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.
Reviewer: S.P.Hurd


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


Zbl 0609.06013
Full Text: DOI EuDML


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