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. (λx)*=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.

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

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

## See also

* Algebra over a field
* Associative algebra
* *-algebra
* C*-algebra.

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