Algebraic orthogonality and commuting projections in operator algebras

Karn, Anil Kumar: Algebraic orthogonality and commuting projections in operator algebras. Acta scientiarum mathematicarum, (84) 1-2. pp. 323-353. (2018)

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Abstract

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 pnormed spaces for 1 ≤ p ≤ ∞ 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.

Item Type: Article
Journal or Publication Title: Acta scientiarum mathematicarum
Date: 2018
Volume: 84
Number: 1-2
Page Range: pp. 323-353
ISSN: 0001-6969
Uncontrolled Keywords: Algebra
Additional Information: Bibliogr.: 353. p. ; Összefoglalás angol nyelven
Official URL: http://www.acta.hu
Date Deposited: 2018. Nov. 10. 10:52
Last Modified: 2019. Sep. 09. 13:44
URI: http://acta.bibl.u-szeged.hu/id/eprint/55817

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