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Normed Right Alternative Algebras Over the Reals

Published online by Cambridge University Press:  20 November 2018

José I. Nieto
José I. Nieto*
Affiliation:
Université de Montréal, Montréal, Québec
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One of the most interesting results on real normed division algebras says that every real normed associative division algebra is finite dimensional [6, Theorem 1.7.6], and hence by a classical theorem of Frobenius either isomorphic to the real field, the complex field, or the algebra of quaternions. Thus the dimension of the algebra can only be either 1, 2 or 4.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 6 , December 1972 , pp. 1183 - 1186
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Albert, A. A., Absolute-valued algebraic algebras, Bull. Amer. Math. Soc. 55 (1949), 763–768.Google Scholar
2. Bruck, R. H. and Kleinfeld, E., The structure of alternative division rings, Proc. Amer. Math. Soc. 2 (1951), 878–890.Google Scholar
3. Kleinfeld, E., A characterization of the Cayley numbers, MAA Studies in Mathematics, Vol. 2: Studies in Modern Algebra. Albert, A. A., editor.Google Scholar
4. Kleinfeld, E., Right alternative rings, Proc. Amer. Math. Soc. 4 (1953), 939–944.Google Scholar
5. Milnor, J., On the parallelizability of the spheres, Bull. Amer. Math. Soc. 64 (1958), 87–89.Google Scholar
6. Rickart, C. E., Banach algebras (Van Nostrand, Princeton, 1960).Google Scholar
7. Skorniakov, L. A., Right alternative division rings (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 15 (1951), 177–184.Google Scholar
8. Skorniakov, L. A., Alternative division rings (Russian), Ukrain. Mat. Z. 2 (1950), 70–85.Google Scholar