Algebraic orthogonality and commuting projections in operator algebras

Abstract

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.

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