Geodesic
A Double-Centralizer Theorem for Simple Associative Algebras
Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 477-478

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Werner, W. L. A Double-Centralizer Theorem for Simple Associative Algebras. Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 477-478. doi: 10.4153/CJM-1969-052-0
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[1] 1. Albert, A. A., Structure of algebras, Amer. Math. Soc. Colloq. Publ., Vol. 24 (Amer. Math. Soc, Providence, R.I., 1939). Google Scholar

[2] 2. Cullen, C. G. and Carlson, R., Commutativity for matrices of quaternions, Can. J. Math. 20 (1968), 21–24. Google Scholar

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