Banach algebra

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.

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 has an identity element 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: for 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.