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Orthomorphisms of archimedean vector lattices

Published online by Cambridge University Press:  24 October 2008

S. J. Bernau
S. J. Bernau
Affiliation:
University of Texas at Austin, Texas, U.S.A.
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Abstract

A linear operator T on a vector lattice L preserves disjointness if Tx ⊥ y whenever x ⊥ y. If such a T is positive it is automatically order bounded. An ortho-morphism is an order bounded disjointness preserving linear operator on L. In this note we show that the theory of orthomorphisms on archimedean vector lattices admits a totally elementary exposition. Elementary methods are also effective in duality considerations when the order dual separates points of L. For the Jordan decomposition T = T+ − T− with T+x = (Tx+)+ − (Tx−)+ we can dtrop the order boundedness assumption if we assume either that T preserves ideals or that L is normed and T is continuous. Alternatively we may keep order boundedness and assume only |Tx| ⊥ |Ty| whenever x ⊥ y. The main duality results show: T preserves ideals if and only if T** does; T is an orthomorphism if and only if T* is; T is central (|T| is bounded by a multiple of the identity) if and only if T* is central if and only if T and T* preserve ideals.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 89 , Issue 1 , January 1981 , pp. 119 - 128
Copyright
Copyright © Cambridge Philosophical Society 1981

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