Generalising Group Algebras

We generalise group algebras to other algebraic objects withbounded Hilbert space representation theory; the generalisedgroup algebras are called ‘host’ algebras. The mainproperty of a host algebra is that its representation theoryshould be isomorphic (in the sense of the Gelfand–Raikovtheorem) to a specified subset of representations of the algebraicobject. Here we obtain both existence and uniqueness theoremsfor host algebras as well as general structure theorems forhost algebras. Abstractly, this solves the question of whena set of Hilbert space representations is isomorphic to therepresentation theory of a C*-algebra. To make contact withharmonic analysis, we consider general convolution algebrasassociated to representation sets, and consider conditions fora convolution algebra to be a host algebra.