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On spectral continuity of positive elements

S. Mouton (2006)

Studia Mathematica

Let x be a positive element of an ordered Banach algebra. We prove a relationship between the spectra of x and of certain positive elements y for which either xy ≤ yx or yx ≤ xy. Furthermore, we show that the spectral radius is continuous at x, considered as an element of the set of all positive elements y ≥ x such that either xy ≤ yx or yx ≤ xy. We also show that the property ϱ(x + y) ≤ ϱ(x) + ϱ(y) of the spectral radius ϱ can be obtained for positive elements y which satisfy at least one of the...

On the ideal structure of algebras of LMC-algebra valued functions

Jorma Arhippainen (1992)

Studia Mathematica

Let X be a completely regular topological space and A a commutative locally m-convex algebra. We give a description of all closed and in particular closed maximal ideals of the algebra C(X,A) (= all continuous A-valued functions defined on X). The topology on C(X,A) is defined by a certain family of seminorms. The compact-open topology of C(X,A) is a special case of this topology.

On the joint spectral radius

Vladimír Müller (1997)

Annales Polonici Mathematici

We prove the p -spectral radius formula for n-tuples of commuting Banach algebra elements

On the ultrametric Stone-Weierstrass theorem and Mahler's expansion

Paul-Jean Cahen, Jean-Luc Chabert (2002)

Journal de théorie des nombres de Bordeaux

We describe an ultrametric version of the Stone-Weierstrass theorem, without any assumption on the residue field. If E is a subset of a rank-one valuation domain V , we show that the ring of polynomial functions is dense in the ring of continuous functions from E to V if and only if the topological closure E ^ of E in the completion V ^ of V is compact. We then show how to expand continuous functions in sums of polynomials.

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