The study of unitary representations of groups naturally leads one to C*-algebras. A C*-algebra is a Banach algebra A together with an involution * which satisfies ||x*x||=||x||², for all x in A. By a fundamental theorem of Gelfand and Naimark, any C*-algebra can be realized as an algebra of bounded operators on a Hilbert space.

On the other hand, Gelfand also proved, for any commutative C*-algebra A, there is a locally compact Hausdorff space D such that A is isomorphic, as a C*-algebra, to C₀(D), the space of complex-valued continuous functions on D that "vanish at infinity".

These two basic theorems come together in the beautiful modern treatment of the spectral theorem for bounded normal operators. Let T be a bounded normal (that is T*T=TT*) operator on some Hilbert space H. Let s(T) denote the spectrum of T, which is a nonempty compact subset of the complex number plane. Let C*(I, T) denote the norm closed algebra of operators on H generated by T and I, the identity operator on H. Now the algebra C(s(T)) is generated, as a C*-algebra, by 1 and i, where 1 is the constantly 1 function and i(z)=z, for all z in s(T). The spectral theorem rests on the fact that there is an isomorphism of C(s(T)) with C*(I, T) that takes 1 to I and i to T.

My own interest lies mainly with C*-algebras that are associated with locally compact groups. If G is a locally compact group, there is a C*-algebra, denoted C*(G), that is defined from G. The representation theory of G is completely mirrored in the representation theory of C*(G).