A general fixed point theorem for isometries in terms of metric functionals is proved under the assumption of the existence of a conical bicombing. It is new for isometries of convex sets of Banach spaces as well as for non-locally compact CAT(0)-spaces and injective spaces. Examples of actions on non-proper CAT(0)-spaces come from the study of diffeomorphism groups, birational transformations, and compact Kähler manifolds. A special case of the fixed point theorem provides a novel mean ergodic theorem that in the Hilbert space case implies von Neumann’s theorem. The theorem accommodates classically fixed-point-free isometric maps such as those of Kakutani, Edelstein, Alspach and Prus. Moreover, from the main theorem together with some geometric arguments of independent interest, one can deduce that every bounded invertible operator of a Hilbert space admits a nontrivial invariant metric functional on the space of positive operators. This is a result in the direction of the invariant subspace problem although its full meaning is dependent on a future determination of such metric functionals.

在圆锥双梳存在的假设下，证明了度量泛函等距的一般不动点定理。对于 Banach 空间凸集的等距以及非局部紧 CAT(0) 空间和单射空间来说，它是新的。非真 CAT(0) 空间上的作用示例来自微分同胚群、双有理变换和紧凯勒流形的研究。不动点定理的一个特例提供了一个新颖的平均遍历定理，该定理在希尔伯特空间的情况下意味着冯·诺依曼定理。该定理适用于经典的无定点等距映射，例如 Kakutani、Edelstein、Alspach 和 Prus 的映射。此外，根据主定理和一些独立感兴趣的几何论证，我们可以推断出希尔伯特空间的每个有界可逆算子都承认正算子空间上的非平凡不变度量泛函。这是不变子空间问题方向的结果，尽管其完整含义取决于此类度量泛函的未来确定。