Title: A fixed-point theorem for isometries

Abstract:

Fixed point theorems, such as those of Brouwer and Banach, are of importance throughout the mathematical sciences. In this talk I will explain a new fixed point theorem for isometries. It states that any isometry of a metric space has a fixed point in a generalized sense, which means that the fixed point may not belong to the space itself only to a natural and canonical compactification of it. This is new even for isometries of Banach spaces, and in this case it for example leads to a new proof of the mean ergodic theorem of von Neumann-Carleman with the feature of remaining valid in any Banach space (contrary to the classical statement of F. Riesz). The notion of metric functionals is central and serves as the metric space analog of linear functionals. Other consequences of the fixed point theorem will be mentioned.