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Strictly positive and strongly positive semigroups

Part of: Individual linear operators as elements of algebraic systems

Published online by Cambridge University Press:  09 April 2009

Adam Majewski  and
Derek W. Robinson
Adam Majewski
Affiliation:
Department of Pure Mathematics University of New South WalesKensington, N.S.W. 2033, Australia
Derek W. Robinson
Affiliation:
Department of Pure Mathematics University of New South WalesKensington, N.S.W. 2033, Australia
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Abstract

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We examine positive semigroups acting on Banach lattices and operator algebras. In the lattice framework we characterize strict positivity and strict ordering of holomorphic semigroups by irreducibility criteria. In the algebraic setting we derive ergodic criteria for irreducibility and discuss various aspects of strict positivity. Finally we examine invariant states of a C*-dynamical system in which the automorphism group is replaced by a strongly positive semigroup. We demonstrate that ergodic states are characterized by a cluster property despite the absence of a covariant implementation law for the semigroup.

MSC classification

Secondary: 47C15: Operators in $C^*$- or von Neumann algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 34 , Issue 1 , February 1983 , pp. 36 - 48
Copyright
Copyright © Australian Mathematical Society 1983

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