Continuity of Lie mappings of the skew elements of Banach algebras with involution
HTML articles powered by AMS MathViewer
- by M. I. Berenguer and A. R. Villena PDF
- Proc. Amer. Math. Soc. 126 (1998), 2717-2720 Request permission
Abstract:
Let $A$ and $B$ be centrally closed prime complex Banach algebras with linear involution. If $A$ is semisimple, then any Lie derivation of the skew elements of $A$ is continuous and any Lie isomorphism from the skew elements of $B$ onto the skew elements of $A$ is continuous.References
- K. I. Beĭdar, W. S. Martindale III, and A. V. Mikhalëv, Lie isomorphisms in prime rings with involution, J. Algebra 169 (1994), no. 1, 304–327. MR 1296596, DOI 10.1006/jabr.1994.1286
- Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
- Pierre de la Harpe, Classical Banach-Lie algebras and Banach-Lie groups of operators in Hilbert space, Lecture Notes in Mathematics, Vol. 285, Springer-Verlag, Berlin-New York, 1972. MR 0476820, DOI 10.1007/BFb0071306
- I. N. Herstein, Topics in ring theory, University of Chicago Press, Chicago, Ill.-London, 1969. MR 0271135
- B. E. Johnson, The uniqueness of the (complete) norm topology, Bull. Amer. Math. Soc. 73 (1967), 537–539. MR 211260, DOI 10.1090/S0002-9904-1967-11735-X
- Wallace S. Martindale III, Lie isomorphisms of prime rings, Trans. Amer. Math. Soc. 142 (1969), 437–455. MR 251077, DOI 10.1090/S0002-9947-1969-0251077-5
- Martin Mathieu, Elementary operators on prime $C^*$-algebras. I, Math. Ann. 284 (1989), no. 2, 223–244. MR 1000108, DOI 10.1007/BF01442873
- T. J. Ransford, A short proof of Johnson’s uniqueness-of-norm theorem, Bull. London Math. Soc. 21 (1989), no. 5, 487–488. MR 1005828, DOI 10.1112/blms/21.5.487
- Gordon A. Swain, Lie derivations of the skew elements of prime rings with involution, J. Algebra 184 (1996), no. 2, 679–704. MR 1409235, DOI 10.1006/jabr.1996.0281
- A. R. Villena, Essentially defined derivations on semisimple Banach algebras, Proc. Edinburgh Math. Soc. (2) 40 (1997), no. 1, 175–179. MR 1437821, DOI 10.1017/S0013091500023531
Additional Information
- M. I. Berenguer
- Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
- A. R. Villena
- Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
- Email: avillena@goliat.ugr.es
- Received by editor(s): February 7, 1997
- Communicated by: Palle E. T. Jorgensen
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 2717-2720
- MSC (1991): Primary 46H40, 17B40
- DOI: https://doi.org/10.1090/S0002-9939-98-04569-9
- MathSciNet review: 1469400