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 ; 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.
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