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Algebraic orthogonality and commuting projections in operator algebras. (arXiv:1704.07631v1 [math.FA])
来源于:arXiv
We describe absolutely ordered $p$-normed spaces, for $1 \le p \le \infty$
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$^{\ast}$-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 spectral
decomposition theory for elements of an absolute order unit space. 查看全文>>