Editing Gelfand representation (section)

== The model algebra ==
For any [[locally compact space|locally compact]] [[Hausdorff space|Hausdorff]] [[topological space]] ''X'', the space ''C''<sub>0</sub>(''X'') of continuous complex-valued functions on ''X'' which [[vanish at infinity]] is in a natural way a commutative C*-algebra:
* The algebra structure over the [[complex number]]s is obtained by considering  the pointwise operations of addition and multiplication.
* The involution is pointwise complex conjugation.
* The norm is the [[uniform norm]] on functions.

The importance of ''X'' being locally compact and Hausdorff is that this turns ''X'' into a [[Tychonoff space|completely regular space]]. In such a space every closed subset of ''X'' is the common zero set of a family of continuous complex-valued functions on ''X'', allowing one to recover the topology of ''X'' from ''C''<sub>0</sub>(''X'').

Note that  ''C''<sub>0</sub>(''X'')  is [[unital algebra|unital]] if and only if ''X'' is [[compact space|compact]], in which  case ''C''<sub>0</sub>(''X'') is equal to ''C''(''X''), the algebra of all continuous complex-valued functions on ''X''.