Two classical theorems in matrix theory, due to Schur and Horn, relate the
eigenvalues of a self-adjoint matrix to the diagonal entries. These have
recently been given a formulation in the setting of operator algebras as the
Schur-Horn problem, where matrix algebras and diagonals are replaced
respectively by finite factors and maximal abelian self-adjoint subalgebras
(masas). There is a special case of the problem, called the carpenter problem,
which can be stated as follows: for a masa A in a finite factor M with
conditional expectation E_A, can each x in A with 0 <= x <= 1 be expressed as
E_A(p) for a projection p in M?
In this paper, we investigate these problems for various masas. We give
positive solutions for the generator and radial masas in free group factors,
and we also solve affirmatively a weaker form of the Schur-Horm problem for the
Cartan masa in the hyperfinite factor.Comment: 15 page