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Mappings which Preserve Idempotents, Local Automorphisms, and Local Derivations

Published online by Cambridge University Press:  20 November 2018

Matej Brešar  and
Peter Šemrl
Matej Brešar
Affiliation:
Department of Mathematics, University of Maribor, PF, Koroška 160, 62000 Maribor, Slovenia
Peter Šemrl
Affiliation:
Department of Mathematics, University of Ljubljana, Jadranska 19, 61000 Ljubljana, Slovenia
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Abstract

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It is proved that linear mappings of matrix algebras which preserve idempotents are Jordan homomorphisms. Applying this theorem we get some results concerning local derivations and local automorphisms. As an another application, the complete description of all weakly continuous linear surjective mappings on standard operator algebras which preserve projections is obtained. We also study local ring derivations on commutative semisimple Banach algebras.

Keywords

15A0446J9947B4747D30
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 3 , 01 June 1993 , pp. 483 - 496
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Brešar, M., Characterizations of derivations on some normed algebras with involution,]. Algebra, to appear.Google Scholar
2. Chernoff, P. R., Representations, automorphisms and derivations of some operator algebras,]. Funct. Anal. 12(1973), 275–289.Google Scholar
3. Herstein, I. N., Topics in ring theory, University of Chicago Press, Chicago, 1969.Google Scholar
4. Jacobson, N. and Rickart, C., Jordan homomorphisms of rings, Trans. Amer. Math. Soc. 69( 1950), 479–502.Google Scholar
5. Jafarian, A. A. and Sourour, A. R., Spectrum-preserving linear maps, J. Funct. Anal. 66(1986), 255–261.Google Scholar
6. Johnson, B. E., Continuity of derivation s on commutative Banach algebras, Amer. J. Math. 91(1969), 1–10.Google Scholar
7. Johnson, B. E. and Sinclair, A. M., Continuity of derivations and a problem ofKaplansky, Amer. J. Math. 90(1968), 1067–1073.Google Scholar
8. Kadison, R. V., Local derivations, J. Algebra 130(1990), 494–509.Google Scholar
9. Larson, D. R. and Sourour, A. R., Local derivations and local automorphisms ofB(X), Proc. Sympos. Pure Math. 51, Providence, Rhode Island, 1990, Part 2, 187-194.Google Scholar
10. Omladic, M., On operators preserving commutativity, J. Funct. Anal. 66(1986), 105–122.Google Scholar
11. Rowen, L., Ring theory, Vol. I, Academic Press, San Diego, 1988.Google Scholar
12. Samuel, P. and Zariski, O., Commutative algebra, Van Nostrand, New York, 1958.Google Scholar