Let $A$ be a finite-dimensional $\*$-algebra over $\mathbb R$. We say that an element $x \in A$ is *positive definite* if $x$ admits an inverse and if $x = y y^\*$ for some $y \in A$. Does every such $x$ admit a $z \in A$ such that $x = z^2$? If someone has references for me to read, I would appreciate that.


I think the Wedderburn-Malcev theorem can help here, as well as Hermite interpolation. But non-commutativity seems like an obstacle.



Do positive-definite elements in finite-dimensional $*$-algebras over $\mathbb R$ always admit square roots?
I'll assume that a $\*$-algebra just means an $\mathbb{R}$-algebra with a linear operation satisfying $(xy)^\*=y^\*x^\*$ and $x^{\*\*}=x$ (as at <https://ncatlab.org/nlab/show/star-algebra>). If so, you can just take $A=\mathbb{R}\times\mathbb{R}$ with $(x\_0,x\_1)^\*=(x\_1,x\_0)$. Then take $x=(-1,-1)$ and $y=(1,-1)$ so $x=yy^\*$ and $x$ is invertible but $x$ has no square root. If you want any more axioms then you should specify them.


