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Algebraic orthogonality and commuting projections in operator algebras. (English) Zbl 1413.46023
Summary: We provide an order-theoretic characterization of algebraic orthogonality among positive elements of a general $$C^*$$-algebra by proving a statement conjectured by A. K. Karn [Positivity 20, No. 3, 607–620 (2016; Zbl 1361.46017)]. Generalizing this idea, we describe absolutely ordered $$p$$ normed spaces for $$1\leq p\leq\infty$$ which present a model for “non-commutative vector lattices”. This notion includes order-theoretic orthogonality. We generalize algebraic orthogonality by introducing the notion of absolute compatibility among positive elements in absolute order unit spaces and relate it to the symmetrized product in the case of a $$C^*$$-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 order projections and extend the results related to projections in a unital $$C^*$$-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.

##### MSC:
 46B40 Ordered normed spaces 46L05 General theory of $$C^*$$-algebras 46L30 States of selfadjoint operator algebras
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