# On a Conjecture of Kaplansky

```@inproceedings{Sakai2007OnAC,
title={On a Conjecture of Kaplansky},
author={Sh{\^o}ichir{\^o} Sakai},
year={2007}
}```
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… CONTINUE READING
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