A Banach algebra is called "unitary" if it has an identity element for the multiplication and "commutative" if its multiplication is commutative.
Banach algebras can also be defined over fields of p-adic numbers. This is part of p-adic analysis.
Several elementary functions which are defined via power series may be defined in any unitary Banach algebra; examples include the exponential function and the trigonometric functions. The formula for the geometric series and the binomial theorem also remain valid in general unitary Banach algebras.
The set of invertible elements in any unitary Banach algebra is an open set, and the inversion operation on this set is continuous, so that it forms a topological group under multiplication.
Unitary Banach algebras provide a natural setting to study general spectral theory. The spectrum of an element x consists of all those scalars λ such that x -λ1 is not invertible. (In the Banach algebra of all n-by-n matrices mentioned above, the spectrum of a matrix coincides with the set of all its eigenvalues.) The spectrum of any element is compact. If the base field is the field of complex numbers, then the spectrum of any element is non-empty.
The various algebras of functions given in the examples above have very different properties from standard examples of algebras such as the reals. For example:
Examples
Properties