Spectrum of A C*-algebra - Other Characterizations of The Spectrum

Other Characterizations of The Spectrum

The hull-kernel topology is easy to describe abstractly, but in practice for C*-algebras associated to locally compact topological groups, other characterizations of the topology on the spectrum in terms of positive definite functions are desirable.

In fact, the topology on  is intimately connected with the concept of weak containment of representations as is shown by the following:

Theorem. Let S be subset of Â. Then the following are equivalent for an irreducile representation π

1. The equivalence class of π in  is in the closure of S
2. Every state associated to π, that is one of the form

with ||ξ||=1, is the weak limit of states associated to representations in S.

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. By one of the basic theorems associated to the GNS construction, a state f is pure if and only if the associated representation π_f is irreducible. Moreover, the mapping κ: PureState(A) → Â defined by f ↦ π_f is a surjective map.

From the previous theorem one can easily prove the following;

Theorem The mapping

given by the GNS construction is continuous and open.