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.
The following remarkable theorem was recently proved by Sinclair [3], THEOREM.The spectral radius of a Hermitian element of a complex unital Banach algebra is equal to its norm.Our purpose is to give an elementary proof of Sinclair's theorem. This will make available an elementary proof of the Vidav-Palmer characterization of C*-algebras [4,2].Let A be a complex unital Banach algebra. The numerical range V(a) of an element a e A is given by V(a) = {f(a): / e D(l)}, where D(\) denotes the set of all continuous linear functionals f on A such that
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