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Pacific Journal of Mathematics, Volume 215, Number 1

Vol. 215, No. 1, 2004

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Moshe Goldberg & W.A.J. Luxemburg

Abstract

Let A be a finite-dimensional, power-associative algebra over a field F, either or , and let S, a subset of A, be closed under scalar multiplication. A real-valued function f defined on S, shall be called a subnorm if f(a) > 0 for all 0a ∈S, and f(αa) = |α|f(a) for all a ∈S and α F. If in addition, S is closed under raising to powers, then a subnorm f shall be called stable if there exists a constant σ > 0 so that

f(am) ≤ σ f (a)m for all a ∈ S and m = 1,2,3....

The purpose of this paper is to provide an updated account of our study of stable subnorms on subsets of finite-dimensional, power-associative algebras over F. Our goal is to review and extend several of our results in two previous papers, dealing mostly with continuous subnorms on closed sets.

Let A be a finite-dimensional, power-associative algebra over a field F, either or , and let S, a subset of A, be closed under scalar multiplication. A real-valued function f defined on S, shall be called a subnorm if f(a) > 0 for all 0a S, and f(αa) = |α|f(a) for all a S and α F. If in addition, S is closed under raising to powers, then a subnorm f shall be called stable if there exists a constant σ > 0 so that

f(am) ≤ σ f (a)m for all a ∈ <b>S</b> and m = 1,2,3....

The purpose of this paper is to provide an updated account of our study of stable subnorms on subsets of finite-dimensional, power-associative algebras over F. Our goal is to review and extend several of our results in two previous papers, dealing mostly with continuous subnorms on closed sets.

Let A be a finite-dimensional, power-associative algebra over a field F, either or , and let S, a subset of A, be closed under scalar multiplication. A real-valued function f defined on S, shall be called a subnorm if f(a) > 0 for all 0a S, and f(αa) = |α|f(a) for all a S and α F. If in addition, S is closed under raising to powers, then a subnorm f shall be called stable if there exists a constant σ > 0 so that

f(am) σf(a)m for alla S andm = 1,2,3.

The purpose of this paper is to provide an updated account of our study of stable subnorms on subsets of finite-dimensional, power-associative algebras over F. Our goal is to review and extend several of our results in two previous papers, dealing mostly with continuous subnorms on closed sets.

Let A be a finite-dimensional, power-associative algebra over a field F, either or , and let S, a subset of A, be closed under scalar multiplication. A real-valued function f defined on S, shall be called a subnorm if f(a) > 0 for all 0a S, and f(αa) = |α|f(a) for all a S and α F. If in addition, S is closed under raising to powers, then a subnorm f shall be called stable if there exists a constant σ > 0 so that

f(am) σf(a)m for alla S andm = 1,2,3.

The purpose of this paper is to provide an updated account of our study of stable subnorms on subsets of finite-dimensional, power-associative algebras over F. Our goal is to review and extend several of our results in two previous papers, dealing mostly with continuous subnorms on closed sets.

Let A be a finite-dimensional, power-associative algebra over a field F, either R or C, and let S, a subset of A, be closed under scalar multiplication. A real-valued function f defined on S, shall be called a subnorm if f(a) > 0 for all 0a in S, and f(αa) = |α|f(a) for all a in S and α in F. If in addition, S is closed under raising to powers, then a subnorm f shall be called stable if there exists a constant σ > 0 so that