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Compressible groups with general comparability

David Foulis

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Rok, strany / Year, pages: 2005, 409-429

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Publikované / Published: 0000-00-00

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Compressible groups generalize the order-unit space of self-adjoint operators on Hilbert space, the directed additive group of self-adjoint elements of a unital $C*$-algebra, and interpolation groups with order units. In a compressible group with general comparability, each element $g$ may be written canonically as a difference $g=g\sp{+}-g\sp{-}$ of elements in the positive cone $G+$, and the absolute value $|g|$ is defined by $|g| :=g++g-$. In such a group $G$, we define and study a ``pseudo-meet'' $g\sqcap h$ and a ``pseudo-join'' $g\sqcup h$. If $G$ is lattice ordered, $g\sqcap h$ and $g\sqcup h$ coincide with the usual meet and join; in the general case, they retain a number of properties of the latter. We also introduce and study a so-called Rickart projection property suggested by an analogous property in Rickart $C*$@-algebras.

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