Strongly smooth paths of idempotents

It is shown that a curve q(t)q(t), t∈It∈I (0∈I0∈I) of idempotent operators on a Banach space XX, which verifies that for each ξ∈Xξ∈X, the map t↦q(t)ξ∈Xt↦q(t)ξ∈X is continuously differentiable, can be lifted by means of a regular curve GtGt, of invertible operators in XX:q(t)=Gtq(0)Gt−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⁎C⁎ algebra AA onto a field of C⁎C⁎-subalgebras Bt⊂ABt⊂A. In this case the invertible operators, restricted to B0B0, induce C⁎C⁎-isomorphisms between B0B0 and BtBt. 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).