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...