# On an Operator Theory on a Banach Space of Countable Type over a Hahn Field

## Abstract

This paper is a survey of the results in Aguayo et al. (J Math Phys 54(2), 2013; Indag Math (N.S.) 26(1):191–205, 2015; *p*-Adic Num Ultrametr Anal Appl 9(2):122–137, 2017) but generalized to the case when the complex Levi-Civita field \(\mathcal {C}\) is replaced by a Hahn field \(\mathbb {K}((G))\) for particular choices of the field \(\mathbb {K}\) and the abelian group *G*. In particular, we consider the Banach space of countable type *c*_{0} consisting of all null sequences of \(\mathbb {K}((G))\), equipped with the supremum norm ∥⋅∥_{∞} and we define a natural inner product on *c*_{0} which induces the norm of *c*_{0}. Then we present characterizations of normal projections and of compact and self-adjoint operators on *c*_{0}. As a new result in this paper, we apply the Hahn–Banach theorem to show the existence of the dual operator of a given continuous linear operator on *c*_{0} and to show that the dual operator and the adjoint operator coincide.

We present some B^{∗}-algebras of operators, including those mentioned above, then we define an inner product on such algebras which induces the usual norm of operators. Finally, we present a study of positive operators on *c*_{0} and use that to introduce a partial order on the set of compact and self-adjoint operators on *c*_{0}.

## Notes

### Acknowledgements

The second author “Changying Ding” was at the time of submission of this paper an undergraduate student supported by a MITACS Globalink summer research internship.

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