  In the first part of this paper the general perspective of history quantum
theories is reviewed. History quantum theories provide a conceptual and
mathematical framework for formulating quantum theories without a globally
defined Hamiltonian time evolution and for introducing the concept of space
time event into quantum theory. On a mathematical level a history quantum
theory is characterized by the space of histories, which represent the space
time events, and by the space of decoherence functionals which represent the
quantum mechanical states in the history approach. The second part of this
paper is devoted to the study of the structure of the space of decoherence
functionals for some physically reasonable spaces of histories in some detail.
The temporal reformulation of standard Hamiltonian quantum theories suggests to
consider the case that the space of histories is given by (i) the lattice of
projection operators on some Hilbert space or -- slightly more general -- (ii)
the set of projection operators in some von Neumann algebra. In the case (i)
the conditions are identified under which decoherence functionals can be
represented by, respectively, trace class operators, bounded operators or
families of trace class operators on the tensor product of the underlying
Hilbert space by itself. Moreover we shall discuss the naturally arising
representations of decoherence functionals as sesquilinear forms. The paper
ends with a discussion of the consequences of the results for the general
axiomatic framework of history theories.
