Banach limits and traces on L1,∞

Abstract

We introduce a new approach to traces on the principal ideal L 1,∞ generated by any positive compact operator whose singular value sequence is the harmonic sequence. Distinct from the well-known construction of J. Dixmier, the new approach provides the explicit construction of every trace of every operator in L 1,∞ in terms of translation invariant functionals applied to a sequence of restricted sums of eigenvalues. The approach is based on a remarkable bijection between the set of all traces on L 1,∞ and the set of all translation invariant functionals on l ∞ . This bijection allows us to identify all known and commonly used subsets of traces (Dixmier traces, Connes-Dixmier traces, etc.) in terms of invariance properties of linear functionals on l ∞ , and definitively classify the measurability of operators in L 1,∞ in terms of qualified convergence of sums of eigenvalues. This classification has led us to a resolution of several open problems (for the class L 1,∞ ) from [7]. As an application we extend Connes' classical trace theorem to positive normalised traces.

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“…Of particular interest is L 1,∞ , and we are concerned with traces on this ideal. For more details, see [37,Section 5.7] and [53]. A functional ϕ : L 1,∞ → C is called a trace if it is unitarily invariant.…”

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“…An extensive discussion of traces, and more recent developments in the theory, may be found in [14] including a discussion of the following facts. traces (see [18]). (4) There exist traces on L 1,∞ which fail to be continuous (see [9]).…”

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