# Banach *-algebra

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

1. (x + y)* = x* + y* for all x, y in A.
2. $\displaystyle{ (\lambda x)^* = \bar{\lambda}x^* }$ for every λ in C and every x in A; here, $\displaystyle{ \bar{\lambda} }$ denotes the complex conjugate of λ.
3. (xy)* = y* x* for all x, y in A.
4. (x*)* = x for all x in A.

In other words, a Banach *-algebra is a Banach algebra over $\displaystyle{ \mathbb{C} }$ 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.