Editing Spectrum of a C*-algebra (section)

== Primitive spectrum ==
The [[topology]] of ''Â'' 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 ideal]]s Prim(''A'') of ''A'', where a primitive ideal is the kernel of a non-zero 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

:<math> \overline{X} = \left \{\rho \in \operatorname{Prim}(A): \rho \supseteq \bigcap_{\pi \in X} \pi \right \}. </math>

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

:<math> \overline{\overline{X}} = \overline{X},</math>

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

:<math> \operatorname{k}: \hat{A} \to \operatorname{Prim}(A). </math>

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

'''Definition'''. The open sets of ''Â'' are inverse images ''k''<sup>−1</sup>(''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 ''Â'' induced from the hull-kernel topology has other characterizations in terms of [[state (functional analysis)|state]]s of ''A''.