Dixmier traces as singular symmetric functionals and applications to measurable operators
By S. Lord, A. Sedaev, Fedor Sukochev. 2005
63 citations recorded by the source corpus.
Abstract We unify various constructions and contribute to the theory of singular symmetric functionals on Marcinkiewicz function/operator spaces. This affords a new approach to the non-normal Dixmier and Connes–Dixmier traces (introduced by Dixmier and adapted to non-commutative geometry by Connes) living on a general Marcinkiewicz space associated with an arbitrary semifinite von Neumann algebra. The corollaries to our approach, stated in terms of the operator ideal L ( 1 , ∞ ) (which is a special example of an operator Marcinkiewicz space), are: (i) a new characterization of the set of all positive measurable operators from L ( 1 , ∞ ) , i.e. those on which an arbitrary Connes–Dixmier trace yields the same value. In the special case, when the operator ideal L ( 1 , ∞ ) is considered on a type I infinite factor, a bounded operator x belongs to L ( 1 , ∞ ) if and only if the sequence of singular numbers { s n ( x ) } n ⩾ 1 (in the descending order and counting the multiplicities) satisfies ∥ x ∥ ( 1 , ∞ ) ≔ sup N ⩾ 1 1 Log ( 1 + N ) ∑ n = 1 N s n ( x ) ∞ . In this case, our characterization amounts to saying that a positive element x ∈ L ( 1 , ∞ ) is measurable if and only if lim N → ∞ 1 Log N ∑ n = 1 N s n ( x ) exists; (ii) the set of Dixmier traces and the set of Connes–Dixmier traces are norming sets (up to equivalence) for the space L ( 1 , ∞ ) / L 0 ( 1 ∞ ) , where the space L 0 ( 1 , ∞ ) is the closure of all finite rank operators in L ( 1 , ∞ ) in the norm ∥ . ∥ ( 1 , ∞ ) .
Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/11ec13f0dc4bf1c8b81f7728b5118b054f9058ca
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Citations in the corpus
- The Dixmier trace and asymptotics of zeta functions (2006, unknown): “Notice that in this definition we have chosen ω to satisfy only the dilation invariance assumption even though Dixmier [20] originally imposed on ω the assumption of dilation and translation invariance (see [33] for more details).”
- Dixmier traces and some applications in non-commutative geometry (2006, methodology): “(ii) The definitions of Dixmier- and Connes–Dixmier-measurable operators extend naturally [60] to the Marcinkiewicz spaces L (1,∞)(N , τ) with (N , τ) an arbitrary semifinite von Neumann algebra, and further, to the operator Marcinkiewicz spaces M(ψ)(N , τ) for all ψ satisfying condition (3).”
- Invariant Banach limits and applications (2010, background): “Since the set {yn} is dense in [1/2,1], the second assertion follows immediately from [9].”