Non-negative observables are squares

For [S],1 whose notation and results we use without further comment, the question of whether or not a non-negative observable is the square of an observable is left open. Our positive solution of this question eliminates one of the hypotheses in a pair of propositions [S,Theorem4 and Corollary4.1 1 in that paperand so nowwe knowthat without exception a pure state of a subsystem may be extended to a pure state of the whole system of observables and that the spectral value of an observable equals the value of the observable in some pure state of the system. This serves to shrink the gap between S, the real linear space of self-adjoint elements of a C* algebra, and 2f, a system of observables. Although every S is isomorphic to an 2X, no examples of an 21 not isomorphic to an 5 have appeared in the literature, so the gap may be just an apparent one.