Abstract
We will extend the main result of [1] to the non-unital case with a totally different proof. More precisely, we give an abstract characterization of an arbitrary self-adjoint weak⁎-closed subspace of L(H) (equipped with the induced matrix norm, the induced matrix cone and the induced weak⁎-topology). In order to do this, we give a matrix analogue of a result of Bonsall for ⁎-operator spaces equipped with closed matrix cones.
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