In the representation theory of locally compact groups the one-to-one passage between unitary group representations and representations of the corresponding group -algebra obtained from is a key tool that makes the rich toolbox of -algebraic techniques available in the group context. For infinite dimensional groups there is no Haar measure and therefore no -algebra that can be used to obtain a universal -algebra. However, under certain semiboundedness requirements on spectra, one can use analytic continuations to obtain -algebras whose representation theory cover the so-called semibounded unitary representations of Lie groups. This technique can even be used to construct -algebras whose representations are precisely the unitary representations of certain Lie supergroups. The CAR algebra of the canonical anticommutation relations is the most basic example.
