In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra 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, i.e. normed and complete. The algebra multiplication and the Banach space norm are required to be related by the following inequality:

(i.e., the norm of the product is less than or equal to the product of the norms). This ensures that the multiplication operation is continuous. This property is found in the real and complex numbers; for instance, |-6×5| = |-6|×|5|.

A Banach *-algebra A is a Banach algebra over the field of complex numbers, together with a map * : A → A, called involution, that has the following properties:

1. (x + y)* = x* + y* for all x, y in A.
2. for every λ in C and every x in A; here, denotes the complex conjugate of λ.
3. (xy)* = y* x* for all x, y in A.
4. (x*)* = x for all x in A.