Editing Gelfand representation (section)

== Applications ==

One of the most significant applications is the existence of a continuous ''functional calculus'' for normal elements in C*-algebra ''A'': An element ''x'' is normal if and only if ''x'' commutes with its adjoint ''x*'', or equivalently if and only if it generates a commutative C*-algebra C*(''x''). By the Gelfand isomorphism applied to C*(''x'') this is *-isomorphic to an algebra of continuous functions on a locally compact space.  This observation leads almost immediately to:

'''Theorem'''.  Let ''A'' be a C*-algebra with identity and ''x'' a normal element of ''A''.  Then there is a *-morphism ''f'' → ''f''(''x'') from the algebra of continuous functions on the spectrum σ(''x'') into ''A'' such that
* It maps 1 to the multiplicative identity of ''A'';
* It maps the identity function on the spectrum to ''x''.

This allows us to apply continuous functions to bounded normal operators on Hilbert space.