Square Roots and Inverses in E-Rings

An e-ring is a pair (R, E) consisting of an associative ring R with unity l together with a subset E ⊆ R of elements, called eflects, with properties suggested by the so-called effect operators on a Hilbert space. Examples are given in which R is a unital C*-algebra, the ring of finite elements in an ordered field, the ring of continuous functions on a compact Hausdorff space, or the ring of measurable functions on a Borel space. We review the basic facts about e-rings and give a structure theorem for the case in which E satisfies the descending chain condition. Motivated by the notion of sequential observation of effects in quantum mechanics, we study the existence and uniqueness of square roots in an e-ring, we apply some of the same techniques to give conditions for the existence of multiplicative inverses, and we make contact with the theory of Jordan algebras.