  A general `quantum history theory' can be characterised by the space of
histories and by the space of decoherence functionals. In this note we consider
the situation where the space of histories is given by the lattice of
projection operators on an infinite dimensional Hilbert space $H$. We study
operator representations for decoherence functionals on this space of
histories. We first give necessary and sufficient conditions for a decoherence
functional being representable by a trace class operator on $H \otimes H$, an
infinite dimensional analogue of the Isham-Linden-Schreckenberg representation
for finite dimensions. Since this excludes many decoherence functionals of
physical interest, we then identify the large and physically important class of
decoherence functionals which can be represented, canonically, by bounded
operators on $H \otimes H$.
