Equivalent complete norms and positivity. (English) Zbl 1182.46006

In the first part of the article, automatic continuity of positive operators is characterized. Namely, it is proved that, if \((E, E_+)\) and \((F, F_+)\) are ordered Banach spaces, then the following (i) and (ii) are equivalent.
At least one of the following is true: (1) \(E\) is finite-dimensional, (2) \(F_+ = \{0\}\), (3) \(\text{codim}(E_+ - E_+)\) is finite and \(F_+\) is proper.
Every positive operator from \(E\) to \(F\) is continuous.
As a consequence of this theorem, the authors obtain that, if \(E\) is an infinite-dimensional ordered Banach space, then \(E_+\) is proper, and \(E_+ - E_+\) has finite codimension if and only if each complete norm for which \(E_+\) is closed is equivalent to the given one.
If \(E\) is an infinite-dimensional Banach space, one can construct infinitely many mutually non-equivalent complete norms on \(E\) for which all the corresponding spaces are isometrically isomorphic to the original one. Moreover, the cardinality of such a set of norms can be chosen to be as large as the dimension of \(E\). Different routes, set theoretic and analytic are taken to prove this. The set theoretical approach allows the authors to construct infinitely many complete norms such that the resulting Banach spaces are mutually non-isomorphic. The paper answers an old question of Laugwitz.


46B40 Ordered normed spaces
47B65 Positive linear operators and order-bounded operators
46B03 Isomorphic theory (including renorming) of Banach spaces
03E75 Applications of set theory
46B26 Nonseparable Banach spaces
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