# Banach *-algebra

From HandWiki

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:

- (
*x*+*y*)* =*x** +*y** for all*x*,*y*in*A*. - [math]\displaystyle{ (\lambda x)^* = \bar{\lambda}x^* }[/math] for every λ in
**C**and every*x*in*A*; here, [math]\displaystyle{ \bar{\lambda} }[/math] denotes the complex conjugate of λ. - (
*xy*)* =*y***x** for all*x*,*y*in*A*. - (
*x**)* =*x*for all*x*in*A*.

In other words, a Banach *-algebra is a Banach algebra over [math]\displaystyle{ \mathbb{C} }[/math] which is also a *-algebra.

In most natural examples, one also has that the involution is isometric, i.e.

- ||
*x**|| = ||*x*||,

Some authors include this isometric property in the definition of a Banach *-algebra.

## See also