Let E and F be Banach lattices. It is shown that if F has the Levi and the Fatou property, then the ordered Banach space 5?1{E,F) of cone absolutely summing operators is a Banach lattice and an order ideal of the Riesz space Sfr{E, F) of regular operators. The same argument yields a Jordan decomposition of F-valued vector measures of bounded variation.
SynopsisAn Archimedean unital f-algebra A is called a U-algebra if, for every a∊A, there exists an invertible element u∊A such that a = u |a|. Characterisations of a U-algebra are established. As an application, an extension theorem of Hahn–Banach type on modules over a U-algebra and over the complexification of a Dedekind complete unital f-algebra is given.
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