A Banach algebra is an algebra over a field endowed with a norm such that is a Banach space under the norm and
is frequently taken to be the complex numbers in order to ensure that the operator spectrum fully characterizes an operator (i.e., the spectral theorems for normal or compact normal operators do not, in general, hold in the operator spectrum over the real numbers).
If is commutative and has a unit, then is invertible iff for all , where is the Gelfand transform.