On nilpotency of the separating ideal of a derivation
HTML articles powered by AMS MathViewer
- by Ramesh V. Garimella PDF
- Proc. Amer. Math. Soc. 117 (1993), 167-174 Request permission
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.References
- Julian Cusack, Automatic continuity and topologically simple radical Banach algebras, J. London Math. Soc. (2) 16 (1977), no. 3, 493–500. MR 461136, DOI 10.1112/jlms/s2-16.3.493
- H. G. Dales, The uniqueness of the functional calculus, Proc. London Math. Soc. (3) 27 (1973), 638–648. MR 333738, DOI 10.1112/plms/s3-27.4.638
- H. G. Dales, Automatic continuity: a survey, Bull. London Math. Soc. 10 (1978), no. 2, 129–183. MR 500923, DOI 10.1112/blms/10.2.129
- J. Esterle, Elements for a classification of commutative radical Banach algebras, Radical Banach algebras and automatic continuity (Long Beach, Calif., 1981), Lecture Notes in Math., vol. 975, Springer, Berlin-New York, 1983, pp. 4–65. MR 697578
- Ramesh V. Garimella, Continuity of derivations on some semiprime Banach algebra, Proc. Amer. Math. Soc. 99 (1987), no. 2, 289–292. MR 870787, DOI 10.1090/S0002-9939-1987-0870787-6
- Sandy Grabiner, The nilpotency of Banach nil algebras, Proc. Amer. Math. Soc. 21 (1969), 510. MR 236700, DOI 10.1090/S0002-9939-1969-0236700-9
- Thomas W. Hungerford, Algebra, Graduate Texts in Mathematics, vol. 73, Springer-Verlag, New York-Berlin, 1980. Reprint of the 1974 original. MR 600654
- Allan M. Sinclair, Automatic continuity of linear operators, London Mathematical Society Lecture Note Series, No. 21, Cambridge University Press, Cambridge-New York-Melbourne, 1976. MR 0487371
- I. M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann. 129 (1955), 260–264. MR 70061, DOI 10.1007/BF01362370
- Marc P. Thomas, The image of a derivation is contained in the radical, Ann. of Math. (2) 128 (1988), no. 3, 435–460. MR 970607, DOI 10.2307/1971432
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 117 (1993), 167-174
- MSC: Primary 46H40; Secondary 46J05
- DOI: https://doi.org/10.1090/S0002-9939-1993-1107920-X
- MathSciNet review: 1107920