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Nombres de Betti L 2 et facteurs de type II 1

Alain Connes (2002/2003)

Séminaire Bourbaki

Damien Gaboriau a montré récemment que les nombres de Betti L 2 des feuilletages mesurés à feuilles contractiles sont des invariants de la relation d’équivalence associée. Sorin Popa a utilisé ce résultat joint à des propriétés de rigidité des facteurs de type II 1 pour en déduire l’existence de facteurs de type II 1 dont le groupe fondamental est trivial.

Non-orbit equivalent actions of 𝔽 n

Adrian Ioana (2009)

Annales scientifiques de l'École Normale Supérieure

For any 2 ≤ n ≤ ∞ , we construct a concrete 1-parameter family of non-orbit equivalent actions of the free group 𝔽 n . These actions arise as diagonal products between a generalized Bernoulli action and the action 𝔽 n ↷ ( 𝕋 2 , λ 2 ) , where 𝔽 n is seen as a subgroup of SL 2 ( ℤ ) .

Notes on automorphisms of ultrapowers of II₁ factors

David Sherman (2009)

Studia Mathematica

In functional analysis, approximative properties of an object become precise in its ultrapower. We discuss this idea and its consequences for automorphisms of II₁ factors. Here are some sample results: (1) an automorphism is approximately inner if and only if its ultrapower is ℵ₀-locally inner; (2) the ultrapower of an outer automorphism is always outer; (3) for unital *-homomorphisms from a separable nuclear C*-algebra into an ultrapower of a II₁ factor, equality of the induced traces implies unitary...

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