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37
Homological algebra of NovikovShubin invariants and Morse inequalities
, 1995
"... Abstract. It is shown in this paper that the topological phenomenon ”zero in the continuous spectrum”, discovered by S.P.Novikov and M.A.Shubin, can be explained in terms of a homology theory on the category of finite polyhedra with values in certain abelian category. This approach implies homotopy ..."
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Cited by 23 (5 self)
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Abstract. It is shown in this paper that the topological phenomenon ”zero in the continuous spectrum”, discovered by S.P.Novikov and M.A.Shubin, can be explained in terms of a homology theory on the category of finite polyhedra with values in certain abelian category. This approach implies homotopy invariance of the NovikovShubin invariants. Its main advantage is that it allows to use the standard homological techniques, such as spectral sequences, derived functors, universal coefficients etc., while studying the NovikovShubin invariants. It also leads to some new quantitative invariants, measuring the NovikovShubin phenomenon in a different way, which are used in the present paper in order to strengthen the Morse type inequalities of S.P. Novikov and M.A. Shubin [NS1]. §0.
Hilbert modules and modules over finite von Neumann algebras and applications to L²invariants
 MATH. ANN. 309, 247285 (1997)
, 1997
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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 10 (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.
Bordism, rhoinvariants and the Baum–Connes conjecture
 J. NONCOMMUT. GEOM.
, 2007
"... Let � be a finitely generated discrete group. In this paper we establish vanishing results for rhoinvariants associated to (i) the spin Dirac operator of a spin manifold with positive scalar curvature and fundamental group �; (ii) the signature operator of the disjoint union of a pair of homotopy e ..."
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Cited by 7 (4 self)
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Let � be a finitely generated discrete group. In this paper we establish vanishing results for rhoinvariants associated to (i) the spin Dirac operator of a spin manifold with positive scalar curvature and fundamental group �; (ii) the signature operator of the disjoint union of a pair of homotopy equivalent oriented manifolds with fundamental group �. The invariants we consider are more precisely theAtiyah–Patodi–Singer ( APS) rhoinvariant associated to a pair of finite dimensional unitary representations 1; 2 W � ! U.d/, the L 2rhoinvariant of Cheeger–Gromov, the delocalized etainvariant of Lott for a nontrivial conjugacy class of � which is finite. We prove that all these rhoinvariants vanish if the group � is torsionfree and the Baum–Connes map for the maximal group C*algebra is bijective. This condition is satisfied, for example, by torsionfree amenable groups or by torsionfree discrete subgroups of SO.n; 1 / and SU.n; 1/. For the delocalized invariant we only assume the validity of the Baum–Connes conjecture for the reduced C*algebra. In addition to the examples above, this condition is satisfied e.g. by Gromov hyperbolic groups or by cocompact discrete subgroups of SL.3; C/. In particular, the three rhoinvariants associated to the signature operator are, for such groups, homotopy invariant. For the APS and the Cheeger–Gromov rhoinvariants the latter result had been established by Navin Keswani. Our proof reestablishes this result and also extends it to the delocalized etainvariant of Lott. The proof exploits in a fundamental way results from bordism theory as well as various generalizations of the APSindex theorem; it also embeds these results in general vanishing phenomena for degree zero higher rhoinvariants (taking values in A=ŒA; A � for suitable C*algebras A). We also obtain precise information about the etainvariants in question themselves, which are usually much more subtle objects than the rhoinvariants.
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.
Adiabatic Limits and Spectral Geometry of Foliations
, 1995
"... We study spectral asymptotics for the Laplace operator on differential forms on a Riemannian foliated manifold equipped with a bundlelike metric in the case when the metric is blown up in directions normal to the leaves of the foliation. The asymptotical formula for the eigenvalue distribution func ..."
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Cited by 6 (2 self)
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We study spectral asymptotics for the Laplace operator on differential forms on a Riemannian foliated manifold equipped with a bundlelike metric in the case when the metric is blown up in directions normal to the leaves of the foliation. The asymptotical formula for the eigenvalue distribution function is obtained. The relationships with the spectral theory of leafwise Laplacian and with the noncommutative spectral geometry of foliations are discussed.
An asymptotic dimension for metric spaces, and the 0th NovikovShubin invariant
"... A nonnegative number d∞, called asymptotic dimension, is associated with any metric space. Such number detects the asymptotic properties of the space (being zero on bounded metric spaces), fulfills the properties of a dimension, and is invariant under rough isometries. It is then shown that for a cl ..."
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Cited by 6 (5 self)
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A nonnegative number d∞, called asymptotic dimension, is associated with any metric space. Such number detects the asymptotic properties of the space (being zero on bounded metric spaces), fulfills the properties of a dimension, and is invariant under rough isometries. It is then shown that for a class of open manifolds with bounded geometry the asymptotic dimension coincides with the 0th NovikovShubin number α0 defined in a previous paper [D. Guido, T. Isola, J. Funct. Analysis, 176 (2000)]. Thus the dimensional interpretation of α0 given in the mentioned paper in the framework of noncommutative geometry is established on metrics grounds. Since the asymptotic dimension of a covering manifold coincides with the polynomial growth of its covering group, the stated equality generalises to open manifolds a result by Varopoulos. 0. Introduction.
Equality of Lifshitz and van Hove exponents on amenable Cayley graphs
"... Abstract. We study the low energy asymptotics of periodic and random Laplace operators on Cayley graphs of amenable, finitely generated groups. For the periodic operator the asymptotics is characterised by the van Hove exponent or zeroth NovikovShubin invariant. The random model we consider is give ..."
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Cited by 5 (3 self)
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Abstract. We study the low energy asymptotics of periodic and random Laplace operators on Cayley graphs of amenable, finitely generated groups. For the periodic operator the asymptotics is characterised by the van Hove exponent or zeroth NovikovShubin invariant. The random model we consider is given in terms of an adjacency Laplacian on site or edge percolation subgraphs of the Cayley graph. The asymptotic behaviour of the spectral distribution is exponential, characterised by the Lifshitz exponent. We show that for the adjacency Laplacian the two invariants/exponents coincide. The result holds also for more general symmetric transition operators. For combinatorial Laplacians one has a different universal behaviour of the low energy asymptotics of the spectral distribution function, which can be actually established on quasitransitive graphs without an amenability assumption. The latter result holds also for long range bond percolation models. 1.