Remarks on commuting exponentials in Banach algebras
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- by Christoph Schmoeger PDF
- Proc. Amer. Math. Soc. 127 (1999), 1337-1338 Request permission
Abstract:
Suppose that $a$ and $b$ are elements of a complex unital Banach algebra such that the spectra of $a$ and $b$ are $2\pi i$-congruence-free. E.M.E. Wermuth has shown that then \[ e^a e^b = e^b e^a \quad \text {implies that} \quad ab = ba. \] In this note we use two elementary facts concerning inner derivations on Banach algebras to give a very short proof of Wermuth’s result.References
- Theodore W. Palmer, Banach algebras and the general theory of $^*$-algebras. Vol. I, Encyclopedia of Mathematics and its Applications, vol. 49, Cambridge University Press, Cambridge, 1994. Algebras and Banach algebras. MR 1270014, DOI 10.1017/CBO9781107325777
- Edgar M. E. Wermuth, A remark on commuting operator exponentials, Proc. Amer. Math. Soc. 125 (1997), no. 6, 1685–1688. MR 1353407, DOI 10.1090/S0002-9939-97-03643-5
Additional Information
- Christoph Schmoeger
- Affiliation: Mathematisches Institut I, Universität Karlsruhe, D-76128 Karlsruhe, Germany
- Email: christoph.schmoeger@math.uni-karlsruhe.de
- Received by editor(s): August 5, 1997
- Published electronically: January 28, 1999
- Communicated by: Theodore W. Gamelin
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 1337-1338
- MSC (1991): Primary 46H99
- DOI: https://doi.org/10.1090/S0002-9939-99-04701-2
- MathSciNet review: 1476391