Editing Spectrum of a C*-algebra (section)

=== Commutative C*-algebras ===
[[File:3-dim commut algebra, subalgebras, ideals.svg|thumb|right|224px|3-dimensional  commutative C*-algebra and its ideals. Each of 8 ideals corresponds to a closed subset of discrete 3-points space (or to an open complement). Primitive ideals correspond to closed [[singleton (mathematics)|singletons]]. See details at the image description page.]]

The spectrum of a commutative C*-algebra ''A'' coincides with the [[Gelfand transformation|Gelfand dual]] of ''A'' (not to be confused with the [[Banach space|dual]] ''A''' of the Banach space ''A'').  In particular, suppose ''X'' is a [[compact space|compact]] [[Hausdorff space]].  Then there is a [[natural transformation|natural]] [[homeomorphism]]

:<math> \operatorname{I}: X \cong \operatorname{Prim}( \operatorname{C}(X)).</math>

This mapping is defined by

: <math>  \operatorname{I}(x) = \{f \in \operatorname{C}(X): f(x) = 0 \}.</math>

I(''x'') is a closed maximal ideal in C(''X'') so is in fact primitive. For details of the proof, see the Dixmier reference.  For a commutative C*-algebra,

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