Smooth Paths of Conditional Expectations

Abstract

Let A be a von Neumann algebra with a finite trace τ , represented in H = L 2 (A, τ ), and let Bt ⊂ A be sub-algebras, for t in an interval I (0 ∈ I). Let Et : A → Bt be the unique τ -preserving conditional expectation. We say that the path t → Et is smooth if for every a ∈ A and ξ ∈ H, the mapIf this operator verifies the additional boundedness condition,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,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. 1 * 2010 MSC. Primary 46L10; Secondary 58B10, 47D06. 1 Therefore a curve of possibly unbounded symmetric operators dE t is defined in H, with common domain A ⊂ H. We shall make the following assumption on dE: