Let si be a C*-algebra acting on the Hilbert space H and let SP be the self-adjoint elements of si. The following characterization of commutativity is due to I. Kaplansky (see Dixmier [3, p. 58] characterize commutativity for si in terms of the usual order structure on £f. We show that Kaplansky's theorem reduces the proofs of these order characterizations to simple computations.
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