Banach star-algebra ($(ab{)}^{\ast }=b^{\ast }{a}^{\ast },(a^{\ast }{)}^{\ast }=a$)

A Banach star-algebra is a complex Banach algebra equipped with a continuous algebra involution. Thus the involution is conjugate-linear, reverses products, and squares to the identity. Some conventions additionally require $\Vert a^{\ast }\Vert =\Vert a\Vert$; this is the isometric convention. In the unital case the involution fixes the unit. The C-star identity is a stronger additional condition. The algebra of continuous complex-valued functions on a compact Hausdorff space, with pointwise operations, complex conjugation and the supremum norm, satisfies both the isometric condition and the C-star identity.