I asked this question in [stackexchange][1], but it flashed and disappeared:

Let $A$ be a C*-algebra and $\exp(x)=\sum_{n=0}\frac{x^n}{n!}$, the usual exponential function from $A$ into $A$. Is it true that if $x\ne y\in A$, $x^*=x$, $y^*=y$, then $\exp(x)\ne\exp(y)$?


  [1]: https://math.stackexchange.com/questions/993861/is-exponential-function-in-a-c-algebra-injective-on-self-adjoint-elements