Editing Banach algebra (section)

==Banach *-algebras==

A Banach *-algebra <math>A</math> is a Banach algebra over the field of [[complex number]]s, together with a map <math>{}^* : A \to A</math> that has the following properties:
# <math>\left(x^*\right)^* = x</math> for all <math>x \in A</math> (so the map is an [[Involution (mathematics)|involution]]).
# <math>(x + y)^* = x^* + y^*</math> for all <math>x, y \in A.</math>
# <math>(\lambda x)^* = \bar{\lambda}x^*</math> for every <math>\lambda \in \Complex</math> and every <math>x \in A;</math> here, <math>\bar{\lambda}</math> denotes the [[complex conjugate]] of <math>\lambda.</math>
# <math>(x y)^* = y^* x^*</math> for all <math>x, y \in A.</math>

In other words, a Banach *-algebra is a Banach algebra over <math>\Complex</math> that is also a [[*-algebra]].

In most natural examples, one also has that the involution is [[isometry|isometric]], that is, 
<math display=block>\|x^*\| = \|x\| \quad \text{ for all } x \in A.</math>
Some authors include this isometric property in the definition of a Banach *-algebra.

A Banach *-algebra satisfying <math>\|x^* x\| = \|x^*\| \|x\|</math> is a [[C*-algebra]].