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Let $I\subset A$ be a ideal of a Banach algebra $A$. Assume $p\in \overline I$ and $p^2=p$.

Show: $p\in I$.

Can someone give me a little hint how to solve this, please?

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  • $\begingroup$ Being complete is the only hypothesis you have on $A$ ? How is its norm ? $\endgroup$
    – Olórin
    Dec 23, 2014 at 10:55
  • $\begingroup$ Well, I have that $A$ is a Banach algebra. What do you mean by "How is its norm?" ? $\endgroup$
    – Peter
    Dec 23, 2014 at 11:12
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    $\begingroup$ There are Banach algebras over anything, over any Banach ring in fact, like $\mathbf{Z}$ or $\mathbf{Z}_p$ ($p$-adic integers) or $\mathbf{Q}(( T ))$, or, why not, over $\mathbf{F}_p$. This wikipedia's article is not accurate at all. $\endgroup$
    – Olórin
    Dec 23, 2014 at 11:45
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    $\begingroup$ Okay, sorry. Since I haven't heard of these algebras so far, I thought my language is clear. Is it at least precise enough now? $\endgroup$
    – Peter
    Dec 23, 2014 at 11:50
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    $\begingroup$ Yeah, no worries ! Now it's clear. $\endgroup$
    – Olórin
    Dec 23, 2014 at 11:50

1 Answer 1

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This is recorded as Lemma 2.7 in

Niels Jakob Laustsen, Thomas Schlumprecht, and András Zsák, The lattice of closed ideals in the Banach algebra of operators on a certain dual Banach space, J. Operator Theory 56 (2006), no. 2, 391–402.

I assume that $A$ is unital, if not, we can pass to the unitisation.

Suppose that $(t_n)_{n=1}^\infty$ is a sequence in $I$ that converges to an idempotent $p$. Since $pAp$ is a Banach algebra with identity $p$ and $pt_np \to p^3=p$, there exists $n$ so that $pt_np$ is invertible in $pAp$. Consequently, $p = (psp)(pt_np)$ for some $s\in A$. This proves that $p\in I$.

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