On a conjecture of Kaplansky
Shôichirô Sakai · Tohoku Mathematical Journal · 1960
Prof. Kaplansky stated a conjecture that any derivation of a C*-algebra would be automatically continuous [1].In this note, we shall show that this conjecture is in fact true. THEOREM. Any derivation of a C*-algebra is automatically continuous.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 A s of A. Therefore if it is not continuous, by the closed graph theorem there is a sequence \x n \ (x n 4= 0) in A s such that x n -»0 and x n -> 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 { -x n }).It is enough to assume that λ = 1.