While attempting to fill in the gaps in a proof of the Gelfand-Naimark-Segal representation theorem that I was given in a course in operator algebras, I found myself wondering whether, if $(u_\lambda)_\lambda$ is an approximate unit of a C*-algebra $\mathcal{A}$, it is true that so is $(u_\lambda^2)_\lambda$. I need this fact to complete my version of the proof of existence of a cyclic vector for the GNS representation of a non-unital C*-algebra.
Clearly, $(u_\lambda^2)_\lambda$ is a net of positive elements, bounded in norm by 1. But how to show that this net is increasing, knowing that $(u_\lambda)_\lambda$ is?. I wouldn't even know how to start proving something like this for an arbitrary approximate unit, but it's enough for my proof to show it for the canonical approximate unit of $\mathcal{A}$ with the usual order.
So this amounts to showing (I cannot find a counterexample) that if $u_\lambda \leq u_{\lambda'}$ then $u_\lambda^2 \leq u_{\lambda'}^2$. More generally, one can ask if $a \leq b$ implies $a^2 \leq b^2$, or even $a^n \leq b^n$.
As I said I'm a bit stuck with this. Thanks for your help!