Abstract
Two operations on sets of bounded operators on Hilbert space are studied: One produces Jordan algebras and the other, its inverse in a certain sense, produces self-adjoint linear spaces. The question of which Jordan algebras arise from the first of these operations is investigated. It is shown that, in many cases, R has this property if and only if the complex associative algebra R + iR is equal to its double commutant. Thus, a by-product of the study is another approach to double commutant theorems.
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