Equivalent complete norms and positivity

Abstract.

In the first part of the article we characterize automatic continuity of positive operators. As a corollary we consider complete norms for which a given cone E₊ in an infinite dimensional Banach space E is closed and we obtain the following result: every two such norms are equivalent if and only if $E_+ \cap (-E_+) = \{0\} \, {\rm and} \, E_+ - E_+$ has finite codimension.

Without preservation of an order structure, on an infinite dimensional Banach space one can always construct infinitely many mutually non-equivalent complete norms. We use different techniques to prove this. The most striking is a set theoretic approach which allows us to construct infinitely many complete norms such that the resulting Banach spaces are mutually non-isomorphic.