Strongly smooth paths of idempotents

It is shown that a curve q ( t ) , t ∈ I ( 0 ∈ I ) of idempotent operators on a Banach space X , which verifies that for each ξ ∈ X , the map t ↦ q ( t ) ξ ∈ X is continuously differentiable, can be lifted by means of a regular curve G t , of invertible operators in X : q ( t ) = G t q ( 0 ) G t − 1 , t ∈ I . This is done by using the transport equation of the Grassmannian manifold, introduced by Corach, Porta and Recht. We apply this result to the case when the idempotents are conditional expectations of a C ⁎ algebra A onto a field of C ⁎ -subalgebras B t ⊂ A . In this case the invertible operators, restricted to B 0 , induce C ⁎ -isomorphisms between B 0 and B t . We examine the regularity condition imposed on the curve of expectations, in the case when these expectations are induced by discrete decompositions of a Hilbert space (also called systems of projectors in the literature).