A correction to “Adjugates in Banach algebras”
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Abstract:
Let $A$ be a semisimple unital Banach algebra. We show that $\operatorname {rank}_A(ab)=\operatorname {rank}_A(ba)$ for all $a,b\in A$ if and only if $\operatorname {soc}(A)$ is contained in the center of $A$, and $ab\in \operatorname {soc}(A)$ implies $ba\in \operatorname {soc}(A)$ for all $a,b\in A$. This corrects an erroneous statement in R.E. Harte and C. Hernández, Adjugates in Banach algebras, Proc. Amer. Math. Soc. 134(5) (2005), 1397–1404.References
- Bernard Aupetit, A primer on spectral theory, Universitext, Springer-Verlag, New York, 1991. MR 1083349, DOI 10.1007/978-1-4612-3048-9
- Bernard Aupetit and H. du T. Mouton, Trace and determinant in Banach algebras, Studia Math. 121 (1996), no. 2, 115–136. MR 1418394
- R. M. Brits, L. Lindeboom, and H. Raubenheimer, Rank and the Drazin inverse in Banach algebras, Studia Math. 177 (2006), no. 3, 211–224. MR 2284455, DOI 10.4064/sm177-3-2
- Robin Harte and Carlos Hernández, Adjugates in Banach algebras, Proc. Amer. Math. Soc. 134 (2006), no. 5, 1397–1404. MR 2199186, DOI 10.1090/S0002-9939-05-08317-6
Additional Information
- R. M. Brits
- Affiliation: Department of Mathematics, University of Johannesburg, P.O. Box 524, Auckland Park, 2006, Johannesburg, South Africa
- Email: rbrits@uj.ac.za
- Received by editor(s): January 15, 2009
- Received by editor(s) in revised form: October 16, 2009
- Published electronically: March 23, 2010
- Communicated by: Nigel J. Kalton
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 3021-3024
- MSC (2010): Primary 46H05, 46H10, 47A10
- DOI: https://doi.org/10.1090/S0002-9939-10-10363-3
- MathSciNet review: 2644913