Spectrum of A C*-algebra - Primitive Spectrum

The topology of  can be defined in several equivalent ways. We first define it in terms of the primitive spectrum .

The primitive spectrum of A is the set of primitive ideals Prim(A) of A, where a primitive ideal is the kernel of an irreducible *-representation. The set of primitive ideals is a topological space with the hull-kernel topology (or Jacobson topology). This is defined as follows: If X is a set of primitive ideals, its hull-kernel closure is

Hull-kernel closure is easily shown to be an idempotent operation, that is

and it can be shown to satisfy the Kuratowski closure axioms. As a consequence, it can be shown that there is a unique topology τ on Prim(A) such that the closure of a set X with respect to τ is identical to the hull-kernel closure of X.

Since unitarily equivalent representations have the same kernel, the map π ↦ ker(π) factors through a surjective map

We use the map k to define the topology on  as follows:

Definition. The open sets of  are inverse images k−1(U) of open subsets U of Prim(A). This is indeed a topology.

The hull-kernel topology is an analogue for non-commutative rings of the Zariski topology for commutative rings.

The topology on  induced from the hull-kernel topology has other characterizations in terms of states of A.