arXiv:math/0604120v4 [math.OA] 12 Jan 2007 A SCHUR-HORN THEOREM IN II 1 FACTORS M. ARGERAMI AND P. MASSEY Abstract. Given a II 1 factor M and a diffuse abelian von Neumann subal- gebra A⊂M, we prove a version of the Schur-Horn theorem, namely E A (U M (b)) σ-sot = {a ∈A sa : a ≺ b}, b ∈M sa , where ≺ denotes spectral majorization, E A the unique trace-preserving condi- tional expectation onto A, and U M (b) the unitary orbit of b in M. This result is inspired by a recent problem posed by Arveson and Kadison. 1. Introduction In 1923, I. Schur [18] proved that if A ∈ M n (C) sa (i.e., A is selfadjoint) then k  j=1 α ↓ j ≤ k  j=1 β ↓ j ,k =1,...,n, with equality when k = n (denoted α ≺ β), where α = diag(A) ∈ R n , β = λ(A) ∈ R n the spectrum (counting multiplicity) of A, and α ↓ ,β ↓ ∈ R n are obtained from α, β by reordering their entries in decreasing order. In 1954, A. Horn [12] proved the converse: given α, β ∈ R n with α ≺ β, there exists a selfadjoint matrix A ∈ M n (C) such that diag(A)= α, λ(A)= β. Since every selfadjoint matrix is diagonalizable, the results of Schur and Horn can be combined in the following assertion: if D denotes the diagonal masa in M n (C) and E D is the compression onto D , then (1) E D ({UM β U ∗ : U ∈ M n (C) unitary})= {M α ∈D : α ≺ β}, where M α is the diagonal matrix with the entries of α in the main diagonal. This combination of the two results, commonly known as Schur-Horn theorem, has played a significant role in many contexts of matrix analysis: although simple, vector majorization expresses a natural and deep relation among vectors, and as such it has been a useful tool both in pure and applied mathematics. We refer to the books [5, 14] and the introductions of [6, 16] for more on this. During the last 25 years, several extensions of majorization have been proposed by, among others, Ando [1] (to selfadjoint matrices), Kamei [13] (to selfadjoint operators in a II 1 factor), Hiai [10, 11] (to normal operators in a von Neumann algebra), and Neumann [16] (to vectors in ℓ ∞ (N)). With these generalizations at hand, it is natural to ask about extensions of the Schur-Horn theorem. 2000 Mathematics Subject Classification. Primary 46L99, Secondary 46L55. Key words and phrases. Majorization, diagonals of operators, Schur-Horn theorem. M. Argerami supported in part by the Natural Sciences and Engineering Research Council of Canada. P. Massey supported in part by CONICET of Argentina and a PIMS Postdoctoral Fellowship. 1