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Order and commutativity in C*-algebras

In this paper, we use the results of (3) to show that a condition which is formally much weaker than that given by Sherman in (11) is necessary and sufficient for the commutativity of a C*-algebra. The basic idea behind the proof is then used again to obtain a characterization in order-theoretic terms of the self-adjoint elements in the centre of a C*-algebra. We shall use the notation and terminology of (3).