Homogeneous decoherence functionals in standard and history quantum mechanics

General history quantum theories are quantum theories without a globally de fined notion of time. Decoherence functionals represent the states in the h istory approach and are defined as certain bivariate complex-valued functio nals on the space of all histories. However, in practical situations - for instance in the history formulation of standard quantum mechanics - there o ften is a global time direction and the homogeneous decoherence functionals are specified by their values on the subspace of homogeneous histories. In this work we study the analytic properties of (i) the standard decoheren ce functional in the history version of standard quantum mechanics and (ii) homogeneous decoherence functionals in general history theories. We restri ct ourselves to the situation where the space of histories is given by the lattice of projections on some Hilbert space H. Among other things we prove the non-existence of a finitely valued extension for the standard decohere nce functional to the space of all histories, derive a representation for t he standard decoherence functional as an unbounded quadratic form with a na tural representation on a Hilbert space and prove the existence of an Isham -Linden-Schreckenberg (ILS) type representation for the standard decoherenc e functional.