Hilbert modules and modules over finite von Neumann algebras and applications to $L^2$ -invariants

Lück, Wolfgang

doi:10.1007/s002080050112

Cited by 52 publications

(50 citation statements)

References 22 publications

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“…This justifies our definition of N G as G-equivariant operators in B(l [11] the following theorem is proved. Theorem 2.1.…”

Section: Finite Von Neumann Algebrassupporting

confidence: 73%

“…The functor ν is such that ν(f * ) = (ν(f )) * (see [11]). Thus, ν(i * (f )) is selfadjoint since f is.…”

Section: Inductionmentioning

confidence: 99%

“…The image of ν(i * (f )) is dense and so the kernel of ν(i * (f )) is trivial. Since ν −1 is exact (see [11]), the kernel of i * (f ) is trivial also. So, 0 → A n → A n → Coker(i * (f )) → 0.…”

Section: Inductionmentioning

confidence: 99%

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Torsion Theories for Finite Von Neumann Algebras

Vaš

2005

Communications in Algebra

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Abstract. The study of modules over a finite von Neumann algebra A can be advanced by the use of torsion theories. In this work, some torsion theories for A are presented, compared and studied. In particular, we prove that the torsion theory (T, P) (in which a module is torsion if it is zero-dimensional) is equal to both Lambek and Goldie torsion theories for A.Using torsion theories, we describe the injective envelope of a finitely generated projective A-module and the inverse of the isomorphism K 0 (A) → K 0 (U ), where U is the algebra of affiliated operators of A. Then, the formula for computing the capacity of a finitely generated module is obtained. Lastly, we study the behavior of the torsion and torsion-free classes when passing from a subalgebra B of a finite von Neumann algebra A to A. With these results, we prove that the capacity is invariant under the induction of a B-module.

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“…This justifies our definition of N G as G-equivariant operators in B(l [11] the following theorem is proved. Theorem 2.1.…”

Section: Finite Von Neumann Algebrassupporting

confidence: 73%

“…The functor ν is such that ν(f * ) = (ν(f )) * (see [11]). Thus, ν(i * (f )) is selfadjoint since f is.…”

Section: Inductionmentioning

confidence: 99%

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Torsion Theories for Finite Von Neumann Algebras

Vaš

2005

Communications in Algebra

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“…In abstract measure theory, Geoghegan has conjectured (see [4] or [13,Question 13]) that F might be a counterexample to the conjecture that any finitely presented group with no non-cyclic free subgroup is amenable (admits a bounded, non-trivial, finitely additive measure on all subsets that is invariant under left multiplication). The contents and bibliographies of [2,3,5,7,8,16,19] give access to all that the authors know concerning Thompson's groups and their connections to other areas of mathematics.…”

Section: Background and Remarksmentioning

confidence: 99%

Higher Dimensional Thompson Groups

Brin

2004

Geometriae Dedicata

156

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We construct a "higher dimensional" version 2V of Thompson's group V. Like V it is an infinite, finitely presented, simple subgroup of the homeomorphism group of the Cantor set, but we show that it is not isomorphic to V by showing that the actions on the Cantor set are not topologically conjugate: 2V has an element with "chaotic" action, while V cannot have such an element. A theorem of Rubin is then applied which shows that for these two groups, isomorphism would imply topological conjugacy.Comment: 27 pages To appear in Geometriae Dedicat

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“…In [Lüc97,Lemma 3.4] it is proved that any finitely generated A-module N splits as N = PN ⊕ TN where PN is finitely generated projective and TN is the kernel of the canonical homomorphism N → N * * into the double dual, mapping Lemma 2.3. Let M be a submodule of a finitely generated projective A-module P .…”

Section: Review Of Dimension Theory Of Finite Von Neumann Algebrasmentioning

confidence: 99%

L²-BETTI NUMBERS OF DISCRETE MEASURED GROUPOIDS

Sauer

2005

Int. J. Algebra Comput.

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Abstract. There are notions of L 2 -Betti numbers for discrete groups (CheegerGromov, Lück), for type II 1 -factors (recent work of Connes-Shlyakhtenko) and for countable standard equivalence relations (Gaboriau). Whereas the first two are algebraically defined using Lück's dimension theory, Gaboriau's definition of the latter is inspired by the work of Cheeger and Gromov. In this work we give a definition of L 2 -Betti numbers of discrete measured groupoids that is based on Lück's dimension theory, thereby encompassing the cases of groups, equivalence relations and holonomy groupoids with an invariant measure for a complete transversal. We show that with our definition, like with Gaboriau's, the L 2 -Betti numbers bn (R) of the orbit equivalence relation R of a free action of G on a probability space. This yields a new proof of the fact the L 2 -Betti numbers of groups with orbit equivalent actions coincide. Introduction and Statement of Results Inn (G) of a group G can be read off from the group homology asMore recently, in an influential paper of Gaboriau [Gab02] the notion of L 2 -Betti numbers for countable standard equivalence relations was introduced. Their construction is motivated by the one of Cheeger and Gromov. This article provides an homological-algebraic definition of L 2 -Betti numbers for countable standard equivalence relations and, more generally, for discrete measured groupoids. In section 3 we will introduce algebraic objects for a discrete measured groupoid G like the groupoid ring CG which are analogous to the group case, and we define theHere G 0 ⊂ G denotes the subset of objects in the groupoid. Denote the source and target maps by s resp. t. We show in 5.3 the following formula for the restriction.2000 Mathematics Subject Classification. Primary: 37A20 Secondary: 46L85.

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Hilbert modules and modules over finite von Neumann algebras and applications to $L^2$ -invariants

Cited by 52 publications

References 22 publications

Torsion Theories for Finite Von Neumann Algebras

Torsion Theories for Finite Von Neumann Algebras

Higher Dimensional Thompson Groups

L²-BETTI NUMBERS OF DISCRETE MEASURED GROUPOIDS

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Hilbert modules and modules over finite von Neumann algebras and applications to $L^2$ -invariants

Cited by 52 publications

References 22 publications

Torsion Theories for Finite Von Neumann Algebras

Torsion Theories for Finite Von Neumann Algebras

Higher Dimensional Thompson Groups

L2-BETTI NUMBERS OF DISCRETE MEASURED GROUPOIDS

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L²-BETTI NUMBERS OF DISCRETE MEASURED GROUPOIDS