Unitary representations and von Neumann's continuous geometries

TL;DR

This paper proves that the unit group of a certain class of continuous rings, as defined by von Neumann, cannot have non-trivial continuous unitary representations in the strong operator topology.

Contribution

It establishes a non-existence result for non-trivial unitary representations of the unit group of non-discrete irreducible continuous rings.

Findings

The unit group of such rings admits no non-trivial continuous unitary representations.

The result applies specifically to rings in the sense of von Neumann.

It advances understanding of the representation theory of continuous rings.

Abstract

We prove that the unit group of a non-discrete irreducible, continuous ring, in the sense of John von Neumann, does not admit any non-trivial unitary representation continuous with respect to the strong operator topology.