Singular symmetric functionals and Banach limits with additional invariance properties

Abstract

For symmetric spaces of measurable functions on the real half-line, we study the problem of existence of positive linear functionals monotone with respect to the Hardy-Littlewood semi-ordering, the so-called symmetric functionals. Two new wide classes of symmetric spaces are constructed which are distinct from Marcinkiewicz spaces and for which the set of symmetric functionals is nonempty. We consider a new construction of singular symmetric functionals based on the translation-invariance of Banach limits defined on the space of bounded sequences. We prove the existence of Banach limits invariant under the action of the Hardy operator and all dilation operators. This result is used to establish the stability of the new construction of singular symmetric functionals for an important class of generating sequences.

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.…”

“…We refer the reader to [9][10][11] for conditions which guarantee the additivity of τ ω . It is well known that τ ω is additive for any ω as above when…”

Dixmier traces and some applications in non-commutative geometry

“…For this and other geometric interpretations of conditions (3) and (4) we refer the reader to [13], [69,II.5.7] and [86]. For various constructions of singular symmetric functionals on M (ψ) (and more generally on fully symmetric spaces and their non-commutative counterparts) we refer to [47], [45], [46]. Constructions relevant to our main topic will be reviewed below, in Section 5.…”

“…[ [25,45,46,74]] For every semifinite von Neumann algebra (N , τ ) and arbitrary states L ∈ BL(R + ) and L ∈ BL(N), the functionals F L and F L are symmetric functionals on…”

“…Many features of the theory may be well understood, even in the most trivial situation, when the von Neumann algebras in question are commutative. In this situation, the theory of Dixmier traces roughly corresponds to the theory of symmetric functionals on rearrangement invariant function spaces [45,46,47] and allows an alternative treatment based on the methods drawn from real analysis.…”