Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question
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. $$a = \pmatrix{1 & -1\cr -1 & 1\cr},\ b = \pmatrix{ 2 & -3\cr -3 & 5\cr}$$

share|cite|improve this answer
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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.