Abstract
This chapter is a pragmatic review of the basic theory of ordered vector spaces. Among other things, we present the notion of positive linear transformations and prove various important properties of these transformations. In view of applications to C∗-algebras, we also discuss ordered vector spaces endowed with norms that are compatible with the order in a natural sense and highlight the crucial relationship between positivity and continuity of linear transformations on ordered normed vector spaces. In fact, C∗-algebras are examples of ordered vector spaces, and various important results regarding these algebras are simple consequences of general results referring to abstract positive linear transformations.
Note that the most well-known and studied class of ordered spaces refers to the Riesz spaces, which are presented in the appendix (Sect. 7.4) for completeness. In fact, commutative C∗-algebras are important examples of this type of ordered space.
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Notes
- 1.
For any \(\lambda \in \mathbb {C}\) and v, w ∈ V , \(\left ( \lambda v\right ) ^{\ast }=\bar {\lambda }v^{\ast }\) and \(\left ( v+w\right ) ^{\ast }=v^{\ast }+w^{\ast }\).
- 2.
For any v ∈ V , \(\left ( v^{\ast }\right ) ^{\ast }=v\).
- 3.
That is, the vector space of all linear transformations from the vector space over \(\mathbb {K=R},\mathbb {C}\) to the field itself \(\mathbb {K}\).
- 4.
A norm \(v\mapsto \left \Vert v\right \Vert \) on a vector space V over a field \(\mathbb {K}\) is a real-valued function satisfying \(\left \Vert v+w\right \Vert \leq \left \Vert v\right \Vert +\left \Vert w\right \Vert \) (the triangle inequality) and \(\left \Vert \lambda v\right \Vert =\left \vert \lambda \right \vert \left \Vert v\right \Vert \) (homogeneity) for any v, w ∈ V and \(\lambda \in \mathbb {K}\), as well as \(\left \Vert v\right \Vert =0\Rightarrow v=0\) (positive definiteness). See the appendix for more details.
- 5.
A seminorm \(v\mapsto \left \Vert v\right \Vert \) on a vector space V is a real-valued function satisfying all properties of a norm except the positive definiteness, i.e., \(\left \Vert v\right \Vert =0\) does not yield v = 0. See the appendix for more details.
- 6.
∥⋅∥(1) and ∥⋅∥(2) are “equivalent norms” if, for some C ∈ [1, ∞). and all v ∈ V , C−1∥v∥(2) ≤∥v∥(1) ≤ C∥v∥(2). See the appendix for more details.
- 7.
A absorbing subset of a vector space V over \(\mathbb {K=R},\mathbb {C}\) is a set W ⊆ V such that for any v ∈ V , there is r > 0 so that v ∈ λW for any \(\lambda \in \mathbb {K}\) with \(\left \vert \lambda \right \vert >r\).
- 8.
See Definition 4.10 and remarks following it.
References
W. Rudin, Functional Analysis (McGraw-Hill Science, New York, 1991)
J.D. Weston, The decomposition of a continuous linear functional into non-negative components. Math. Scand. 5, 54–56 (1957)
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Bru, JB., Alberto Siqueira Pedra, W.d. (2023). Ordered Vector Spaces and Positivity. In: C*-Algebras and Mathematical Foundations of Quantum Statistical Mechanics. Latin American Mathematics Series(). Springer, Cham. https://doi.org/10.1007/978-3-031-28949-1_1
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