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**Inverses of disjointness preserving operators.**
*(English)*
Zbl 0974.47032

Mem. Am. Math. Soc. 679, 164 p. (2000).

Let \(X\) and \(Y\) be vector lattices. A linear map \(T:X\to Y\) is disjointness preserving if \(Tx\perp Ty\) whenever \(x\perp y\) in \(X\); if \(T\) is bijective and \(T^{-1}\) is also disjointness-preserving then \(T\) is said to be a \(d\)-isomorphism. The main results of this monograph include:

(A) a characterization of those Dedekind-complete lattices \(X\) having the property that every disjointness-preserving bijection with domain \(X\) is a \(d\)-isomorphism, and

(B) a theorem to the effect that for Dedekind-complete vector lattices, \(d\)-isomorphism implies order isomorphism.

Partial results along these lines were announced in [the authors, “Functional analysis and economic theory”. Based on the special session of the conference on nonlinear analysis and its applications in engineering and economics, Samos, Greece, July 1996, Berlin: Springer, 1-8 (1998; Zbl 0916.47026)]. Besides being more definitive and complete, the present account also initiates a new perspective: rather than regarding disjointness preserving bijections which fail to be \(d\)-isomorphisms as aberrant curiosities, their existence is now taken as an opportunity to learn about the structures of domain and range spaces. This new perspective is reflected in the rich variety of counterexamples presented and in the full treatments of concepts like ‘determining/cofinal families of band projections’, ‘\(d\)-splitting numbers’, ‘essentially constant functions’, and ‘cofinal universal completeness’ leading up to (A).

(A) a characterization of those Dedekind-complete lattices \(X\) having the property that every disjointness-preserving bijection with domain \(X\) is a \(d\)-isomorphism, and

(B) a theorem to the effect that for Dedekind-complete vector lattices, \(d\)-isomorphism implies order isomorphism.

Partial results along these lines were announced in [the authors, “Functional analysis and economic theory”. Based on the special session of the conference on nonlinear analysis and its applications in engineering and economics, Samos, Greece, July 1996, Berlin: Springer, 1-8 (1998; Zbl 0916.47026)]. Besides being more definitive and complete, the present account also initiates a new perspective: rather than regarding disjointness preserving bijections which fail to be \(d\)-isomorphisms as aberrant curiosities, their existence is now taken as an opportunity to learn about the structures of domain and range spaces. This new perspective is reflected in the rich variety of counterexamples presented and in the full treatments of concepts like ‘determining/cofinal families of band projections’, ‘\(d\)-splitting numbers’, ‘essentially constant functions’, and ‘cofinal universal completeness’ leading up to (A).

Reviewer: H.G.Nutzinger (Kassel)

### MSC:

47B60 | Linear operators on ordered spaces |

46B40 | Ordered normed spaces |

46A40 | Ordered topological linear spaces, vector lattices |

47B65 | Positive linear operators and order-bounded operators |

46B42 | Banach lattices |

54G05 | Extremally disconnected spaces, \(F\)-spaces, etc. |

47B38 | Linear operators on function spaces (general) |