A host algebra generalises the concept of a group algebra as follows. Let F
be a unital C*-algebra, and let S_0 be a proper subset of its states within
which one wants to keep the analysis (e.g. F is the group algebra of a discrete
group G, and S_0 is the set of states continuous w.r.t. some nondiscrete
topology of G). Then a host algebra is a C*-algebra L for which we have
embeddings of F and L into a larger C*-algebra E, such that the states on L
extend uniquely to F, and this extension defines a norm continuous affine
bijection between S_0 and the whole state space of L. The main examples (but
not the only ones) are group and covariance algebras. Here we study existence
questions of a host algebra for a given pair (F,S_0), we show that if a host
algebra exists, we can do integral decompositions of states in S_0 in terms of
other states in S_0, and we show that if one does induction of representations
via host algebras one stays within the class of representations with the right
continuity properties w.r.t. S_0. Moreover, if S_0 is a folium, then up to a
central algebra one can always construct a host algebra, but this central
algebra can be an obstruction to the existence of a host algebra. These results
should be interesting to anyone who wants to construct group algebras for
general topological groups, and they are also useful for quantum physics due to
some selection criteria for physically acceptable states.