In a $C^\*$-algebra ${\cal A}$, a **positive** element is a one of the form $aa^\*$, for some $a \in {\cal A}$. It is known that the set of positive elements is a [cone](https://en.wikipedia.org/wiki/Cone_%28linear_algebra%29), and that for $a,b$ two non-zero elements, and $\lambda \in {\bf R}\_{>0}$, we have $a + \lambda b \neq 0$. 


If we generalize to $\ast$-algebras, do these facts still hold: is the set of positive elements still a cone, and are sums of the form $a + \lambda b$ still non-zero. 



Positive Elements of a $\ast$-Algebra
The answer to the second part is also no. That is, $aa^\*+bb^\*$ could be zero for nonzero $a$ and $b$. Look at the $\*$-algebra $\mathbb{C}^2$ with component-wise addition and multiplcation and $\*$ given by $(x,y)^\* = (\overline{y},\overline{x})$. It's straightforward to calculate that anything of the form $aa^\*$ will look like $(z,\overline{z})$ for some $z\in\mathbb{C}$. Conversely, $(z,\overline{z})=(z,1)(z,1)^\*$. We see $(z,\overline{z})$ and $(w,\overline{w})$ sum to $(z+w, \overline{z+w})$, so $aa^\*+bb^\*$ is still of the form $cc^\*$. But taking $w=-z$ gives a sum of zero.


