L 2-homology for von Neumann algebras
By Alain Connes, D. Shlyakhtenko. 2003
99 citations recorded by the source corpus.
The aim of this paper is to introduce a notion of L-homology in the context of von Neumann algebras. Finding a suitable (co)homology theory for von Neumann algebras has been a dream for several generations (see [KR71a, KR71b, JKR72, SS95] and references therein). One’s hope is to have a powerful invariant to distinguish von Neumann algebras. Unfortunately, little positive is known about the Kadison-Ringrose cohomology H∗ b (M,M), except that it vanishes in many cases. Furthermore, there does not seem to be a good connection between the bounded cohomology theory of a group and of the bounded cohomology of its von Neumann algebra. Our interest in developing an L-cohomology theory was revived by the introduction of Lcohomology invariants in the field of ergodic equivalence relations in the paper of Gaboriau [Gab02]. His results in particular imply that L-Betti numbers β (2) i (Γ) of a discrete group are the same for measure-equivalent groups (i.e., for groups that can generate isomorphic ergodic measure-preserving equivalence relations). Parallels between the “worlds” of von Neumann algebras and measurable equivalence relations have been noted for a long time (starting with the parallel between the work of Murray and von Neumann [MvN] and that of H. Dye [Dy]). Thus there is hope that an invariant of a group that “survives” measure equivalence will survive also “von Neumann algebra equivalence”, i.e., will be an invariant of the von Neumann algebra of the group. The original motivation for our construction comes from the well understood analogy between the theory of II1-factors and that of discrete groups, based on the theory of correspondences [Con, Con94]. This analogy has been remarkably efficient to transpose analytic properties such as “amenability” or “property T” from the group context to the factor context [Con80] [CJ] and more recently in the breakthrough work of Popa [Popa] [Con03]. We use the theory of correspondences together with the algebraic description of L-Betti numbers given by Luck. His definition involves the computation of the algebraic group homology with coefficients in the group von Neumann algebra. Following the guiding idea that the category of bimodules over a von Neumann algebra is the analogue of the category of modules over a group, we are led to the following algebraic definition of L-homology of a von Neumann algebra M :
Subfields: Algebra and Number Theory, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/2428aa60664f14dab61032597afc526d25bc153e
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Citations in the corpus
- A free stochastic partial differential equation (2010, background): “state free entropy dimensions (respectively written δ0(X1,...,Xn) and δ∗(X1,...,Xn)) have been introduced to generalize results known for finite entropy. Surprisingly, Connes and Shlyakhtenko found in [10] a relation between those entropy dimensions and the first L2-Betti numbers they defined for finite”
- L2-rigidity in von Neumann algebras (2009, unknown): “Derivations into the coarse correspondence appear naturally in the context of Voiculescu’s nonmicrostates approach to free entropy [V], and also play a central role in studying the first L2-Betti number of a von Neumann algebra as introduced by Connes and Shlyakhtenko [ CSh ] (see also [T]).”
- On Ozawa's Property for Free Group Factors (2006, methodology): “cation of free entropy to free group factors, namely the non-existence of Cartan subalgebras in LFn, could not be shown (so far) by the new methods of Ozawa and Peterson ([O], [Pe]), nor by Connes-Shlyakhtenko’s cohomological invariants ([ CS ]).”
- Noncommutative Riesz transforms -- a probabilistic approach (2008, methodology): “y positive maps on von Neumann algebras has been impressively demonstrated by the recent work of Popa [Pop06], Peterson [Pel06], Popa and Ozawa [OP] and also occurs in the work of Shlyahktenko/Connes [CS05] on Betti numbers for von Neumann algebras. A common thread in this analysis is to adapt some ”
- Free entropy dimension in amalgamated free products (2006, background): “Bounds and computations with δ0 have also been made for subfactors of finite index, property T factors, group generators of a discrete group, and free products of certain von Neumann algebras with amalgamation over a diffuse subalgebra ([20], [22], [ 7 ] [21]).”
- Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, I (2003, result): “By now, several results in von Neumann algebras ([CoHa], [Po8], [ CSh ]) and ergodic theory ([Zi], [CoZi], [GeGo], [GoNa], [Fu1,2]) provide supporting evidence towards a positive answer to the conjecture (see V.F.R.”