Editing Gelfand representation (section)

=== Statement of the commutative Gelfand–Naimark theorem ===
Let ''A'' be a commutative C*-algebra and let ''X'' be the spectrum of ''A''. Let 
:<math>\gamma:A \to C_0(X)</math>

be the Gelfand representation defined above.

'''Theorem'''. The Gelfand map γ is an isometric *-isomorphism from ''A'' onto ''C''<sub>0</sub>(''X'').

See the Arveson reference below.

The spectrum of a commutative C*-algebra can also be viewed as the set of all [[maximal ideal]]s ''m'' of ''A'', with the [[hull-kernel topology]]. (See the earlier remarks for the general, commutative Banach algebra case.) For any such ''m'' the quotient algebra ''A/m'' is  one-dimensional (by the Gelfand-Mazur theorem), and therefore any ''a'' in ''A'' gives rise to a complex-valued function on ''Y''.

In the case of C*-algebras with unit, the spectrum map gives rise to a contravariant [[functor]] from the category of commutative C*-algebras with unit and unit-preserving continuous *-homomorphisms, to the category of compact Hausdorff spaces and continuous maps. This functor is one half of a [[Equivalence of categories|contravariant equivalence]] between these two categories (its [[adjoint functor|adjoint]] being the functor that assigns to each compact Hausdorff space ''X'' the C*-algebra ''C''<sub>0</sub>(''X'')). In particular, given compact Hausdorff spaces ''X'' and ''Y'', then ''C''(''X'') is isomorphic to ''C''(''Y'') (as a C*-algebra) if and only if ''X'' is [[homeomorphic]] to ''Y''.

The 'full' [[Gelfand–Naimark theorem]] is a result for arbitrary (abstract) [[noncommutative]] C*-algebras ''A'', which though not quite analogous to the Gelfand representation, does provide a concrete representation of ''A'' as an algebra of operators.