Abstract. We provide an order-theoretic characterization of algebraic orthogonality among positive elements of a general C$^{\ast }$-algebra by proving a statement conjectured in [12]. Generalizing this idea, we describe absolutely ordered $p$-normed spaces for $1 \le p \le \infty $ which present a model for ``non-commutative vector lattices''. This notion includes order-theoretic orthogonality. We generalize algebraic orthogonality by introducing the notion of {\it absolute compatibility} among positive elements in absolute order unit spaces and relate it to the 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 the spectral decomposition theory for elements of an absolute order unit space.

DOI: 10.14232/actasm-017-574-3

AMS Subject Classification (1991): 46B40; 46L05, 46L30

Keyword(s): absolute $\infty $-orthogonality, absolute order unit space, absolute compatibility, order projection

Received November 19, 2017, and in revised form February 1, 2018. (Registered under 74/2017.)