## Noncommutative Riemann integration and Novikov-Shubin invariants for Open Manifolds (2001)

Citations: | 6 - 3 self |

### BibTeX

@MISC{Guido01noncommutativeriemann,

author = {Daniele Guido and Tommaso Isola},

title = {Noncommutative Riemann integration and Novikov-Shubin invariants for Open Manifolds},

year = {2001}

}

### OpenURL

### Abstract

Given a C ∗-algebra A with a semicontinuous semifinite trace τ acting on the Hilbert space H, we define the family A R of bounded Riemann measurable elements w.r.t. τ as a suitable closure, à la Dedekind, of A, in analogy with one of the classical characterizations of Riemann measurable functions [26], and show that A R is a C ∗-algebra, and τ 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 τ-a.e. bimodule on A R, denoted by A R, and such bimodule contains the functional calculi of selfadjoint elements of A R under unbounded Riemann measurable functions. Besides, τ extends to a bimodule trace on A R.