On infinite direct products of continuous unitary one-parameter groups

We discuss the infinite product of unitary operators in an incomplete direct product of Hilbert spaces. Necessary and sufficient conditions are derived under which this infinite product leads to a continuous unitary one-parameter group provided each factor is assumed to have this property. A certain minimal condition guarantees the existence of a renormalized unitary group. An application is made to product representations of the canonical commutation relations in order to determine the admissible test functions.