Vanishing of the First Continuous L2-Cohomology for II1 Factors

Abstract

We prove that the continuous version of the Connes-Shlyakhtenko first L-cohomology for II1 factors, as proposed by A. Thom in [Th06], always vanishes. In [CS03], A. Connes and D. Shlyakhtenko developed an L2-cohomology theory for finite von Neumann algebras M , and more generally for weakly dense ∗-subalgebras A ⊂ M of such von Neumann algebras. Then in [Th06], A. Thom provided an alternative, Hochschild-type characterization of the first such L2-cohomology of M as the quotient of the space of derivations δ : M → Aff(M ⊗Mop) by the space of inner derivations, where Aff(M ⊗Mop) denotes the ∗-algebra of operators affiliated with M ⊗Mop. Thom also proposed in [Th06] a continuous version of the first L2-cohomology, by considering the (smaller) space of derivations δ that are continuous from M with the operator norm to Aff(M ⊗Mop) with the topology of convergence in measure. He noted that in many cases (e.g., when M has a Cartan subalgebra, or when M is not prime), this cohomology vanishes, i.e. any continuous derivation of M into Aff(M ⊗Mop) is inner. Following up on this work, V. Alekseev and D. Kyed have shown in [AK11] that the first continuous L2-cohomology also vanishes whenM has property (T), whenM is finitely generated with nontrivial fundamental group, or when M has property Gamma. Recently, V. Alekseev proved in [Al13] that this is also the case for the free group factors L(Fn). In this article, we prove that in fact the first continuous L2-cohomology vanishes for all finite von Neumann algebras. The starting point of our proof is a key calculation in the proof of [Al13, Proposition 3.1], which provides a concrete sequence of elements yn in the II1 factor M = L(F3) of the free group F3 with generators a, b, c, that tends to 0 in operator norm, but has the property that if a derivation δ : M → Aff(M ⊗Mop) satisfies δ(ua) = ua ⊗ 1 and δ(ub) = δ(uc) = 0, then δ(yn) does not tend to 0 in measure. More precisely, the yn’s in [Al13] are scalar multiples of words wn in a, b, c with the property that δ(wn) is a larger and larger sum of free independent Haar unitaries. In the case of an arbitrary II1 factor M , we fix a hyperfinite II1 factor R ⊂ M with trivial relative commutant, and then use [Po92] to “simulate” (in distribution) L(F3) inside M , with a any fixed unitary in M and bm, cm Haar unitaries in R such that a, bm, cm are asymptotically free. If now δ is a continuous derivation on M , then by subtracting an inner derivation, we may assume δ vanishes on R, thus on bm, cm. If δ(a) 6= 0, and if we formally define yn’s via the same formula as Alekseev’s, with a, bm, cm in lieu of a, b, c, then a careful estimation of norms of yn and δ(yn), which uses results in [HL99], shows that one still has ‖yn‖ → 0, while δ(yn) 6→ 0 in measure. Mathematics Department, UCLA, CA 90095-1555 (United States), popa@math.ucla.edu Supported in part by NSF Grant DMS-1101718 KU Leuven, Department of Mathematics, Leuven (Belgium), stefaan.vaes@wis.kuleuven.be Supported by Research Programme G.0639.11 of the Research Foundation – Flanders (FWO) and KU Leuven BOF research grant OT/13/079.