Stable subnorms revisited

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 0 not equal a is an element of S, and f(alpha a) = |alpha| f(a) for all a is an element of S and alpha is an element of F. If in addition, S is closed under raising to powers, then a subnorm f shall be called stable if there exists a constant sigma > 0 so that f(a(m)) less than or equal to sigma f(a)(m) for all a is an element of 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

Sundual characterizations of the translation group of R

We characterize the first three sundual spaces of C-0(R), with respect to the translation group of R

On the convergence of successive approximations in the theory of ordinary differential equations

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An extension of the concept of the order dual of a Riesz space

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Positive representations of finite groups in Riesz spaces

In this paper, which is part of a study of positive representations of locally compact groups in Banach lattices, we initiate the theory of positive representations of finite groups in Riesz spaces. If such a representation has only the zero subspace and possibly the space itself as invariant principal bands, then the space is Archimedean and finite dimensional. Various notions of irreducibility of a positive representation are introduced and, for a finite group acting positively in a space with sufficiently many projections, these are shown to be equal. We describe the finite dimensional positive Archimedean representations of a finite group and establish that, up to order equivalence, these are order direct sums, with unique multiplicities, of the order indecomposable positive representations naturally associated with transitive $G$-spaces. Character theory is shown to break down for positive representations. Induction and systems of imprimitivity are introduced in an ordered context, where the multiplicity formulation of Frobenius reciprocity turns out not to hold.Comment: 23 pages. To appear in International Journal of Mathematic

Stable seminorms revisited

A seminorm S on an algebra A is called stable if for some constant σ > 0 , S(x^k) ≤ σS(x)^k for all x ∈ A and all k = 1, 2, 3,.... We call S strongly stable if the above holds with σ = 1 . In this note we use several known and new results to shed light on the concepts of stability. In particular, the interrelation between stability and similar ideas is discussed

Non-Standard Analysis: Lectures on A. Robinson's Theory of Infinitesimals and Infinitely Large Numbers

The present lecture notes have grown from a series of three lectures which were given by the author at the California Institute of Technology in December 1961. The purpose of these lectures was to give a discussion of A. Robinson's theory of infinitesimals and infinitely large numbers which had just appeared in print under the title "Non-Standard Analysis". The title "Non-Standard Analysis" refers to the fact that this theory is an interpretation of analysis in a non-standard model of the arithmetic of the real numbers

A remark on Sikorski's extension theorem for homomorphisms in the theory of Boolean algebras

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On Finitely Additive Measures in Boolean Algebras

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A remark on Sikorski's extension theorem for homomorphisms in the theory of Boolean algebras

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