On Ozawa's Property for Free Group Factors

By Sorin Popa. 2006

93 citations recorded by the source corpus.

We give a new proof of a result of Ozawa showing that if a von Neumann subalgebra $Q$ of a free group factor $L\Bbb F_n, 2\leq n\leq \infty$ has relative commutant diffuse (i.e. without atoms), then $Q$ is amenable.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a4c410e2d14b8ebf8370c642c8482b22c2c87640

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In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

On the superrigidity of malleable actions with spectral gap (2006, methodology): “The starting point of all deformation/spectral gap rigidity arguments we have used in this paper and in [P6], [ P7 ] is the following observation, which can be viewed as a general “spectral gap rigidity principle”:”
One-cohomology and the uniqueness of the group measure space decomposition of a II1 factor (2010, methodology): “n’s [Pe06, Theorem 1.3] on the solidity of the group factors LΓ when Γ is an icc group that admits a proper(4) 1-cocycle into a multiple of the regular representation. Prior to this Popa had given in [Po06b] a new proof for the solidity of the free group factors LFn using the malleable deformation c”
Ergodic Subequivalence Relations Induced by a Bernoulli Action (2008, unknown): “ L2(hM,Kii,Tr) ≺ L2(M)⊗L2(M) and by combining this fact with (1) and (2), we get the desired weak containment of Hilbert bimodules. From this it follows, by the argument in the proof of Lemma 2.2. in [P6], that P′ ∩ M˜ω ⊂ Mω.6 We end this section by recalling Popa’s technique of intertwining subalge”
L2-rigidity in von Neumann algebras (2009, result): “We also note that S. Popa in [ P6 ] has recently given another proof of Ozawa’s theorem for specific case of the free group factors.”
Cartan subalgebras of amalgamated free product II$_1$ factors (2012, unknown): “h that ξm = δt m(xm) satisﬁes kξmk2 >c. Fix y∈ Mand z∈ (Q)1. Then we have that kyξmk2 = k(1 −EMω)(yθt m(xn))k2 6 kyθtm(xm)k2 6 kyk2. Also, since zxm = xmz, by using S. Popa’s spectral gap argument [Po06b] we get that kzξm −ξmzk2 = k(1 −EM)(zθtm(xm)−θtm(xm)z)k2 6 kzθtm(xm)−θtm(xm)zk2 = kθ−tm(z)xm ”
Classification and rigidity for von Neumann algebras (2012, methodology): “ on closable derivations, Peterson was able to show that II1 factors arising from icc groups with positive ﬁrst ℓ2-Betti number are prime [Pe06]. A new proof of solidity of L(Fn) was found by Popa in [Po06c], while II1 factors coming from icc groups Γ admitting a proper cocycle into ℓ2(Γ) were shown”
Bass-Serre rigidity results in von Neumann algebras (2008, methodology): “We prove the following analog of Lemma 2 in [31]:”
ON THE STRUCTURAL THEORY OF II1 FACTORS OF NEGATIVELY CURVED GROUPS, II. ACTIONS BY PRODUCT GROUPS (2011, methodology): “The result and its proof are inspired by Ozawa’s general semi-solidity theorem (Theorem 15.1.5 in [5]) and Ozawa and Popa’s Theorem B in [46], viewed through the framework developed by the first two authors in [9].”
On a class of $\mathrm{II}_1$ factors with at most one Cartan subalgebra (2007, methodology): “ither, i.e. whether it does not have Cartan decomposition. Corollary 2 shows that indeed it doesn’t. The proof of the Theorem follows a “deformation/rigidity” strategy, being inspired by arguments in [Po5] and [Po3]. The proof is in two parts: First we show that if a factor M has the complete metric”
Type III factors with unique Cartan decomposition (2012, background): “unitary conjugacy. The key points of [OP07] were the weak compactness for arbitrary Cartan subalgebras of II1 factors with the complete metric approximation property, and the malleable deformation of [Po06b] for arbitrary crossed products by the free group. 2010 Mathematics Subject Classiﬁcation. 46”