Published online by Cambridge University Press: 18 May 2009
Two numerical characterizations of commutativity for C*-algebra (acting on the Hilbert space H) were given in [1]; one used the norms of self-adjoint operators in
(Theorem 2), and the other the numerical index of
(Theorem 3). In both cases the proofs were based on the result of Kaplansky which states that if the only nilpotent operator in
is 0, then
is commutative ([2] 2.12.21, p. 68). Of course the converse also holds.