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Some classes of smooth bimodules over II1 factors and their associated 1-cohomology spaces

https://doi.org/10.1016/j.jfa.2024.110452

Hiatt, Patrick; Peterson, Jesse; Popa, Sorin (August 2024, Journal of Functional Analysis)

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On the singular abelian rank of ultraproduct II1 factors

Hiatt, Patrick; Popa, Sorin (March 2024, Journal of Operator Theory)

We prove that, under the continuum hypothesis c = א1, any ul- traproduct II1 factor M = ∏ Mn of separable finite factors Mn contains more ω than c many mutually disjoint singular MASAs, in other words the singular abelian rank of M, r(M), is larger than c. Moreover, if the strong continuum hypothesis 2c = א2 is assumed, then r(M) = 2c. More generally, these results hold true for any II1 factor M with unitary group of cardinality c that satisfies the bicommutant condition (A0′ ∩ M)′ ∩ M = M, for all A0 ⊂ M separable abelian.
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Non-isomorphism of A∗n,2≤n≤∞, for a non-separable abelian von Neumann algebra A

https://doi.org/10.1007/s00039-024-00669-8

Boutonnet, Rémi; Drimbe, Daniel; Ioana, Adrian; Popa, Sorin (April 2024, Geometric and Functional Analysis)

Abstract We prove that ifAis a non-separable abelian tracial von Neuman algebra then its free powersA^∗n,2≤n≤∞, are mutually non-isomorphic and with trivial fundamental group,$$\mathcal{F}(A^{*n})=1$$, whenever 2≤n<∞. This settles the non-separable version of the free group factor problem.
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