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Automatic Continuity of Homomorphisms in Non-associative Banach Algebras

Published online by Cambridge University Press:  20 November 2018

C-H. Chu  and
M. V. Velasco
C-H. Chu
Affiliation:
School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS, UK, e-mail: c.chu@qmul.ac.uk
M. V. Velasco
Affiliation:
Dpto. de Analisis Matematico, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain, e-mail: vvelasco@ugr.esc.chu@qmul.ac.uk
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Abstract

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We introduce the concept of a rare element in a non-associative normed algebra and show that the existence of such an element is the only obstruction to continuity of a surjective homomorphism from a non-associative Banach algebra to a unital normed algebra with simple completion. Unital associative algebras do not admit any rare elements, and hence automatic continuity holds.

Keywords

46H4046H70automatic continuitynon-associative algebraspectrumrare operatorrare element
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 65 , Issue 5 , 01 October 2013 , pp. 989 - 1004
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Chu, C.-H., Jordan structures in geometry and analysis. Cambridge Tracts in Mathematics, 190, Cambridge University Press, Cambridge, 2012.Google Scholar
[2] Dales, H. G., Banach algebras and automatic continuity. LondonMathematical SocietyMonographs, 24, Clarendon Press, Oxford University Press, New York, 2000.Google Scholar
[3] Dieudonné, J., Sur les homomorphismes d’espaces normés. Bull. Sci. Math. 67(1943), 72–84.Google Scholar
[4] Downey, L. and Enflo, Per, Operators with eigenvalues and extreme cases of stability. Proc. Amer. Math. Soc. 132(2004), 719–724. http://dx.doi.org/10.1090/S0002-9939-03-07059-X Google Scholar
[5] Johnson, B. E., The uniqueness of (complete) norm topology. Bull. Amer. Math. Soc. 73(1967), 537–539. http://dx.doi.org/10.1090/S0002-9904-1967-11735-X Google Scholar
[6] Kato, T., Perturbation theory for linear operators. Die Grundlehren der mathematischen Wissenschaften, 132, Springer-Verlag, New York, 1966.Google Scholar
[7] Laursen, K. B. and Neumann, M. M., An introduction to local spectral theory. London Mathematical Society Monographs. New Series, 20, Clarendon Press, Oxford University Press, New York, 2000.Google Scholar
[8] Marcos, J. C. and Velasco, M. V., The Jacobson radical of a non-associative algebra and the uniqueness of the complete norm topology. Bull. Lond. Math. Soc. 42(2010), no. 6, 1010–1020. http://dx.doi.org/10.1112/blms/bdq060 Google Scholar
[9] Marcos, J. C., Continuity of homomorphisms into power-associative complete normed algebras. Forum. Math., to appear.Google Scholar
[10] Mena Jurado, J. F. and Palacios, A. Rodriguez, Weakly compact operators on non-complete normed spaces. Expo. Math. 27(2009), no. 2, 143–151. http://dx.doi.org/10.1016/j.exmath.2008.10.005Google Scholar
[11] Palmer, T., Banach algebras and the general theory of *-algebras. I. Encyclopedia of Mathematics and its Applications, 49, Cambridge University Press, Cambridge, 2009.Google Scholar
[12] Rickart, C. E., The uniqueness of the norm problem in Banach algebras. Ann. of Math. 51(1950), 615–628. http://dx.doi.org/10.2307/1969371 Google Scholar
[13] Rodriguez-Palacios, A., The uniqueness of complete norm topology in complete normed non-associative algebras. J. Funct. Anal. 60(1985), no. 1, 1–15. http://dx.doi.org/10.1016/0022-1236(85)90055-2 Google Scholar
[14] Rodriguez-Palacios, A., Continuity of densely valued homomorphisms into H*-algebras. Quart. J. Math. Oxford Ser. (2) 46(1995), no. 181, 107–118. http://dx.doi.org/10.1093/qmath/46.1.107 Google Scholar
[15] Rodriguez-Palacios, A. and M. V. Velasco, A non-associative Rickart's dense-range-homomorphism theorem. Q. J. Math. 54(2003), no. 3, 367–376. http://dx.doi.org/10.1093/qmath/hag015 Google Scholar
[16] Spurný, J., A note on compact operators on normed linear spaces. Expo. Math. 25(2007), no. 3, 261–263. http://dx.doi.org/10.1016/j.exmath.2006.11.002 Google Scholar
[17] Tian, J. P., Evolution algebras and their applications. Lecture Notes in Mathematics, 1921, Springer, Berlin, 2008.Google Scholar
[18] Upmeier, H., Symmetric Banach manifolds and Jordan C*-algebras. North-Holland Mathematics Studies, 104, North-Holland, Amsterdam, 1985.Google Scholar
[19] Velasco, M. V., Spectral theory for non-associative complete normed algebras and automatic continuity. J. Math. Anal. Appl. 351(2009), no. 1, 97–106. http://dx.doi.org/10.1016/j.jmaa.2008.09.036 Google Scholar
[20] Villena, A. R., Automatic continuity in associative and nonassociative context. Irish Math. Soc. Bull. 46(2001), 43–76.Google Scholar