Scalar characterization in Banach Jordan algebras

Year 2018, , 121 - 124, 26.06.2018

Abdelaziz Maouche

https://doi.org/10.32323/ujma.399537

Cited By: 1

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

Jordan algebra, Lipschitzian, scalar element, power bounded

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.