On Kadison’s Carpenters theorem

Abstract

Let M be a II1 factor with trace τ, A⊆M a masa and EA the unique conditional expectation onto A. Under some technical assumptions on the inclusion A⊆M, which hold true for any semiregular masa of a separable factor, we show that for elements a in certain dense families of the positive part of the unit ball of A, it is possible to find a projection p ∈ M such that EA(p) = a. This shows a new family of instances of a conjecture by Kadison, the so-called “carpenter’s theorem”.