Banach * algebra

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. ( λ x ) ∗ = λ ¯ x ∗ for every λ in C and every x in A; here, λ ¯ 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 C which is also a *-algebra.

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

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