Pinchings and positive linear maps

We employ the pinching theorem, ensuring that some operators A admit any sequence of contractions as an operator diagonal of A, to deduce/improve two recent theorems of Kennedy–Skoufranis and Loreaux–Weiss for conditional expectations onto a masa in the algebra of operators on a Hilbert space. Similarly, we obtain a proof of a theorem of Akeman and Anderson showing that positive contractions in a continuous masa can be lifted to a projection. We also discuss a few corollaries for sums of two operators in the same unitary orbit.