12 research outputs found
Algebraic orthogonality and commuting projections in operator algebras
We describe absolutely ordered -normed spaces, for
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-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-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.Comment: 2
A generalization of spin factors
Using the technique of adjoining an order unit to a normed linear space, we have characterized strictly convex spaces among normed linear spaces and Hilbert spaces among strictly convex Banach spaces, respectively. This leads to a generalization of spin factors and provides a new class of absolute order unit spaces
CM-ideals and L1-matricial split faces
We discuss the order-theoretic properties of CM-ideals in matricially order smooth∞-normed spaces. We study the relation between CM-ideals and CL-summands in the matrix duality setup. We introduce the notion of L1-matricial split faces in an L1-matricially normed space and characterize CM-ideals in a matricially order smooth∞-normed space Vin terms of the L1-matricial split face of the L1-matrix convex set{Qn(V)}