Noncommutative Riemann integration and Novikov-Shubin invariants for open manifolds

Given a C*-algebra A with a semicontinuous semifinite trace tau acting on the Hilbert space H, we define the family A(R) of bounded Riemann measurable elements w.r.t. tau as a suitable closure, a la Dedekind of A, in analogy with one of the classical characterizations of Riemann measurable functions, and show that A(R) is a C*-algebra, and tau extends to a semicontinuous semifinite trace on A(R). Then, unbounded Riemann measurable operators are defined as the closed operators on H which are affiliated to A(") and can be approximated in measure by operators in A(R), in analogy with unbounded Riemann integration. Unbounded Riemann measurable operators form a tau-a.e biomodule on A(R), denoted by <(A(R))over bar>, and such a bimodule contains the functional calculi of selfadjoint elements of A(R) under unbounded Riemann measurable functions. Besides, tau extends to a bimodule trace on <(A(R))over bar>. Type II1 singular traces for C*-algebras can be defined on the bimodule of unbounded Riemann-measurable operators. Noncommutative Riemann integration and singular traces for C*-algebras are then used to define Novikov Shubin numbers for amenable open manifolds, to show their invariance under quasi-isometries, and to prove that they are (noncommutative) asymptotic dimensions.