A fixed point theorem for isometries (Geometry Seminar)

A new fixed-point theorem will be explained asserting that every isometry of a metric space has a fixed-point in the metric compactification of (the injective hull of) the space. In case the metric space admits a conical bicombing there is no need for passing to the injective hull, examples of such spaces include all Banach spaces, CAT(0)-spaces, injective metric spaces, spaces of positive operators, as well as convex subsets and products thereof. The central notion is that of a metric functional which is an extension of Busemann’s and Gromov’s horofunctions. The result is in particular new for infinite-dimensional Banach spaces and non-proper CAT(0)-spaces. As a consequence, well-known fixed-point free examples get their fixed-point as it were. A new mean ergodic theorem generalizing von Neumann’s is another direct consequence. A more elaborate corollary is that every invertible bounded linear operator of a Hilbert space admits a non-trivial invariant metric functional on the symmetric space of positive operators.