Abstract m-space

Short description: Concept in order theory

In mathematics, specifically in order theory and functional analysis, an abstract m-space or an AM-space is a Banach lattice $(X,\Vert \cdot \Vert )$ whose norm satisfies $\Vert \sup \{x,y\}\Vert =\sup \{\Vert x\Vert ,\Vert y\Vert \}$ 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 p_u is the Minkowski functional of $[u,-u]:=\{x\in X:-u\le x\text{ and }x\le x\},$ then the complete of the semi-normed space (X, p_u) 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 $C_{\mathbb{ℝ}}\left(X\right)$. 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. $\sigma (X^{\prime },X)$-compact) subset of $X^{\prime }$ and furthermore, the evaluation map $I:X\to C_{\mathbb{ℝ}}\left(K\right)$ defined by $I(x):=I_x$ (where $I_x:K\to \mathbb{ℝ}$ is defined by $I_x(t)=⟨x,t⟩$) is an isomorphism.

See also

• Vector lattice
• AL-space

Bibliography

• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.