Spectral isometries on non-simple \textsl{C*}-algebras

Mathieu, Martin; Sourour, Ahmed

doi:10.1090/S0002-9939-2013-11704-1

Spectral isometries on non-simple C*-algebras
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by Martin Mathieu and Ahmed R. Sourour PDF

Proc. Amer. Math. Soc. 142 (2014), 129-135 Request permission

Abstract:

We prove that unital surjective spectral isometries on certain non-simple unital C*-algebras are Jordan isomorphisms. Along the way, we establish several general facts in the setting of semisimple Banach algebras.

References

Graham R. Allan, Introduction to Banach spaces and algebras, Oxford Graduate Texts in Mathematics, vol. 20, Oxford University Press, Oxford, 2011. Prepared for publication and with a preface by H. Garth Dales. MR 2761146
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, Spectral characterization of the radical in Banach and Jordan-Banach algebras, Math. Proc. Cambridge Philos. Soc. 114 (1993), no. 1, 31–35. MR 1219911, DOI 10.1017/S0305004100071371
Bernard Aupetit, Spectrum-preserving linear mappings between Banach algebras or Jordan-Banach algebras, J. London Math. Soc. (2) 62 (2000), no. 3, 917–924. MR 1794294, DOI 10.1112/S0024610700001514
M. Bendaoud, A. Bourhim, and M. Sarih, Linear maps preserving the essential spectral radius, Linear Algebra Appl. 428 (2008), no. 4, 1041–1045. MR 2382111, DOI 10.1016/j.laa.2007.09.010
Constantin Costara, A Cartan type theorem for finite-dimensional algebras, Linear Algebra Appl. 426 (2007), no. 2-3, 299–304. MR 2347842, DOI 10.1016/j.laa.2007.04.022
C. Costara, Commuting holomorphic maps on the spectral unit ball, Bull. Lond. Math. Soc. 41 (2009), no. 1, 57–62. MR 2481989, DOI 10.1112/blms/bdn104
Constantin Costara and Dušan Repovš, Spectral isometries onto algebras having a separating family of finite-dimensional irreducible representations, J. Math. Anal. Appl. 365 (2010), no. 2, 605–608. MR 2587062, DOI 10.1016/j.jmaa.2009.11.040
N. Jacobson and C. E. Rickart, Jordan homomorphisms of rings, Trans. Amer. Math. Soc. 69 (1950), 479–502. MR 38335, DOI 10.1090/S0002-9947-1950-0038335-X
Ying-Fen Lin and Martin Mathieu, Jordan isomorphism of purely infinite $C^*$-algebras, Q. J. Math. 58 (2007), no. 2, 249–253. MR 2334865, DOI 10.1093/qmath/hal024
Martin Mathieu, Spectrally bounded operators on simple $C^*$-algebras. II, Irish Math. Soc. Bull. 54 (2004), 33–40. MR 2138429
Martin Mathieu, Towards a non-selfadjoint version of Kadison’s theorem, Ann. Math. Inform. 32 (2005), 87–94. MR 2264870
Martin Mathieu, A collection of problems on spectrally bounded operators, Asian-Eur. J. Math. 2 (2009), no. 3, 487–501. MR 2568008, DOI 10.1142/S1793557109000418
Martin Mathieu and Conal Ruddy, Spectral isometries. II, Function spaces, Contemp. Math., vol. 435, Amer. Math. Soc., Providence, RI, 2007, pp. 301–309. MR 2359437, DOI 10.1090/conm/435/08385
Martin Mathieu and Gerhard J. Schick, First results on spectrally bounded operators, Studia Math. 152 (2002), no. 2, 187–199. MR 1916549, DOI 10.4064/sm152-2-6
Martin Mathieu and Gerhard J. Schick, Spectrally bounded operators from von Neumann algebras, J. Operator Theory 49 (2003), no. 2, 285–293. MR 1991740
Martin Mathieu and Ahmed Ramzi Sourour, Hereditary properties of spectral isometries, Arch. Math. (Basel) 82 (2004), no. 3, 222–229. MR 2053626, DOI 10.1007/s00013-003-0595-5
Gert K. Pedersen, $C^{\ast }$-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR 548006