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On Normed Algebras Whose Norms Satisfy Polynomial Identities

Published online by Cambridge University Press:  20 November 2018

Silvio Aurora
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In this note we are concerned with normed algebras over a non-discrete field with absolute value. The norm, N, of a normed algebra A is said to satisfy a polynomial identity on a subset B if there is a polynomial such that P(N(x1), … , N(xr)) = N(P(x1 … , xr)) whenever x1, … , xr are in B, where the polynomial has rational integer coefficients, degree greater than 1, constant term zero, and non-negative coefficients for each term of highest degree. It is shown in Theorem 1, following a method of proof used by Kadison in (6, § 7), that if the norm of a normed algebra satisfies a polynomial identity on the entire algebra, then the norm is power multiplicative. (That is, then N(x)2 = N(x2) for all x.)

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 13 , 1961 , pp. 664 - 674
Copyright
Copyright © Canadian Mathematical Society 1961

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