Noncommutative Riemann integration and and Novikov-Shubin invariants for open manifolds

Details Noncommutative Riemann integration and and Novikov-Shubin invariants for open manifolds Noncommutative Riemann integration and and Novikov-Shubin invariants for open manifolds

1998-02-03

arxiv.org/abs/math/9802015v3

J. Funct. Anal. 176, (2000), 115-152

Mathematics

Operator Algebras

34 pages, LaTeX, a new section has been added, concerning an application to Novikov-Shubin invariants, the title changed accor

Scientific paper

10.1006/jfan.2000.3609

Given a C*-algebra A with a semicontinuous semifinite trace tau acting on the Hilbert space H, we define the family 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 R is a C*-algebra, and tau extends to a semicontinuous semifinite trace on 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 R, in analogy with unbounded Riemann integration. Unbounded Riemann measurable operators form a tau-a.e. bimodule on R, denoted by R^, and such bimodule contains the functional calculi of selfadjoint elements of R under unbounded Riemann measurable functions. Besides, tau extends to a bimodule trace on R^. Type II_1 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, show their invariance under quasi-isometries, and prove that they are (noncommutative) asymptotic dimensions.

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