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The natural ordering on a strictly real Banach algebra

Published online by Cambridge University Press:  24 October 2008

John Boris Miller
Affiliation:
Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia

Abstract

A strictly real unital Banach algebra is one in which every element has real spectrum. An antilattice partial order and its associated open-interval topology are defined on by taking as positive cone the principal component of the maximal group of the algebra, and their properties are studied. The topology coincides with the semimetric topology of the spectral radius, which is a seminorm, making into a locally convex partially ordered topological algebra with continuous inversion and normal cone. Every positive element has a unique positive square root, and logarithm, and these functions and exp are continuous and monotone. The cone has the same closure in both open-interval and normal topologies, the closure being a wedge for an associated preorder on , this preorder coinciding with the preorder of Kelley and Vaught. The partial order and the associated preorder have the same dual cone. There is a modified Gelfand representation for (possibly non-commutative) , and a boundary integral representation for functionals in the order dual. The compact convex subset in the dual is metrizable if the cone of the partial order in is separable.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1990

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References

REFERENCES

[1]Alfsen, E. M.. Compact Convex Sets and Boundary Integrals. Ergeb. Math. Grenzgeb. Band 57 (Springer-Verlag, 1971).CrossRefGoogle Scholar
[2]Aupetit, B.. Propriétés Spectrales des Algèbres de Banach. Lecture Notes in Math. vol. 735 (Springer-Verlag, 1975).Google Scholar
[3]Bonsall, F. F. and Duncan, J.. Complete Normed Algebras. Ergeb. Math. Grenzgeb. Band 80 (Springer-Verlag, 1973).CrossRefGoogle Scholar
[4]Bonsall, F. F. and Duncan, J.. Numerical Ranges of Operators on Normed Spaces and Elements of Normed Algebras (Cambridge University Press, 1971).CrossRefGoogle Scholar
[5]Fuchs, L.. Riesz Vector Spaces and Riesz Algebras (Queen's University).Google Scholar
[6]Halmos, P.. Measure Theory (Van Nostrand, 1959).Google Scholar
[7]Hervé, M.. Sur les représentations intégrales à l'aide des points extrémeaux dans un ensemble compact convexe métrisable. C. R. Acad. Sci. Paris 253 (1961), 366368.Google Scholar
[8]Inglestam, L.. Real Banach algebras. Ark. Mat. 5 (1964), 239270.CrossRefGoogle Scholar
[9]Kadison, R. V.. A Representation Theory for Commutative Topological Algebra. Memoirs Amer. Math. Soc. no. 7 (American Mathematical Society, 1951).CrossRefGoogle Scholar
[10]Kaplansky, I.. Normed algebras. Duke Math. J. 16 (1949), 399418.CrossRefGoogle Scholar
[11]Kelley, J. L. and Namioka, I.. Linear Topological Spaces (Van Nostrand, 1963).CrossRefGoogle Scholar
[12]Kelley, J. L. and Vaught, R. L.. The positive cone in Banach algebras. Trans. Amer. Math. Soc. 74 (1953), 4455.CrossRefGoogle Scholar
[13]Phelps, R. R.. Lectures on Choquet's Theorem (Van Nostrand, 1966).Google Scholar
[14]Raikov, D. A.. To the theory of normed rings with involution. Dokl. Acad. Sci. URSS 54 (1946), 387390.Google Scholar
[15]Rickart, C. E.. General Theory of Banach Algebras (Van Nostrand, 1960).Google Scholar
[16]Schaefer, H. H.. Topological Vector Spaces (Macmillan, 1966).Google Scholar