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On Nilpotency of the Separating Ideal of a Derivation

Ramesh V. Garimella
Proceedings of the American Mathematical Society
Vol. 117, No. 1 (Jan., 1993), pp. 167-174
DOI: 10.2307/2159712
Stable URL: http://www.jstor.org/stable/2159712
Page Count: 8
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On Nilpotency of the Separating Ideal of a Derivation
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Abstract

We prove that the separating ideal S(D) of any derivation D on a commutative unital algebra B is nilpotent if and only if $S(D) \cap (\bigcap R^n)$ is a nil ideal, where R is the Jacobson radical of B. Also we show that any derivation D on a commutative unital semiprime Banach algebra B is continuous if and only if $\bigcap(S(D))^n = \{0\}$. Further we show that the set of all nilpotent elements of S(D) is equal to $\bigcap(S(D) \cap P)$, where the intersection runs over all nonclosed prime ideals of B not containing S(D). As a consequence, we show that if a commutative unital Banach algebra has only countably many nonclosed prime ideals then the separating ideal of a derivation is nilpotent.

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