A metric fixed point theorem and some of its applications

Abstract

A general fixed point theorem for isometries in terms of metric functionals is proved under the assumption of the existence of a conical bicombing. One special case provides a new mean ergodic theorem that in the Hilbert space case implies von Neumann's theorem. For CAT(0)-spaces and injective spaces the fixed point theorem is new for non-locally compact spaces, and implies the usual result for proper CAT(0)-spaces. For Banach spaces the therorem accomodates classically fixed-point-free isometric maps such as those of Kakutani, Edelstein, Alspach and Prus. Furthermore, 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.

A non-compact convex hull in general non-positive curvature

“…In other words, σ * is a Γequivariant concial bicombing, as desired. We remark that additional fixedpoint results for spaces with a conical bicombing can be found in [28,30].…”