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**The dual space of a totally ordered Abelian group.**
*(English)*
Zbl 0645.06007

In this paper the author continues his earlier work [“Dual spaces of ordered groups”, in: Algebra and Order, Res. Expo. Math. 14, 95-103 (1986; Zbl 0609.06013)] which explores the extent to which methods of real analysis can be applied when one has only the group operation and the order properties to work with.

The dual space \(T^\wedge\) of an abelian totally ordered group T is defined (slightly differently from the earlier paper cited) so that \(T^\wedge\) will not be Archimedian when T is not. The homomorphisms singled out for \(T^\wedge\) form a partially ordered group which is locally order-preservingwith respect to a fixed but arbitrary Banaschewski function which, in turn, has a dual Banaschewski function, and so one can form higher dual spaces in the same way. The construction allows a homomorphism between base groups to lift to a homomorphism of their dual spaces. The evaluation map into the second dual is a one-to- one homomorphism, and, interestingly, all the odd-numbered higher dual spaces will be isomorphic as will all the even numbered ones. The group of eventually constant sequences has two non-isomorphic dual spaces arising from two different Banaschewski functions. In a forthcoming paper (Dual spaces of totally ordered rings, to appear) the author points out that the modified definition allows convolution to be a well-defined operation on the second dual. Many interesting examples are given.

The dual space \(T^\wedge\) of an abelian totally ordered group T is defined (slightly differently from the earlier paper cited) so that \(T^\wedge\) will not be Archimedian when T is not. The homomorphisms singled out for \(T^\wedge\) form a partially ordered group which is locally order-preservingwith respect to a fixed but arbitrary Banaschewski function which, in turn, has a dual Banaschewski function, and so one can form higher dual spaces in the same way. The construction allows a homomorphism between base groups to lift to a homomorphism of their dual spaces. The evaluation map into the second dual is a one-to- one homomorphism, and, interestingly, all the odd-numbered higher dual spaces will be isomorphic as will all the even numbered ones. The group of eventually constant sequences has two non-isomorphic dual spaces arising from two different Banaschewski functions. In a forthcoming paper (Dual spaces of totally ordered rings, to appear) the author points out that the modified definition allows convolution to be a well-defined operation on the second dual. Many interesting examples are given.

Reviewer: S.P.Hurd

### MSC:

06F20 | Ordered abelian groups, Riesz groups, ordered linear spaces |

### Keywords:

ordered abelian group; dual space; Banaschewski function; group of eventually constant sequences### Citations:

Zbl 0609.06013### References:

[1] | C D. Aliprantis O. Burkinshaw: Locally Solid Riesz Spaces. Academic Press, New York, 1978. · Zbl 0402.46005 |

[2] | B. Banaschewski: Totalgeordnete moduln. Arch. Math. 7 (1956), 430-440. · Zbl 0208.03702 · doi:10.1007/BF01899022 |

[3] | G. Birkhoff: Lattice Theory. Arner. Math. Soc. Coll. Pub. 25, third (new) edition. Providence, 1967. · Zbl 0153.02501 |

[4] | L. Fuchs: Partially Ordered Algebraic Systems. Pergamon Press, Oxford, 1963. · Zbl 0137.02001 |

[5] | H. Hahn: Über die nichtarchimedischen Grossensysteme. S.-B. Wiener Akad. Math.-Nat. Klasse Abt. II a, 116 (1907), 601-653. |

[6] | J. L. Kelley: General Topology. Graduate Texts in Mathematics 27, Springer-Verlag, New York, reprint of D. Van Nostrand Co. edition, 1955. · Zbl 0066.16604 |

[7] | J. L. Kelley I. Namioka, etc.: Linear Topological Spaces. D. Van Nostrand Co., Inc., Princeton, 1963. · Zbl 0318.46001 |

[8] | J. B. Miller: The order-dual of a TRL group I. J. Australian Math. Soc. 25 (Series A) (1978), 129-141. · Zbl 0381.06023 · doi:10.1017/S1446788700038714 |

[9] | R.H. Redfield: Dual spaces of ordered groups. Algebra and Order, Helderman Verlag, Berlin, 1986, 95-103. · Zbl 0609.06013 |

[10] | R. H. Redfiled: Dual spaces of totally ordered rings. to appear. |

[11] | H. H. Schaefer: Banach Lattices and Positive Operators. Die Grundlehren der mathematischen Wissenschaften, 215, Springer-Verlag, New York, 1974. · Zbl 0296.47023 |

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