Editing C*-algebra (section)

=== Commutative C*-algebras ===
Let ''X'' be a [[locally compact]] Hausdorff space.  The space <math>C_0(X)</math> of complex-valued continuous functions on ''X'' that ''vanish at infinity'' (defined in the article on [[locally compact|local compactness]]) forms a commutative C*-algebra <math>C_0(X)</math> under pointwise multiplication and addition. The involution is pointwise conjugation. <math>C_0(X)</math> has a multiplicative unit element if and only if <math>X</math> is  compact.  As does any C*-algebra, <math>C_0(X)</math> has an [[approximate identity]]. In the case of <math>C_0(X)</math> this is immediate: consider the directed set of compact subsets of <math>X</math>, and for each compact <math>K</math> let <math>f_K</math> be a function of compact support which is identically 1 on <math>K</math>.  Such functions exist by the [[Tietze extension theorem]], which applies to locally compact Hausdorff spaces. Any such sequence of functions <math>\{f_K\}</math> is an approximate identity.

The [[Gelfand representation]] states that every commutative C*-algebra is *-isomorphic to the algebra <math>C_0(X)</math>, where <math>X</math> is the space of [[Character (mathematics)|characters]] equipped with the [[Weak topology|weak* topology]]. Furthermore, if <math>C_0(X)</math> is [[isomorphism|isomorphic]] to <math>C_0(Y)</math> as C*-algebras, it follows that <math>X</math> and <math>Y</math> are [[homeomorphism|homeomorphic]]. This characterization is one of the motivations for the [[noncommutative topology]] and [[noncommutative geometry]] programs.