Derivations on commutative normed algebras. (English) Zbl 0067.35101

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[1] Silov showed in his paper ?On a property of rings of functions?, Doklady Akad. Nauk SSSR. (N.S.) 58, 985-988 (1947), that the algebra of all infinitely differentiable functions on an interval cannot be normed so as to be a Banach algebra. Prof.I. Kaplansky conjectured that the ?reason? for this was that non-zero derivations could not exist on a commutative semisimple Banach algebra. Theorem 1 proves this conjecture for bounded derivations. It seems probable that hypothesis (iv) is superfluous.
[2] SeeChevalley: ?Theory of Lie Groups?, p. 137. Princeton Univ. Press. (1946).
[3] Originally proved bySilov, cf. footnote 1).
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