In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra
over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, that is, a normed space that is complete in the metric induced by the norm.
This ensures that the multiplication operation is continuous with respect to the metric topology.
A Banach algebra is called unital if it has an identity element for the multiplication whose norm is
Often one assumes a priori that the algebra under consideration is unital because one can develop much of the theory by considering
and then applying the outcome in the original algebra.
For example, one cannot define all the trigonometric functions in a Banach algebra without identity.
For example, the spectrum of an element of a nontrivial complex Banach algebra can never be empty, whereas in a real Banach algebra it could be empty for some elements.
The prototypical example of a Banach algebra is
(In particular, the exponential map can be used to define abstract index groups.)
The formula for the geometric series remains valid in general unital Banach algebras.
The binomial theorem also holds for two commuting elements of a Banach algebra.
The set of invertible elements in any unital Banach algebra is an open set, and the inversion operation on this set is continuous (and hence is a homeomorphism), so that it forms a topological group under multiplication.
For example: Unital Banach algebras over the complex field provide a general setting to develop spectral theory.
is non-empty and satisfies the spectral radius formula:
of bounded linear operators on a complex Banach space
(for example, the algebra of square matrices), the notion of the spectrum in
coincides with the usual one in operator theory.
This generalizes an analogous fact for normal operators.
be a complex unital Banach algebra in which every non-zero element
is then a commutative ring with unit, every non-invertible element of
is a Banach algebra that is a field, and it follows from the Gelfand–Mazur theorem that there is a bijection between the set of all maximal ideals of
since the kernel of a character is a maximal ideal, which is closed.
of complex continuous functions on the compact space
As an algebra, a unital commutative Banach algebra is semisimple (that is, its Jacobson radical is zero) if and only if its Gelfand representation has trivial kernel.
An important example of such an algebra is a commutative C*-algebra.
is a commutative unital C*-algebra, the Gelfand representation is then an isometric *-isomorphism between
is a Banach algebra over the field of complex numbers, together with a map
In most natural examples, one also has that the involution is isometric, that is,
Some authors include this isometric property in the definition of a Banach *-algebra.