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A Double-Centralizer Theorem for Simple Associative Algebras

Published online by Cambridge University Press:  20 November 2018

W. L. Werner
W. L. Werner*
Affiliation:
College of Southern Utah, Cedar City, Utah
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Consider the following result.

PROPOSITION. Let D be a finite-dimensional central division algebra over a field F, and let Dn be the algebra (over F) of all n × n matrices with entries in D. Let A and B be in Dn, and suppose that BX = XB for every X in Dn such that XA = AX. Then B is a polynomial in A with coefficients in F.

The case D = F is a well-known classical result. Recently, the particular case where D is the algebra of real quaternions was established by Cullen and Carlson (2). In this note, the general proposition is proved by reduction to the classical case by way of tensor products.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 477 - 478
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Albert, A. A., Structure of algebras, Amer. Math. Soc. Colloq. Publ., Vol. 24 (Amer. Math. Soc, Providence, R.I., 1939).Google Scholar
2. Cullen, C. G. and Carlson, R., Commutativity for matrices of quaternions, Can. J. Math. 20 (1968), 21–24.Google Scholar