A linear form $f$ on an involutive algebra $A$ is said to be positive if $f(x^\ast x)\geq 0$ for every $x$ in $A$.
To be useful, this definition requires that is not always possible to write $-(x^\ast x)=y^\ast y$ for some y.
How can I prove this fact?
In such generality, what you want is not true. You can take $A=\mathbb C$, and take the trivial involution $z^*=z$. Then $$ -(2i)^*2i=2^*2. $$