Scalar characterization in Banach Jordan algebras
Abstract
Using a Diagonalization Theorem obtained when the spectrum is Lipschitzian, we extend a result of G. Braatvedt on scalar characterization in Banach algebras to Banach-Jordan algebras. We also establish that any element of a semisimple Banach-Jordan algebra with the property that all elements in some neighbourhood of the identity are spectrally invariant under multiplication by the quadratic U operator, has analogs with the identity.
Keywords
References
- [1] Aupetit Bernard, A Primer on spectral theory, Universitext, Springer, New York, 1991.
- [2] Aupetit Bernard, “Spectral characterisation of the radical in Banach and Jordan-Banach algebras”, Math. Proc. Camb. Philos. Soc. 114 (1993), 31-35.
- [3] Aupetit Bernard, “Spectrum-preserving linear mappings between Banach algebras and Jordan-Banach algebras”, J. London Math. Soc. 262 (2000), 917-924.
- [4] Aupetit Bernard and Abdelaziz Maouche, “Trace and determinant in Jordan-Banach algebras”, Publ. Mat. 46 (2002), 3-16.
- [5] Braatvedt , G., Brits, R., and Raubeuheimer, H., “Spectral characterization of scalars in a Banach algebra”, Bull. London Math. Soc. 41 (2009), 1095-1104.
- [6] Miguel Cabrera Garcia and Angel Rodriguez Palacios, Non-Associative Normed Algebras, Encyclopedia of Mathematics and its Applications 154, Cambridge, 1994.
- [7] O. Loos, “On the set of invertible elements in Banach-Jordan algebras”, Results in Math. 29 (1996), no. 1-2, 111-114.
- [8] Kevin McCrimmon, A Taste of Jordan algebras, Universitext, Springer, New York, 2004.
- [9] Abdelaziz Maouche, “Diagonalization in Jordan-Banach algebras”, Extracta Mathematicae. Vol. 19, Num. 2 (2004), 257-260.
- [10] K.A. Zhevlakov, A.M. Slin’ko, I.P. Shestakov and A. I. Shirshov, Rings that are nearly associative, Academic Press, New York, 1982.
