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Journal of the London Mathematical Society 2005 72(3):742-762; doi:10.1112/S0024610705007003
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© The London Mathematical Society

Generalising Group Algebras

Hendrik Grundling

Department of Mathematics, University of New South Wales Sydney, NSW 2052, Australia hendrik{at}maths.unsw.edu.au

Received 11 November 2004. Revision received 2 June 2005.

We generalise group algebras to other algebraic objects with bounded Hilbert space representation theory; the generalised group algebras are called ‘host’ algebras. The main property of a host algebra is that its representation theory should be isomorphic (in the sense of the Gelfand–Raikov theorem) to a specified subset of representations of the algebraic object. Here we obtain both existence and uniqueness theorems for host algebras as well as general structure theorems for host algebras. Abstractly, this solves the question of when a set of Hilbert space representations is isomorphic to the representation theory of a C*-algebra. To make contact with harmonic analysis, we consider general convolution algebras associated to representation sets, and consider conditions for a convolution algebra to be a host algebra.


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