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On certain pairs of automorphisms of rings

Part of: Selfadjoint operator algebras Rings and algebras with additional structure

Published online by Cambridge University Press:  09 April 2009

Matej Brešar
Matej Brešar
Affiliation:
University of MariborPF, Koroška 160 62000 Maribor Slovenia
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Abstract

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In this paper we prove algebraic generalizations of some results of C. J. K. Batty and A. B. Thaheem, concerned with the identity α + α−1 = β + β−1 where α and β are automorphisms of a C*-algebra. The main result asserts that if automorphisms α, β of a semiprime ring R satisfy α + α-1 = β + β−1 then there exist invariant ideals U1, U2 and U3 of R such that Ui ∩ Uj = 0, i ≠ j, U1 ⊕ U2 ⊕ U3 is an essential ideal, α = β on U1, α = β−1 on U2, and α2 = β2 = α−2 on U3. Furthermore, if the annihilator of any ideal in R is a direct summand (in particular, if R is a von Neumann algebra), then U1 ⊕ U2 ⊕ U3 = R.

Keywords

automorphismsemiprime ringprime ringC*-algebravon Neumann algebraideal.

MSC classification

Secondary: 16W20: Automorphisms and endomorphisms 46L40: Automorphisms
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 54 , Issue 1 , February 1993 , pp. 29 - 38
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Batty, C. J. K., ‘On certain pairs of automorphisms of C*-algebras’, J. Austral. Math. Soc. (Series A) 46 (1989), 197–211.CrossRefGoogle Scholar
[2]Brešar, M., ‘On the compositions of (α, β)-derivations of rings, and applications to von Neumann algebras’, preprint.Google Scholar
[3]Thaheem, A. B., ‘On the operator equation α + α-1 = β + β-1’, Internat. J. Math. & Math. Sci. 9 (1986), 767–770.CrossRefGoogle Scholar
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