In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra ${\displaystyle A}$ 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. The norm is required to satisfy $${\displaystyle \|x\,y\|\ \leq \|x\|\,\|y\|\quad {\text{ for all }}x,y\in A.}$$

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 ${\displaystyle 1,}$ and commutative if its multiplication is commutative. Any Banach algebra ${\displaystyle A}$ (whether it is unital or not) can be embedded isometrically into a unital Banach algebra ${\displaystyle A_{e}}$ so as to form a closed ideal of ${\displaystyle A_{e}}$. Often one assumes a priori that the algebra under consideration is unital because one can develop much of the theory by considering ${\displaystyle A_{e}}$ and then applying the outcome in the original algebra. However, this is not the case all the time. For example, one cannot define all the trigonometric functions in a Banach algebra without identity.

The theory of real Banach algebras can be very different from the theory of complex Banach algebras. 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.

Banach algebras can also be defined over fields of ${\displaystyle p}$-adic numbers. This is part of ${\displaystyle p}$-adic analysis.

The prototypical example of a Banach algebra is ${\displaystyle C_{0}(X)}$, the space of (complex-valued) continuous functions, defined on a locally compact Hausdorff space ${\displaystyle X}$, that vanish at infinity. ${\displaystyle C_{0}(X)}$ is unital if and only if ${\displaystyle X}$ is compact. The complex conjugation being an involution, ${\displaystyle C_{0}(X)}$ is in fact a C*-algebra. More generally, every C*-algebra is a Banach algebra by definition.

Several elementary functions that are defined via power series may be defined in any unital Banach algebra; examples include the exponential function and the trigonometric functions, and more generally any entire function. (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.