The Local Form of Doubly Stochastic Maps and Joint Majorization in II 1 Factors

Abstract

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.

Convex hulls of unitary orbits of normal elements in C⁎-algebras with tracial rank zero

“…Using these results, in this paper, we characterize the normal elements in the closure of convex hull of unitary orbits of normal elements in a general simple C * -algebra with tracial rank zero (see 4.6 below). This keeps the same sprit of results in II 1 -factors as in [9] and [4] even though the simple C * -algebra A may have rich tracial simplex. On the other hand, say, if we assume that A also has a unique tracial state, then a purely measure theoretical description of normal elements in the closure of convex full of normal elements can be presented (see 5.11 below).…”

“…Suppose that x is a normal element in the closure of convex hull of the unitary orbit of a normal element y and y is in the closure of convex hull of the unitary orbit of x. Then, in a II 1 -factor M, x and y are approximately unitarily equivalent (see Theorem 5.1 of [4]). In a general unital simple C * -algebra A with tracial rank zero, this no longer holds simply because the presence of non-trivial K 1 as well as infinitesimal elements in K 0 (A).…”

Interpolation by completely positive maps

“…Condition (b) in Theorem 2.1 has been studied in [5,17,1] with row (column, doubly, respectively) D. An analog of (3) for II 1 factor is also proven in [1].…”

Multivariable Schur–Horn theorems

“…When both A and S are tuples of commuting hermitians, the relation A ≺ S is equivalent to any of the following conditions (see [2]):…”

“…In case both A and S are tuples of commuting hermitian operators in a type II 1 factor, the relation A ≺ S was considered in [2] (based on Hiai's work on majorization between normal operators [12]), where several characterizations of this notion were shown; indeed, joint majorization between tuples of commuting hermitian operators can be characterized in terms Choquet's notion of comparison of measures and also in terms of tracial inequalities using convex functions (that we discuss in the next section). Apart from this, the fact that the set {A : A ≺ S} has a pleasing topological structure (it is weak* closed as well as convex) makes this a natural notion.…”