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 \(\mu : P(M)\rightarrow X\), where P(M) is a projection lattice of a von Neumann algebra M with no \(\sigma \)-finite direct summand, and X is a normed space with weak\(^*\) separable dual, has a convex range. Similar result is obtained for non \(\sigma \)-finite JW factor. Further results along this line are proved for weak* continuous countably dimensional affine maps on closed unit balls of nonatomic \(\hbox {JBW}^*\) triples and on positive parts of nonatomic von Neumann algebras and \(\hbox {JBW}^*\) algebras.

我们根据出现在冯·诺依曼代数理论和更一般算子上的重要凸集的极限点集（封闭单位球，封闭单位球的正部分，状态空间）的各种仿射变换的自动凸度建立Lyapunov型定理。结构。除其他外，我们证明了每个有界有限加法\（\ mu：P（M）\ rightarrow X \），其中P（M）是不带\（\ sigma \）的冯·诺伊曼代数M的投影格-有限直接求和，并且X是具有弱\（^ * \）可分离对偶的范数空间，具有凸范围。对于非\（\ sigma \）可获得类似的结果-有限的JW因子。证明了在非原子\（\ hbox {JBW} ^ * \）三元组的封闭单位球上以及非原子冯·诺依曼代数和\（\ hbox {JBW}的正部分上的弱*连续可维数仿射图的进一步证明^ * \）代数。