Abstract m-space

In mathematics, specifically in order theory and functional analysis, an abstract m-space or an AM-space is a Banach lattice $$(X, \| \cdot \|)$$ whose norm satisfies $$\left\| \sup \{ x, y \} \right\| = \sup \left\{ \| x \|, \| y \| \right\}$$ for all x and y in the positive cone of X. We say that an AM-space X is an AM-space with unit if in addition there exists some $u ≥ 0$ in X such that the interval $[−u, u] := { z ∈ X : −u ≤ z and z ≤ u }$ is equal to the unit ball of X; such an element u is unique and an order unit of X.

Examples
The strong dual of an AL-space is an AM-space with unit.

If X is an Archimedean ordered vector lattice, u is an order unit of X, and pu is the Minkowski functional of $$[u, -u] := \{ x \in X : -u \leq x \text{ and } x \leq x \},$$ then the complete of the semi-normed space (X, pu) is an AM-space with unit u.

Properties
Every AM-space is isomorphic (as a Banach lattice) with some closed vector sublattice of some suitable $$C_{\R}\left( X \right)$$. The strong dual of an AM-space with unit is an AL-space.

If X ≠ { 0 } is an AM-space with unit then the set K of all extreme points of the positive face of the dual unit ball is a non-empty and weakly compact (i.e. $$\sigma\left( X^{\prime}, X \right)$$-compact) subset of $$X^{\prime}$$ and furthermore, the evaluation map $$I : X \to C_{\R} \left( K \right)$$ defined by $$I(x) := I_x$$ (where $$I_x : K \to \R$$ is defined by $$I_x(t) = \langle x, t \rangle$$) is an isomorphism.