Continuity properties of the representations of the canonical commutation relations

We prove that a given representation of the canonical commutation relations can be extended uniquely by continuity to larger test function spaces which are maximal in the sense that no further extension is possible. For irreducible tensor product representations of the canonical commutation relations we give a necessary and a sufficient condition for the admissible test functions. We consider the problem of finding topologies on the test function spaces such that this extension can be obtained by a topological completion. Various examples are discussed.Supported in part by the National Research Council of Canada.An earlier version of the present work was distributed as a preprint entitled Topologies for Test Function Spaces for Representations of the Canonical Commutation Relations.