Abstract. Given a II 1 factor M and a diffuse abelian von Neumann subalgebra A ⊂ M, we prove a version of the Schur-Horn theorem, namelywhere ≺ denotes spectral majorization, E A the unique trace-preserving conditional 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.

We describe majorization between selfadjoint operators in a σ-finite II ∞ factor (M, τ ) in terms of simple spectral relations. For a diffuse abelian von Neumann subalgebra A ⊂ M with trace-preserving conditional expectation E A , we characterize the closure in the measure topology of the image through E A of the unitary orbit of a selfadjoint operator in M in terms of majorization (i.e., a Schur-Horn theorem). We also obtain similar results for the contractive orbit of positive operators in M and for the unitary and contractive orbits of τ -integrable operators in M.

Abstract. Let M be a II 1 factor with trace τ , A ⊆ M a masa and E A 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 E A (p) = a. This shows a new family of instances of a conjecture by Kadison, the so-called "carpenter's theorem".

Let N ⊆ M be von Neumann algebras and E : M → N a faithful normal conditional expectation. In this work it is shown that the similarity orbit S(E) of E by the natural action of the invertible group of G M of M has a natural complex analytic structure and the map given by this action: G M → S(E) is a smooth principal bundle. It is also shown that if N is finite then S(E) admits a Reductive Structure. These results were known previously under the conditions of finite index and N ′ ∩ M ⊆ N , which are removed in this work. Conversely, if the orbit S(E) has an Homogeneous Reductive Structure for every expectation defined on M , then M is finite. For every algebra M and every expectation E, a covering space of the unitary orbit U (E) is constructed in terms of the connected component of 1 in the normalizer of E. Moreover, this covering space is the universal covering in any of the following cases: 1) M is a finite factor and Ind(E) < ∞; 2) M is properly infinite and E is any expectation; 3) E is the conditional expectation onto the centralizer of a state. Therefore, in those cases, the fundamental group of U (E) can be characterized as the Weyl group of E.

Abstract. We find a description of the restriction of doubly stochastic maps to separable abelian C * -subalgebras of a II 1 factor M. We use this local form of doubly stochastic maps to develop a notion of joint majorization between n-tuples of mutually commuting self-adjoint operators that extends those of Kamei (for single self-adjoint operators) and Hiai (for single normal operators) in the II 1 factor case. Several characterizations of this joint majorization are obtained. As a byproduct we prove that any separable abelian C * -subalgebra of M can be embedded into a separable abelian C * -subalgebra of M with diffuse spectral measure.