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Noncommutative Riemann integration and singular traces for C ∗  algebras
"... 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 [1 ..."
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Cited by 4 (4 self)
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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 τa.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 (nonatomic) trace τ have been defined as traces on M − Mbimodules of unbounded τmeasurable operators [5], type II1 singular traces for a C ∗algebra A with a semicontinuous semifinite (nonatomic) trace τ are defined here as traces on A − Abimodules 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].
de Rham cohomology of configuration spaces with Poisson measure
 J. Funct. Anal
, 1995
"... The space ΓX of all locally finite configurations in a Riemannian manifold X of infinite volume is considered. The deRham complex of squareintegrable differential forms over ΓX, equipped with the Poisson measure, and the corresponding deRham cohomology are studied. The latter is shown to be unitari ..."
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Cited by 4 (0 self)
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The space ΓX of all locally finite configurations in a Riemannian manifold X of infinite volume is considered. The deRham complex of squareintegrable differential forms over ΓX, equipped with the Poisson measure, and the corresponding deRham cohomology are studied. The latter is shown to be unitarily isomorphic to a certain Hilbert tensor algebra generated by the L 2cohomology of the underlying manifold X.
A Semicontinuous Trace for Almost Local Operators on an Open Manifold
, 2001
"... A semicontinuous semifinite trace is constructed on the C*algebra ..."
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Cited by 3 (3 self)
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A semicontinuous semifinite trace is constructed on the C*algebra
L²Homology over Traced *Algebras
, 1996
"... Given a unital complex *algebra A, a tracial positive linear functional ø on A that factors through a *representation of A on Hilbert space, and an A module M possessing a resolution by finitely generated projective Amodules, we construct homology spaces H k (A; ø; M ) for k = 0; 1; : : : . Ea ..."
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Cited by 2 (0 self)
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Given a unital complex *algebra A, a tracial positive linear functional ø on A that factors through a *representation of A on Hilbert space, and an A module M possessing a resolution by finitely generated projective Amodules, we construct homology spaces H k (A; ø; M ) for k = 0; 1; : : : . Each is a Hilbert space equipped with a *representation of A, independent (up to unitary equivalence) of the given resolution of M . A short exact sequence of Amodules gives rise to a long weakly exact sequence of homology spaces. There is a Kunneth formula for tensor products. The von Neumann dimension which is defined for Ainvariant subspaces of L 2 (A; ø ) n gives wellbehaved Betti numbers and an Euler characteristic for M with respect to A and ø .
Twisted L² Invariants Of NonSimply Connected Manifolds And Asymptotic L² Morse Inequalities
"... We develop the theory of twisted L²cohomology and twisted spectral invariants for at Hilbertian bundles over compact manifolds. They can be viewed as functions on H¹(M, R) and they generalize the standard notions. A new feature of the twisted L²cohomology theory is that in addition ..."
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Cited by 1 (1 self)
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We develop the theory of twisted L²cohomology and twisted spectral invariants for at Hilbertian bundles over compact manifolds. They can be viewed as functions on H¹(M, R) and they generalize the standard notions. A new feature of the twisted L²cohomology theory is that in addition to satisfying the standard L² Morse inequalities, they also satisfy certain asymptotic L² Morse inequalities. These reduce to the standard Morse inequalities in the finite dimensional case, and when the Morse 1form is exact. We de ne the extended twisted L² de Rham cohomology and prove the asymptotic L² MorseFarber inequalities, which give quantitative lower bounds for the Morse numbers of a Morse 1form on M.
harmonic forms
"... Let X be a Riemannian manifold endowed with a cocompact isometric action of an infinite discrete group. We consider L 2 spaces of harmonic vectorvalued forms on the product manifold X N, which are invariant with respect to an action of the braid group BN, and compute their von Neumann dimensions ( ..."
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Let X be a Riemannian manifold endowed with a cocompact isometric action of an infinite discrete group. We consider L 2 spaces of harmonic vectorvalued forms on the product manifold X N, which are invariant with respect to an action of the braid group BN, and compute their von Neumann dimensions (the braided L 2 Betti numbers).