Vector spaces with an order unit

Understand

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.