equivalence of properties. Is the restriction in (ii) redundant?

Question

I have a question about the claim, which I found in a paper:

Let $\phi:A\to B$ a linear map between $C^*$-algebras $A$ and $B$. The following are equivalent:

1. $\phi(\frac{1}{2}(ab+ba))=\frac{1}{2}(\phi(a)\phi(b)+\phi(b)\phi(a))$ for all $a,b\in A$, and
2. $\phi(a^2)=\phi(a)^2$ for all $a\in A$ such that $0\le a\le 1$.

My question is, why the condition $0\le a\le 1$ in the second property is necessary. In my opinion it is true for all $a\in A$. Look at this question, Jordan-homomorphism; equivalent properties, the situation is very simliar.

Do you have an idea, why there is the restriction "$0\le a\le 1$" ?.