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 Vector spaces with an order unit

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We develop a theory of ordered *-vector spaces with an order unit. We prove fundamental results concerning positive linear functionals and states, and we show that the order seminorm on the space of self-adjoint elements admits multiple extensions to an order seminorm on the entire space. We single out three of these seminorms for further study and discuss their significance for operator algebras and operator systems. In addition, we introduce a functorial method for taking an ordered space with an order unit and forming an Archimedean ordered space. We then use this process to describe an appropriate notion of quotients in the category of Archimedean ordered spaces.

Author: Vern Paulsen; Mark Tomforde



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