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62
Hilbert modules and modules over finite von Neumann algebras and applications to L²invariants
 MATH. ANN. 309, 247285 (1997)
, 1997
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The relation between the BaumConnes conjecture and the trace conjecture
 Invent. Math
"... We prove a version of the L 2index Theorem of Atiyah, which uses the universal centervalued trace instead of the standard trace. We construct for Gequivariant Khomology an equivariant Chern character, which is an isomorphism and lives over the ring Z ⊂ Λ G ⊂ Q obtained from the integers by inver ..."
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Cited by 17 (10 self)
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We prove a version of the L 2index Theorem of Atiyah, which uses the universal centervalued trace instead of the standard trace. We construct for Gequivariant Khomology an equivariant Chern character, which is an isomorphism and lives over the ring Z ⊂ Λ G ⊂ Q obtained from the integers by inverting the orders of all finite subgroups of G. We use these two results to show that the BaumConnes Conjecture implies the modified Trace Conjecture, which says that the image of the standard trace K0(C ∗ r (G)) → R takes values in Λ G. The original Trace Conjecture predicted that its image lies in the additive subgroup of R generated by the inverses of all the orders of the finite subgroups of G, and has been disproved by Roy [13].
ON THE DIFFERENTIAL FORM SPECTRUM OF HYPERBOLIC MANIFOLDS
, 2003
"... Abstract. We give a lower bound for the bottom of the L 2 differential form spectrum on hyperbolic manifolds, generalizing thus a wellknown result due to Sullivan and Corlette in the function case. Our method is based on the study of the resolvent associated with the Hodgede Rham Laplacian and lea ..."
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Cited by 15 (3 self)
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Abstract. We give a lower bound for the bottom of the L 2 differential form spectrum on hyperbolic manifolds, generalizing thus a wellknown result due to Sullivan and Corlette in the function case. Our method is based on the study of the resolvent associated with the Hodgede Rham Laplacian and leads to applications for the (co)homology and topology of certain classes of hyperbolic manifolds. 1.
L²Invariants of Locally Symmetric Spaces
 DOCUMENTA MATH.
, 2002
"... Let X = G/K be a Riemannian symmetric space of the noncompact type, Γ ⊂ G a discrete, torsionfree, cocompact subgroup, and let Y = Γ\X be the corresponding locally symmetric space. In this paper we explain how the HarishChandra Plancherel Theorem for L²(G) and results on (g, K)cohomology can be u ..."
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Cited by 10 (0 self)
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Let X = G/K be a Riemannian symmetric space of the noncompact type, Γ ⊂ G a discrete, torsionfree, cocompact subgroup, and let Y = Γ\X be the corresponding locally symmetric space. In this paper we explain how the HarishChandra Plancherel Theorem for L²(G) and results on (g, K)cohomology can be used in order to compute the L²Betti numbers, the NovikovShubin invariants, and the L²torsion of Y in a uniform way thus completing results previously obtained by Borel, Lott, Mathai, Hess and Schick, Lohoue and Mehdi. It turns out that the behaviour of these invariants is essentially determined by the fundamental rank m = rkCG−rkCK of G. In particular, we show the nonvanishing of the L²torsion of Y whenever m = 1.
Singular traces, dimensions, and NovikovShubin invariants.
 Proceedings of the 17th OT Conference,
, 2000
"... Abstract In Alain Connes noncommutative geometry, the question of the existence of a nontrivial integral can be described in terms of the singular traceability of the compact operator D −d , D being the Dirac operator. In this paper we give a nontriviality condition different from the cohomolog ..."
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Cited by 7 (7 self)
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Abstract In Alain Connes noncommutative geometry, the question of the existence of a nontrivial integral can be described in terms of the singular traceability of the compact operator D −d , D being the Dirac operator. In this paper we give a nontriviality condition different from the cohomological one used by Connes, namely we show that, under suitable regularity conditions on the eigenvalue sequence of D, the dimension d can be uniquely determined by imposing that D −d is singularly traceable, thus providing a geometric measure theoretic definition for d. In the second part of the paper we discuss large scale counterparts of this notion of dimension for the case of covering manifolds. We show that ∆ −1/2 p , raised to power αp, the pth NovikovShubin number, is singularly traceable. As a consequence, NovikovShubin numbers can be considered as (asymptotic) dimensions in the sense of geometric measure theory. Finally we show that the (lower) NovikovShubin number α p coincides with (the supremum of) the dimension at ∞ of the semigroup generated by the Laplacian on pforms introduced in
Noncommutative Riemann integration and NovikovShubin invariants for Open Manifolds
, 2001
"... 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 [2 ..."
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Cited by 7 (3 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 [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 τa.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.