Gelfand transform

Summary

The Gelfand transform turns each element $a$ of a commutative complex Banach algebra into a continuous function $\hat a$ on the character space, by $\hat a(\varphi) = \varphi(a)$. It is an algebra homomorphism into the functions vanishing at infinity, and it does not increase norms. For a unital algebra, the range of $\hat a$ is the spectrum of $a$. For commutative C*-algebras it is an isometric isomorphism onto all such functions, by the commutative Gelfand–Naimark theorem.