Lyapunov Theorems for Operator Algebras

In 1940, A A Lyapunov published his celebrated discovery that the range of a nonatomic vector-valued measure is convex and compact. This book presents the result of a systematic generalization of Lyapunov's theorem to the setting of operator algebras. The author's point of view follows that of Lindenstrauss, so that, in their terminology, Lyapunov's theorem asserts that if *v is a weak* continuous map of a nonatomic abelian von Neumann algebra *N into Cn, and B denotes the positive part of the unit ball of *N, then for each a *e B there is an extreme point p of B (i.e., a projection) with *v (p) = *V (a). We begin by studying an affine map *v of a convex subset Q of a linear space X into a linear space y. If *v (E) = *v (Q), where E donotes the extreme points of Q, we say a Lyapunov theorem of type 1 holds. If y is a normed space, we say a Lyapunov theorem of type 2 (resp. type 3) holds if *v (E) is norm dense (resp. *e-dense) in *v (Q). Roughly speaking, a Lyapunov theorem of type 4 asserts that *v maps at least one element of E "away from the boundary". Results of all four types are obtained. In some cases (notably, when x is a nonatomic von Neumann algebra), Q may be a face of B, or the entire unit ball of x. If *v is a singular map and is a purely infinite, countably decomposable von Neumann algebra, then the range of *v may be infinite-dimensional.