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A spectral problem in ordered Banach algebras

Published online by Cambridge University Press:  17 April 2009

S. Mouton
S. Mouton
Affiliation:
Department of Mathematics, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa, e-mail: [email protected]
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Abstract

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We recall the definition and properties of an algebra cone C of a complex unital Banach algebra A. It can be shown that C induces on A an ordering which is compatible with the algebraic structure of A, and A is then called an ordered Banach algebra. The Banach algebra ℒ(E) of all bounded linear operators on a complex Banach lattice E is an example of an ordered Banach algebra, and an interesting aspect of research in ordered Banach algebras is that of investigating in an ordered Banach algebra-context certain problems that originated in ℒ(E). In this paper we investigate the problems of providing conditions under which (1) a positive element a with spectrum consisting of 1 only will necessarily be greater than or equal to 1, and (2) f (a) will be positive if a is positive, where f (a) is the element defined by the holomorphic functional calculus.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 67 , Issue 1 , February 2003 , pp. 131 - 144
Copyright
Copyright © Australian Mathematical Society 2003

References

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