A result concerning derivations in Banach algebras

Vukman, J.

doi:10.1090/S0002-9939-1992-1079710-7

A result concerning derivations in Banach algebras
HTML articles powered by AMS MathViewer

by J. Vukman

Proc. Amer. Math. Soc. 116 (1992), 971-975

DOI: https://doi.org/10.1090/S0002-9939-1992-1079710-7

PDF | Request permission

Abstract:

The main result: Let $A$ be a Banach algebra over the complex field $C$. Suppose there exists a continuous derivation $D:A \to A$, such that $\alpha {D^3} + {D^2}$ is a derivation for some $\alpha \in C$. In this case $D$ maps $A$ into its radical.

References

I. N. Herstein, Rings with involution, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1976. MR 0442017
B. E. Johnson and A. M. Sinclair, Continuity of derivations and a problem of Kaplansky, Amer. J. Math. 90 (1968), 1067–1073. MR 239419, DOI 10.2307/2373290
Edward C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093–1100. MR 95863, DOI 10.1090/S0002-9939-1957-0095863-0
A. M. Sinclair, Continuous derivations on Banach algebras, Proc. Amer. Math. Soc. 20 (1969), 166–170. MR 233207, DOI 10.1090/S0002-9939-1969-0233207-X
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
J. Vukman, Commuting and centralizing mappings in prime rings, Proc. Amer. Math. Soc. 109 (1990), no. 1, 47–52. MR 1007517, DOI 10.1090/S0002-9939-1990-1007517-3
J. Vukman, A result concerning derivations in noncommutative Banach algebras, Glas. Mat. Ser. III 26(46) (1991), no. 1-2, 83–88 (English, with English and Serbo-Croatian summaries). MR 1269177