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**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.

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.

- (i)
- 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.
- (ii)
- Every positive operator from \(E\) to \(F\) is continuous.

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.

Reviewer: Şafak Alpay (Ankara)

### MSC:

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 |