Lord, S.Sukochev, F.2011-05-192011-05-192011Proceedings of the American Mathematical Society, 2011; 139(1):243-2570002-99391088-6826http://hdl.handle.net/2440/63726We 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.en© 2010 American Mathematical Society. The copyright for this article reverts to public domain after 28 years from publication.Journal article00201032472011051911434210.1090/S0002-9939-2010-10472-00002866488000232-s2.0-7865075397832117Lord, S. [0000-0002-6142-5358]