Lyapunov Convexity Theorem for von Neumann Algebras and Jordan Operator Structures
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Lyapunov Convexity Theorem for von Neumann Algebras and Jordan Operator Structures Jan Hamhalter Abstract. We establish Lyapunov type theorems on automatic convexity of various affine transformations of the set of extreme points of important convex sets (closed unit ball, positive part of the closed unit ball, state space) appearing in the theory of von Neumann algebras and more general operator structures. Among others, we have shown that every bounded finitely additive measure μ : P (M ) → X, where P (M ) is a projection lattice of a von Neumann algebra M with no σ-finite direct summand, and X is a normed space with weak∗ separable dual, has a convex range. Similar result is obtained for non σ-finite JW factor. Further results along this line are proved for weak* continuous countably dimensional affine maps on closed unit balls of nonatomic JBW∗ triples and on positive parts of nonatomic von Neumann algebras and JBW∗ algebras. Mathematics Subject Classification. 46L51, 46L10, 46L30, 17C65. Keywords. Noncommutative Lyapunov theorems, affine maps on convex sets, von Neumann algebras, JBW∗ algebras, JBW∗ triples.
1. Introduction The aim of this paper is to study Lyapunov theorem for affine maps (resp. measures) on distinguished convex sets (resp. order structures) that arise naturally in the theory of von Neumann algebras and Jordan structures. Starting point of our research was an interesting paper by Azarnia nad Wright [3] and beautiful monograph by Akemann and Anderson [2]. We would like to develop further certain aspects of these stimulating works. In 1940, A.A.Lyapunov published his famous result saying that nonatomic finite dimensional vector valued measure has range that is convex and compact [19]. This crucial measure theoretic result was put in functional analytic perspective by Lindenstrauss [18]. Azarnia and Wright pioneered generalization of Lyapunov Theorem to von Neumann algebras in a far reaching paper 0123456789().: V,-vol
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[3]. They characterized completely the weak∗ —weak continuous operator T from the self-adjoint part of a von Neumann algebra M into a general Banach space Y , for which the following version of Lyapunov Theorem holds: T transforms any principal ideal in the projection lattice of M into a compact convex subset of Y . Based on this result they recovered classical Lyapunov theorem and previous advances of Kingman and Robertson [16] and Knowles [17]. The key ingredient of the above mentioned characterization is a high nonsingularity of T (more precisely nonsingularity on each hereditary von Neumann subalgebra). The connections between nonsingularity and the Lyapunov and Darboux properties of functionals and operators have influenced highly subsequent research. A thorough treatment of Lyapunov Theorem for von Neumann algebras and C∗ -algebras was presented by Akemann and Anderson in [2] where a lot of noncommutative versions of Lyapunov Theorem were shown. The following reformulation of classical Lyapunov Theorem will be central for our investigation (let us r
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