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.

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Abstract. A class of CW-complexes, called self-similar complexes, is introduced, together with C ∗-algebras Aj of operators, endowed with a finite trace, acting on square-summable cellular j-chains. Since the Laplacian ∆j belongs to Aj, L 2-Betti numbers and Novikov-Shubin numbers are defined for such complexes in terms of the trace. In particular a relation involving the Euler-Poincaré characteristic is proved. L 2-Betti and Novikov-Shubin numbers are computed for some self-similar complexes arising from self-similar fractals. 1. Introduction. In this paper we address the question of the possibility of extending the definition of some L 2-invariants, like the L 2-Betti numbers and Novikov-Shubin numbers, to geometric structures which are not coverings of compact spaces. The first attempt in this sense is due to John Roe [29], who defined a trace