supplied with a reliable proof (e.g.…”

“…A literature devoted to general singular traces on L(H) is tremendous. We limit our list of papers from this area to the following articles [58,59,2,3,4,26,92]. In many cases, the theory of singular traces runs in parallel with the theory of symmetric functionals.…”

“…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].…”

Majorization and arithmetic mean ideals

“…This representation is directly related to diagonals of operators in unitary orbits (see Lemma 2.6 and the succeeding comment). Majorization at infinity, aka "tail majorization," was first introduced and studied for finite sequences and appears in [1], [29], [13], [46] among others. Here it provides the natural characterization of the notion of arithmetic mean at infinity closure for operator ideals contained in the trace class (see Proposition 4.3 and [22]- [26]).…”

Soft Ideals and Arithmetic Mean Ideals

“…The "right" tool for ideals in the trace-class is IEOT the arithmetic mean at infinity which was employed for sequences in [1,7,14,22] among others. For every summable sequence η,…”