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36
Hilbert modules and modules over finite von Neumann algebras and applications to L²invariants
 MATH. ANN. 309, 247285 (1997)
, 1997
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Various L 2 –signatures and a topological L 2 –signature theorem
"... For a normal covering over a closed oriented topological manifold we give a proof of the L 2signature theorem with twisted coefficients, using Lipschitz structures and the Lipschitz signature operator introduced by Teleman. We also prove that the Ltheory isomorphism conjecture as well as the C ∗ m ..."
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Cited by 17 (2 self)
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For a normal covering over a closed oriented topological manifold we give a proof of the L 2signature theorem with twisted coefficients, using Lipschitz structures and the Lipschitz signature operator introduced by Teleman. We also prove that the Ltheory isomorphism conjecture as well as the C ∗ maxversion of the BaumConnes conjecture imply the L 2signature theorem for a normal covering over a Poincaré space, provided that the group of deck transformations is torsionfree. We discuss the various possible definitions of L 2signatures (using the signature operator, using the cap product of differential forms, using a cap product in cellular L 2cohomology,...) in this situation, and prove that they all coincide.
FINITE GROUP EXTENSIONS AND THE ATIYAH CONJECTURE
"... In 1976, Atiyah [2] constructed the L2Betti numbers of a compact Riemannian manifold. They are defined in terms of the spectrum of the Laplace operator on the universal covering of M. ByAtiyah’sL2index theorem [2], they can be used e.g. to compute the Euler characteristic of M. ..."
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Cited by 11 (5 self)
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In 1976, Atiyah [2] constructed the L2Betti numbers of a compact Riemannian manifold. They are defined in terms of the spectrum of the Laplace operator on the universal covering of M. ByAtiyah’sL2index theorem [2], they can be used e.g. to compute the Euler characteristic of M.
Spectral asymptotics of percolation Hamiltoninas on amenable Cayley graphs
 In Methods of Spectral Analysis in Mathematical Physics (Lund, 2006), Volume 186 of Oper. Theory Adv. Appl
, 2008
"... Abstract. In this paper we study spectral properties of adjacency and Laplace operators on percolation subgraphs of Cayley graphs of amenable, finitely generated groups. In particular we describe the asymptotic behaviour of the integrated density of states (spectral distribution function) of these r ..."
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Cited by 8 (2 self)
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Abstract. In this paper we study spectral properties of adjacency and Laplace operators on percolation subgraphs of Cayley graphs of amenable, finitely generated groups. In particular we describe the asymptotic behaviour of the integrated density of states (spectral distribution function) of these random Hamiltonians near the spectral minimum. The first part of the note discusses various aspects of the quantum percolation model, subsequently we formulate a series of new results, and finally we outline the strategy used to prove our main theorem. 1.
L 2 Cohomology of Geometrically Infinite Hyperbolic 3Manifolds
, 1997
"... Abstract. We give results on the following questions about a topologically tame hyperbolic 3manifold M: 1. Does M have nonzero L 2harmonic 1forms? 2. Does zero lie in the spectrum of the Laplacian acting on Λ 1 (M)/Ker(d)? 1. ..."
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Cited by 6 (3 self)
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Abstract. We give results on the following questions about a topologically tame hyperbolic 3manifold M: 1. Does M have nonzero L 2harmonic 1forms? 2. Does zero lie in the spectrum of the Laplacian acting on Λ 1 (M)/Ker(d)? 1.
Singular traces, dimensions, and NovikovShubin invariants
 Proceedings of the 17th OT Conference, Theta
, 2000
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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 6 (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.
L²Invariants from the Algebraic Point of View
, 2008
"... We give a survey on L²invariants such as L²Betti numbers and L²torsion taking an algebraic point of view. We discuss their basic definitions, properties and applications to problems arising in topology, geometry, group theory and Ktheory. ..."
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Cited by 5 (3 self)
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We give a survey on L²invariants such as L²Betti numbers and L²torsion taking an algebraic point of view. We discuss their basic definitions, properties and applications to problems arising in topology, geometry, group theory and Ktheory.