Smooth paths of conditional expectations

Abstract

Let A be a von Neumann algebra with a finite trace , represented in H = L2(A, ), and let Bt ⊂ A be sub- algebras, for t in an interval I (0 ∈ I). Let Et : A → Bt be the unique -preserving conditional expectation. We say that the path t 7→ Et is smooth if for every a ∈ A and ∈ H, the map I ∋ t 7→ Et(a) ∈ H is continuously differentiable. This condition implies the existence of the derivative operator dEt(a) : H → H, dEt(a) = d dt Et(a). If this operator satifies the additional boundedness condition, ZJ kdEt(a)k2 2dt ≤ CJ kak2 2, for any closed bounded sub-interval J ⊂ I, and CJ > 0 a constant depending only on J, then the algebras Bt are ∗-isomorphic. More precisely, there exists a curve Gt : A → A, t ∈ I of unital, ∗-preserving linear isomorphisms which intertwine the expectations, Gt ◦ E0 = Et ◦ Gt. The curve Gt is weakly continuously differentiable. Moreover, the intertwining property in particular implies that Gt maps B0 onto Bt. We show that this restriction is a multiplicative isomorphism.