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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!

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Doesn't this follow from $$\Vert xu_{\lambda}^2 - x \Vert \le \Vert xu_{\lambda}^2 - xu_\lambda \Vert + \Vert xu_\lambda - x\Vert = \Vert (xu_{\lambda} - x)u_\lambda \Vert + \Vert xu_\lambda - x\Vert \le \Vert xu_{\lambda} - x \Vert(\Vert u_\lambda\Vert + 1)$$ ? I mean $u_\lambda^2$ being an approximate unit. – Sam May 4 '12 at 19:45
Ah yes sure, but I could do that part :) What I am stuck with is showing that the net is increasing! – Umberto Lupo May 4 '12 at 19:49
The problem arose because I would like it to be true that for a positive linear functional $\tau$, $\lim_\lambda \tau(u_\lambda^2) = \| \tau \|$ (and a bounded positive linear functional has $\lim_{\lambda} \tau(u_\lambda) = \| \tau \|$ for any approximate unit $(u_\lambda)_\lambda$). – Umberto Lupo May 4 '12 at 19:59

The general statement ($0 \le a \le b$ implies $a^2 \le b^2$) is false, even for $2 \times 2$ matrices. Consider e.g. $$a = \pmatrix{1 & -1\cr -1 & 1\cr},\ b = \pmatrix{ 2 & -3\cr -3 & 5\cr}$$

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Thank you Robert. I am going to have to change my argument then, somehow! – Umberto Lupo May 4 '12 at 20:49

It is true in a commutative unital C* algebra because you use the gel'fand transform to relate the abstract notion of positivity to one on functions mapping from the maximal ideal space into C, and this is a completely concrete and usual thing that is easy to deal with.

Even more in [1] it is proved that a $C^*$-algebra is commutative iff this implication is always true.

[1] T. Ogasawara, A Theorem on Operator Algebras, Journal of Science of The Hiroshima University, Ser A, vol 18, No. 3, 1955, pp. 307-309.

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