Editing Gelfand representation (section)

=== The spectrum of a commutative C*-algebra ===
{{See also|Spectrum of a C*-algebra}}

The '''spectrum''' or '''Gelfand space''' of a commutative C*-algebra ''A'', denoted ''Â'', consists of the set of ''non-zero'' *-homomorphisms from ''A'' to the complex numbers. Elements of the spectrum are called '''characters''' on ''A''. (It can be shown that every algebra homomorphism from ''A'' to the complex numbers is automatically a [[*-algebra|*-homomorphism]], so that this definition of the term 'character' agrees with the one above.)

In particular, the spectrum of a commutative C*-algebra is a locally compact Hausdorff space: In the unital case, i.e. where the C*-algebra has a multiplicative unit element 1, all characters ''f'' must be unital, i.e. ''f''(1) is the complex number one. This excludes the zero homomorphism. So ''Â'' is closed under weak-* convergence and the spectrum is actually ''compact''. In the non-unital case, the weak-* closure of ''Â'' is ''Â'' ∪ {0}, where 0 is the zero homomorphism, and the removal of a single point from a compact Hausdorff space yields a locally compact Hausdorff space.

Note that ''spectrum'' is an overloaded word.  It also refers to the spectrum σ(''x'') of an element ''x'' of an algebra with unit 1, that is the set of complex numbers ''r'' for which ''x''&nbsp;−&nbsp;''r'' 1 is not invertible in ''A''. For unital C*-algebras, the two notions are connected in the following way: σ(''x'') is the set of complex numbers ''f''(''x'') where ''f'' ranges over Gelfand space of ''A''. Together with the [[spectral radius|spectral radius formula]], this shows that ''Â'' is a subset of the unit ball of ''A*'' and as such can be given the relative weak-* topology. This is the topology of pointwise convergence. A [[net (mathematics)|net]] {''f''<sub>''k''</sub>}<sub>''k''</sub> of elements of the spectrum of ''A'' converges to ''f'' [[if and only if]] for each ''x'' in ''A'', the net of complex numbers {''f''<sub>''k''</sub>(''x'')}<sub>''k''</sub> converges to ''f''(''x'').

If ''A'' is a [[separable space|separable]] C*-algebra, the weak-* topology is [[metrizable]] on bounded subsets.  Thus the spectrum of a separable commutative C*-algebra ''A'' can be regarded as a metric space. So the topology can be characterized via convergence of sequences.

Equivalently, σ(''x'') is the [[range of a function|range]] of γ(''x''), where γ is the Gelfand representation.