Spectrum of a C*-algebra

In mathematics, the spectrum of a C*-algebra or dual of a C*-algebra A, denoted Â, is the set of unitary equivalence classes of irreducible *-representations of A.

One of the most important applications of this concept is to provide a notion of dual object for any locally compact group.

This dual object is suitable for formulating a Fourier transform and a Plancherel theorem for unimodular separable locally compact groups of type I and a decomposition theorem for arbitrary representations of separable locally compact groups of type I.

Then there is a natural homeomorphism This mapping is defined by I(x) is a closed maximal ideal in C(X) so is in fact primitive.

It is known A is isomorphic to a finite direct sum of full matrix algebras: where min(A) are the minimal central projections of A.

The spectrum of A is canonically isomorphic to min(A) with the discrete topology.

In fact, the topology on  is intimately connected with the concept of weak containment of representations as is shown by the following: The second condition means exactly that π is weakly contained in S. The GNS construction is a recipe for associating states of a C*-algebra A to representations of A.

This conjecture was proved by James Glimm for separable C*-algebras in the 1961 paper listed in the references below.

A C*-algebra A is of type I if and only if any separable factor representation of A is a finite or countable multiple of an irreducible one.

Since a C*-algebra A is a ring, we can also consider the set of primitive ideals of A, where A is regarded algebraically.

For a ring an ideal is primitive if and only if it is the annihilator of a simple module.