Noncommutative Riemann Integration and Novikov–Shubin Invariants for Open Manifolds

Abstract

Given a C*-algebra A with a semicontinuous semifinite trace τ acting on the Hilbert space H, we define the family AR of bounded Riemann measurable elements w.r.t. τ as a suitable closure, á la Dedekind, of A, in analogy with one of the classical characterizations of Riemann measurable functions, and show that AR is a C*-algebra, and τ extends to a semicontinuous semifinite trace on AR. 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 AR, in analogy with unbounded Riemann integration. Unbounded Riemann measurable operators form a τ-a.e. bimodule on AR, denoted by AR, and such a bimodule contains the functional calculi of selfadjoint elements of AR under unbounded Riemann measurable functions. Besides, τ extends to a bimodule trace on AR. 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.