On a Conjecture of Kaplansky

PROOF. Let A be a C*-algebra, ' a derivation of A. It is enough to show that the derivation is continuous on the self-adjoint portion As of A. Therefore if it is not continuous, by the closed graph theorem there is a sequence \xn\ (xn 4= 0) in As such that xn -»0 and xn -> a + ίέ(φθ), where a and b are self-adjoint. First, suppose that a =f= 0 and there exists a positive number λ(> 0) in the spectrum of a (otherwise consider { —xn}). It is enough to assume that λ = 1. Then there is a positive element Λ(||/*|| = 1) of A such that hah S — h 2 . Put yn = xn + 3 \\xn\\ 7, then yn -> 0, yn = xn and (hyji)' = tiyji + hyn h + hynh' hence (hynh)' —> h(a + ib)h. Therefore