Caractérisations de différents types d’algèbres de Banach par le calcul fonctionnel. (Different characterizations of Banach algebras by functional calculi). (French) Zbl 0732.46031

Akkar, M.; El Kinani, A.; Oudadess, M.

Publ. Math., Ec. Norm. Supér. Takaddoum 5, 1-6 (1989).

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

Keywords:

characterizations of Banach algebras by functional calculi; \(C^ *\)- algebras; continuous functional calculus; Banach algebras with continuous involution; hermitian commutative Banach algebras; real analytic functional calculus