Positive elements in * Algebras If one has a * Algebra $W$ one can define the notion of a positive element via the spectrum $\sigma$:
$A \in W$ is positive if $A^* = A$ and $\sigma(A) \subset \mathbb{R}_{≥0}$.
If $W$ can be given a norm so that it becomes a  C* Algebra then this is equivalent to the notion that there exists a $B \in W$ so that $A=B^* B$.
The proof of existence of such a $B$ that I have seen require completeness of $W$, and it doesn't seem unlikely that you can construct * Algebras where you have positive elements that are not decomposable in this way.
But is it true in a general * Algebra that if $A=B^*B$ that then $A$ is a positive element?
 A: The answer is no: $B^*B$ need not be positive, even if the algebra is unital and commutative and $B$ is self-adjoint. Consider the $\mathbb C$-algebra $\mathbb C[X]$ of polynomials in one indeterminate with coefficients in $\mathbb C$. We equip $\mathbb C[X]$ with involution $p \mapsto \overline{p}$, that is, complex conjugation of the coefficients. This turns $\mathbb C[X]$ into a unital $*$-algebra over the complex numbers. Note that the only invertible elements in $\mathbb C[X]$ are the non-zero constant polynomials. Now the degree one polynomial $f := X$ is self-adjoint, so we have $f^*f = X^2$. Note however that $X^2 - \lambda$ is not invertible for any choice of $\lambda \in \mathbb C$, so we have $\sigma(X^2) = \mathbb C$.
This example shows that spectral theory fails dramatically in the absence of a complete norm. Recall that the spectrum of an element in a Banach algebra is compact, so the above example shows that there is no Banach algebra norm on $\mathbb C[X]$.
There seems to be only one thing we can say with certainty: an element of the form $B^*B$ is always self-adjoint, since we have $(B^*B)^* = B^*B^{**} = B^*B$.
