Operator Algebras

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).

For Operator Algebra resources, visit the site maintained by N. C. Phillips. You may also find the Directory of Operator Algebraists' home pages useful. One can also find recent papers related to operator algebras on the Functional Analysis preprint server.

Here is a picture of most of the participants in the Canadian Operator Theory and Operator Algebras Symposium, the 1998 version held in Edmonton in May.

Here is a picture of the participants in the Martina Franca workshop on NonCommutative Geometry in September, 2000.

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Last update: September 18, 2000