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40
Hilbert modules and modules over finite von Neumann algebras and applications to L²invariants
 MATH. ANN. 309, 247285 (1997)
, 1997
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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 12 (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.
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 6 (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.
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.
Amenable groups, topological entropy and Betti numbers
, 1999
"... Abstract. We investigate an analogue of the L 2Betti numbers for amenable linear subshifts. The role of the von Neumann dimension shall be played by the topological entropy. Partially supported by OTKA grant T 25004 and the Bolyai Fellowship ..."
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Cited by 3 (0 self)
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Abstract. We investigate an analogue of the L 2Betti numbers for amenable linear subshifts. The role of the von Neumann dimension shall be played by the topological entropy. Partially supported by OTKA grant T 25004 and the Bolyai Fellowship
Amenable Covers, Volume and L 2 Betti Numbers of Aspherical Manifolds, available at math.AT/0605627
"... Abstract. We provide a proof for an inequality between volume and L 2Betti numbers of aspherical manifolds for which Gromov outlined a strategy based on general ideas of Connes. The implementation of that strategy involves measured equivalence relations, Gaboriau’s theory of L 2Betti numbers of R ..."
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Cited by 3 (2 self)
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Abstract. We provide a proof for an inequality between volume and L 2Betti numbers of aspherical manifolds for which Gromov outlined a strategy based on general ideas of Connes. The implementation of that strategy involves measured equivalence relations, Gaboriau’s theory of L 2Betti numbers of Rsimplicial complexes, and other themes of measurable group theory. Further, we prove new vanishing theorems for L 2Betti numbers that generalize a classical result of Cheeger and Gromov. As one of the corollaries, we obtain a gap theorem which implies vanishing of L 2Betti numbers of an aspherical manifold when its minimal volume is sufficiently small.