Contraction elements in unital *-rings

Question

Let $A$ be a unital *-ring. Let us put, $A^+=\{\sum_1^N x^*_ix_i: x_i\in A , N\in \mathbb{N} \}$. We write $x\geq0$ if $x\in A^+$.

Suppose that for every $n\in \mathbb{N}$ and $a\in A$, there exsits $x\in A$ with $a=nx$.

Let us assume for natural numbers $m,n$ we have $mx^*x-n\geq0$. Can we conclude that $mxx^*-n\geq0$ as well?

Answer 1

Assume WLOG that $n,m>0 $ and note that $mx^*x-n\geq0$ iff $x^*x\geq n/m$ iff the spectrum $\sigma(x^*x)\subseteq[n/m,\infty)$ and since $\sigma(x^*x)$\{0}$=\sigma(xx^*)$\{0} it follows that $\sigma(xx^*)\subseteq [n/m,\infty)$ so $mxx^*-n\geq 0$.