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.

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 [16], 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 improper Riemann integration. Unbounded Riemann measurable operators form a τ-a.e. bimodule on A R, denoted by AR, 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 AR. As type II1 singular traces for a semifinite von Neumann algebra M with a normal semifinite faithful (non-atomic) trace τ have been defined as traces on M − M-bimodules of unbounded τ-measurable operators [5], type II1 singular traces for a C ∗-algebra A with a semicontinuous semifinite (non-atomic) trace τ are defined here as traces on A − A-bimodules of unbounded Riemann measurable operators (in AR) for any faithful representation of A. An application of singular traces for C ∗-algebras is contained in [6].

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Abstract. We prove in an elementary fashion that the image of a commutative monotone σ-complete C ∗-algebra under a σ-normal morphism is again monotone σ-complete and give an application of this result in spectral theory. 1.

A trace on the C ∗-algebra A of quasi-local operators on an open manifold is described, based on the results in [36]. It allows a description à la Novikov-Shubin [31] of the low frequency behavior of the Laplace-Beltrami operator. The 0-th Novikov-Shubin invariant defined in terms of such a trace is proved to coincide with a metric invariant, which we call asymptotic dimension, thus giving a large scale “Weyl asymptotics ” relation. Moreover, in analogy with the Connes-Wodzicki result [7, 8, 45], the asymptotic dimension d measures the singular traceability (at 0) of the Laplace-Beltrami operator, namely we may construct a (type II1) singular trace which is finite on the ∗-bimodule over A generated by ∆ −d/2. 1 Asymptotic dimension and Novikov-Shubin invariants 2 0 Introduction. The inspiration of this paper came from the idea of Connes ’ [8] of defining the dimension of a noncommutative compact manifold in terms of the Weyl asymptotics,

We present the mathematical setting of the theory of W ∗-algebras, including the Tomita– Takesaki modular theory, the Connes–Araki relative modular theory, the theory of W ∗-dynamical systems (with derivations, liouvilleans, and crossed products), and noncommutative integration up to the Falcone–Takesaki theory of noncommutative Lp(N) spaces. We pay special attention to abstract algebraic formulation of all structural properties (avoiding