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Algebraic orthogonality and commuting projections in operator algebras

2018; Birkhäuser; Volume: 84; Issue: 1-2 Linguagem: Inglês

10.14232/actasm-017-574-3

2064-8316

Anil Kumar Karn,

Advanced Topics in Algebra

We provide an order-theoretic characterization of algebraic orthogonality among positive elements of a general C * -algebra by proving a statement conjectured in [12].Generalizing this idea, we describe absolutely ordered p-normed spaces, for 1 ≤ p ≤ ∞ which present a model for "non-commutative vector lattices".Thid 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 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 spectral decomposition theory for elements of an absolute order unit space.