We establish the Schwarz inequality for a class of Banach *-algebras, and use it to derive some consequences, for example when a linear map between Banach *-algebras is a Jordan homomorphism. This applies to the class of Banach *-algebras l(1) (G, A; alpha) arising from C*-dynamical systems (A, G, alpha) with alpha an action of a discrete group G on a separable C*-algebra A.
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