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A note on invertibility preservers on Banach algebras

Author(s): Matej Bresar; Ajda Fosner; Peter Semrl
Journal: Proc. Amer. Math. Soc. 131 (2003), 3833-3837.
MSC (2000): Primary 46H05, 46H10, 47B48
Posted: July 9, 2003
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Abstract: Let ${\mathcal{A}}$ be ${\mathcal{B}}$be semisimple Banach algebras and let $\phi:\mathcal{A}\to\mathcal{B}$be a unital bijective linear operator that preserves invertibility. If the socle of ${\mathcal A}$ is an essential ideal of ${\mathcal A}$, then $\phi$ is a Jordan isomorphism.

References:

1.
B. Aupetit, The uniqueness of the complete norm topology in Banach algebras and Banach Jordan algebras, J. Funct. Anal. 47 (1982), 1-6. MR 83g:46044

2.
B. Aupetit, ``A primer on spectral theory", Springer-Verlag, New York, 1991. MR 92c:46001

3.
B. Aupetit, Spectrum-preserving linear mappings between Banach algebras or Jordan-Banach algebras, J. London Math. Soc. (2) 62 (2000), 917-924. MR 2001h:46078

4.
B. Aupetit and H. du Mouton, Spectrum preserving linear mappings in Banach algebras, Studia Math. 109 (1994), 91-100. MR 95c:46070

5.
M. Bresar and P. Semrl, Finite rank elements in semisimple Banach algebras, Studia Math. 128 (1998), 287-298. MR 99a:46089

6.
M. Bresar and P. Semrl, Spectral characterization of idempotents and invertibility preserving linear maps, Expo. Math. 17 (1999), 185-192. MR 2000d:16050

7.
I. N. Herstein, Jordan homomorphisms, Trans. Amer. Math. Soc. 81 (1956), 331-341. MR 17:938f

8.
A. A. Jafarian and A. R. Suorour, Spectrum preserving linear maps, J. Funct. Anal. 66 (1986), 255-261. MR 87m:47011

9.
I. Kaplansky, ``Algebraic and analytic aspects of operator algebras", Regional Conference Series in Mathematics 1, Amer. Math. Soc., Providence, RI, 1970. MR 47:845

10.
A. R. Sourour, Invertibility preserving linear maps on ${\mathcal L}(X)$, Trans. Amer. Math. Soc. 348 (1996), 13-30. MR 96f:47069

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Additional Information:

Matej Bresar
Affiliation: Department of Mathematics, University of Maribor, PF, Koroska 160, SI-2000 Maribor, Slovenia
Email: bresar@uni-mb.si

Ajda Fosner
Affiliation: Department of Mathematics, University of Maribor, PF, Koroska 160, SI-2000 Maribor, Slovenia
Email: ajda.fosner@uni-mb.si

Peter Semrl
Affiliation: Department of Mathematics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
Email: peter.semrl@fmf.uni-lj.si

DOI: 10.1090/S0002-9939-03-07192-2
PII: S 0002-9939(03)07192-2
Received by editor(s): July 25, 2002
Posted: July 9, 2003
Additional Notes: Partially supported by a grant from the Ministry of Science of Slovenia
Communicated by: David R. Larson
Copyright of article: Copyright 2003, American Mathematical Society

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