Host Algebras

Abstract

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.