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Caractérisations de différents types d’algèbres de Banach par le calcul fonctionnel. (Different characterizations of Banach algebras by functional calculi). (French) Zbl 0732.46031
Summary: We give different characterizations of Banach algebras by functional calculi. We prove that $$C^*$$-algebras for an equivalent norm are the involutive Banach algebras which admit the continuous functional calculus. We then characterize the Banach algebras with continuous involution by the harmonic one. We also show that the hermitian commutative Banach algebras are exactly the Banach algebras that admit the real analytic functional calculus.
##### MSC:
 46H30 Functional calculus in topological algebras 46H05 General theory of topological algebras