Vector Spaces with an Order Unit

By Vern I. Paulsen, M. Tomforde. 2007

107 citations recorded by the source corpus.

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 (semi)norm on the space of self-adjoint elements admits multiple extensions to an order (semi)norm on the entire space. We single out three of these (semi)norms 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.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1249a3ca136db4d2a3103ce17da7d24f53c1244f

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In Algebra and Number Theory

In Mathematical Physics

Citations in the corpus

Quotients, exactness, and nuclearity in the operator system category (2010, background): “For an x ∈ Qh, set ‖x‖o = inf{λ ≥ 0 : λ1±x ∈ D1}; then ‖ · ‖o is a seminorm on the real vector space Qh, called the order seminorm in [20].”
Tensor products of operator systems (2009, background): “This was first introduced in [22] for ordered spaces and extended to matrix ordered spaces in [21].”
Noncommutative Choquet Theory (2019, unknown): “This means that it is an ordered complex *-vector space with a distinguished archimedean order unit [43].”