SMOOTH PATHS OF CONDITIONAL EXPECTATIONS

Abstract

Let [Formula: see text] be a von Neumann algebra with a finite trace τ, represented in [Formula: see text], and let [Formula: see text] be sub-algebras, for t in an interval I (0 ∈ I). Let [Formula: see text] be the unique τ-preserving conditional expectation. We say that the path t ↦ Et is smooth if for every [Formula: see text] and [Formula: see text], the map [Formula: see text] is continuously differentiable. This condition implies the existence of the derivative operator [Formula: see text] If this operator satisfies the additional boundedness condition, [Formula: see text] for any closed bounded subinterval J ⊂ I, and CJ > 0 a constant depending only on J, then the algebras [Formula: see text] are *-isomorphic. More precisely, there exists a curve [Formula: see text], t ∈ I of unital, *-preserving linear isomorphisms which intertwine the expectations, [Formula: see text] The curve Gt is weakly continuously differentiable. Moreover, the intertwining property in particular implies that Gt maps [Formula: see text] onto [Formula: see text]. We show that this restriction is a multiplicative isomorphism.