Measures from Dixmier traces and zeta functions

For L∞-functions on a (closed) compact Riemannian manifold, the noncommutative residue and the Dixmier trace formulation of the noncommutative integral are shown to equate to a multiple of the Lebesgue integral. The identifications are shown to continue to, and be sharp at, L2-functions. For functions strictly in Lp, 1⩽p<2, symmetrised noncommutative residue and Dixmier trace formulas must be introduced, for which the identification is shown to continue for the noncommutative residue. However, a failure is shown for the Dixmier trace formulation at L1-functions. It is shown the noncommutative residue remains finite and recovers the Lebesgue integral for any integrable function while the Dixmier trace expression can diverge. The results show that a claim in the monograph [J.M. Gracia-Bondía, J.C. Várilly, H. Figueroa, Elements of Noncommutative Geometry, Birkhäuser Adv. Texts, Birkhäuser, Boston, 2001], that the equality on C∞-functions between the Lebesgue integral and an operator-theoretic expression involving a Dixmier trace (obtained from Connes' Trace Theorem) can be extended to any integrable function, is false. The results of this paper include a general presentation for finitely generated von Neumann algebras of commuting bounded operators, including a bounded Borel or L∞ functional calculus version of C∞ results in IV.2.δ of [A. Connes, Noncommutative Geometry, Academic Press, New York, 1994].