NONCOMMUTATIVE RESIDUES AND A CHARACTERISATION OF THE NONCOMMUTATIVE INTEGRAL

作者：Lord Steven^*; Sukochev Fedor A

来源：Proceedings of the American Mathematical Society, 2011, 139(1): 243-257.

DOI：10.1090/S0002-9939-2010-10472-0

• Dixmier trace
• zeta functions
• noncommutative integral
• noncommutative geometry
• normal
• noncommutative residue

We continue the study of the relationship between Dixmier traces and noncommutative residues initiated by A. Connes. The utility of the residue approach to Dixmier traces is shown by a characterisation of the noncommutative integral in Connes' noncommutative geometry (for a wide class of Dixmier traces) as a generalised limit of vector states associated to the eigenvectors of a compact operator (or an unbounded operator with compact resolvent). Using the characterisation, a criteria involving the eigenvectors of a compact operator and the projections of a von Neumann subalgebra of bounded operators is given so that the noncommutative integral associated to the compact operator is normal, i.e. satisfies a monotone convergence theorem, for the von Neumann subalgebra. Flat tori, noncommutative tori, and a link with the QUE property of manifolds are given as examples.

• 出版日期2011-1