Amalgamated free product type III factors with at most one Cartan subalgebra

Abstract

International audienceWe investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras M1 * B M2 over an amenable von Neumann subalgebra B. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free product von Neumann algebras. Namely, we show that any nonamenable free product von Neumann algebra (M1, ϕ1) * (M2, ϕ2) with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established in [Io12a]. Next, we prove that any countable nonsingular ergodic equivalence relation R defined on a standard measure space and which splits as the free product R = R1 * R2 of recurrent subequivalence relations gives rise to a nonamenable factor L(R) with a unique Cartan subalgebra, up to unitary conjugacy. Finally, we prove unique Cartan decomposition for a class of group measure space factors L ∞ (X) ⋊ Γ arising from nonsingular free ergodic actions Γ (X, µ) on standard measure spaces of amalgamated groups Γ = Γ1 * Σ Γ2 over a finite subgroup Σ

Some prime factorization results for free quantum group factors

“…On the other hand, by Lemma 3.4.3, for any i there is some j such that N i⊗ LRp = pC φ (N i )p C h (M ) C h (M j ) = M j⊗ LR for any Tr-finite projection p ∈ LR N . By [1,Proposition 2.10], this implies N i M M j . So by the uniqueness, we get σ(i) = j and σ is determined from…”

Absence of Cartan subalgebras in continuous cores of free product von Neumann algebras

“…Thus one may expect that the continuous core of any 'type III 1 free product factor' has no regular diffuse hyperfinite von Neumann subalgebra. This is unclear at the moment of this writing, but the next weaker assertion follows directly from [8], [1].…”

“…It is well-known that if A is a Cartan subalgebra in a von Neumann algebra N , then so is A ⋊ σ ψ R = A⊗ L(R) in the continuous core N = N ⋊ σ ψ R with a faithful normal state ψ = ψ • E on N , where E denotes the unique normal conditional expectation from N onto A. The work [1] actually shows only the absence of such special Cartan subalgebras in the continuous core M (even when both M 1 , M 2 are hyperfinite). Hence Theorem 1 is seemingly stronger than the original one [1, Theorem A].…”

“…See [8, Lemma 2.2],[1, Lemma 2.3] (due to Vaes) for its equivalent conditions. The main, necessary ingredients from [8], [1] are in order.…”

“…[21, Lemma 4.1 (i)] and[8, Proposition 2.7]). Here, we have known by[19, Theorem 4.1] that M = M c is a non-hyperfinite factor of type III, and hence it is standard, see e.g [1,. Proposition 2.8], that q M q has no hyperfinite direct summand.…”