An algebra can not be normed so that it becomes a Banach algebra.

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A = C^{∞}([0, 1]), the algebra of all complex valued infinitely many times continuously differentiable functions on [0, 1] is semisimple, for Rad(A) = ∩_{t∈[0,1]}{f ∈ C^{∞}([0, 1]); f(t) = 0} = 0. f ↦ f′ is a derivation on A. The Johnson theorem says that 0 is the only derivation on a semisimple Banach algebra ( cf. [B&D, Theorem 18.21]). It follows that A = C^{∞}([0, 1]) is not a Banach algebra under any norm. For a proof based on the Singer-Wermer theorem see [Sak2, Corollary 2.2.4]). In addition a direct proof can be found in [Aup, Corollary 4.1.12]. This example is due to Šilov([sil]).

Ref.

[Aup] B. Aupetit,A primer on spectral theory, Springer-Verlag, 1991. [B&D] F.F. Bonsall, J.Duncan, complet normed algebras, Springer-Verlag, 1973.

[Sil] G.E. Silov, On a property of rings of functions, DoKl. AKad. Nauk. SSSR, 58(1974),985-8.