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173
On a class of type II1 factors with Betti numbers invariants
, 2002
"... We prove that a type II1 factor M can have at most one Cartan subalgebra A satisfying a combination of rigidity and compact approximation properties. We use this result to show that within the class HT of factors M having such Cartan subalgebras A ⊂ M, the Betti numbers of the standard equivalence ..."
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Cited by 97 (23 self)
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We prove that a type II1 factor M can have at most one Cartan subalgebra A satisfying a combination of rigidity and compact approximation properties. We use this result to show that within the class HT of factors M having such Cartan subalgebras A ⊂ M, the Betti numbers of the standard equivalence relation associated with A ⊂ M ([G2]), are in fact isomorphism invariants for the factors M, β HT n (M), n ≥ 0. The class HT is closed under amplifications and tensor products, with the Betti numbers satisfying β HT n (Mt) = β HT n (M)/t, ∀t> 0, and a Künneth type formula. An example of a factor in the class HT is given by the group von Neumann factor M = L(Z2 ⋊ SL(2, Z)), for which β HT 1 (M) = β1(SL(2, Z)) = 1/12. Thus, Mt ̸ ≃ M, ∀t ̸ = 1, showing that the fundamental group of M is trivial. This solves a long standing problem of R.V. Kadison. Also, our results bring some insight into a recent problem of A. Connes and answer a number of open questions on von Neumann algebras.
The Novikov conjecture and groups with finite asymptotic dimension
, 1995
"... this paper we shall prove the coarse BaumConnes conjecture for proper metric spaces with nite asymptotic dimension. Combining this result with a certain descent principle we obtain the following application to the Novikov conjecture on homotopy invariance of higher signatures. Theorem 1.1 Let be a ..."
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Cited by 53 (5 self)
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this paper we shall prove the coarse BaumConnes conjecture for proper metric spaces with nite asymptotic dimension. Combining this result with a certain descent principle we obtain the following application to the Novikov conjecture on homotopy invariance of higher signatures. Theorem 1.1 Let be a nitely presented group whose classifying space B
A survey of foliations and operator algebras
 Proc. Sympos. Pure
, 1982
"... 1 Transverse measure for flows 4 2 Transverse measure for foliations 6 ..."
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Cited by 53 (5 self)
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1 Transverse measure for flows 4 2 Transverse measure for foliations 6
Eta invariants as sliceness obstructions and their relation to CassonGordon invariants
, 2004
"... ..."
The BaumConnes and the FarrellJones conjectures in K and Ltheory
 Preprintreihe SFB 478 — Geometrische Strukturen in der Mathematik, Heft 324
, 2004
"... Summary. We give a survey of the meaning, status and applications of the BaumConnes Conjecture about the topological Ktheory of the reduced group C ∗algebra and the FarrellJones Conjecture about the algebraic K and Ltheory of the group ring of a (discrete) group G. Key words: K and Lgroups o ..."
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Cited by 32 (24 self)
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Summary. We give a survey of the meaning, status and applications of the BaumConnes Conjecture about the topological Ktheory of the reduced group C ∗algebra and the FarrellJones Conjecture about the algebraic K and Ltheory of the group ring of a (discrete) group G. Key words: K and Lgroups of group rings and group C ∗algebras, BaumConnes
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
"... ..."
Quantum Hall Effect on the hyperbolic plane
 Commun. Math. Physics
, 1997
"... Abstract. We study both the continuous model and the discrete model of the quantum Hall effect (QHE) on the hyperbolic plane in the presence of disorder, extending the results of an earlier paper [CHMM]. Here we model impurities, that is we consider the effect of a random or almost periodic potentia ..."
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Cited by 20 (13 self)
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Abstract. We study both the continuous model and the discrete model of the quantum Hall effect (QHE) on the hyperbolic plane in the presence of disorder, extending the results of an earlier paper [CHMM]. Here we model impurities, that is we consider the effect of a random or almost periodic potential as opposed to just periodic potentials. The Hall conductance is identified as a geometric invariant associated to an algebra of observables, which has plateaus at gaps in extended states of the Hamiltonian. We use the Fredholm modules defined in [CHMM] to prove the integrality of the Hall conductance in this case. We also prove that there are always only a finite number of gaps in extended states of any random discrete Hamiltonian.
Geometry of growth: approximation theorems for L 2 invariants
, 1998
"... Abstract. In this paper we study the problem of approximation of the L 2topological invariants by their finite dimensional analogues. We obtain generalizations of the theorem of Lück [L], dealing with towers of finitely sheeted normal coverings. We prove approximation theorems, establishing relatio ..."
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Cited by 20 (1 self)
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Abstract. In this paper we study the problem of approximation of the L 2topological invariants by their finite dimensional analogues. We obtain generalizations of the theorem of Lück [L], dealing with towers of finitely sheeted normal coverings. We prove approximation theorems, establishing relations between the homological invariants, corresponding to infinite dimensional representations and sequences of finite dimensional representations, assuming that their normalized characters converge. Also, we find an approximation theorem for residually finite pgroups (p is a prime), where we use the homology with coefficients in a finite field Fp. We view sequences of finite dimensional flat bundles of growing dimension as examples of growth processes. We study a von Neumann category with a Dixmier type trace, which allows to describe the asymptotic invariants of growth processes. We introduce a new invariant of torsion objects, the torsion dimension. We show that the torsion dimension appears in general as an additional correcting term in the approximation theorems; it vanishes under some arithmeticity assumptions. We also show that the torsion dimension allows to establish nontriviality of the Grothendieck group of torsion