The Duality of Partially Ordered Banach Spaces

Structure in Real Banach Spaces. Part II

background

/pdf/the-ideal-center-of-partially-ordered-vector-spaces-1e1q8hgyh9.pdf

The ideal center of partially ordered vector spaces

Order unit and base norm spaces

/pdf/on-the-facial-structure-of-the-unit-balls-in-a-gl-space-and-1hfchhwxy3.pdf

On the facial structure of the unit balls in a GL-space and its dual

We describe absolutely ordered $p$-normed spaces, for $1 \le p \le \infty$ which presents a model for "non-commutative" vector lattices and includes order theoretic orthogonality. To demonstrate its relevance, we introduce the notion of {\it absolute compatibility} among positive elements in absolute order unit spaces and relate it to symmetrized product in the case of a C$^{\ast}$-algebra. In the latter case, whenever one of the elements is a projection, the elements are absolutely compatible if and only if they commute. We develop an order theoretic prototype of the results. For this purpose, we introduce the notion of {\it order projections} and extend the results related to projections in a unital C$^{\ast}$-algebra to order projections in an absolute order unit space. As an application, we describe spectral decomposition theory for elements of an absolute order unit space.

The Duality of Partially Ordered Banach Spaces

Citations

Structure in Real Banach Spaces. Part II

The ideal center of partially ordered vector spaces

Order unit and base norm spaces

On the facial structure of the unit balls in a GL-space and its dual

Algebraic orthogonality and commuting projections in operator algebras

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