I have the following lemma 4.2.4 on page 80 in the book (we have locally compact abelian topological groups $G_1, G_2$ and their duals $\Gamma_1, \Gamma_2$):

Suppose $E$ is a coset in $\Gamma_2$ and $\alpha$ is an affine map of $E$ into $\Gamma_1$. THen $\alpha$ can be extended to an affine map of the closure $\overline{E}$ of $E$, and $\alpha(\overline{E})$ is a closed coset in $\Gamma_1$

I think $\alpha(\overline{E})$ is not necessary a closed coset in $\Gamma_1$. The book just states that it is a corollary of the uniform continuity of $\alpha$ but consider $E = \Gamma_2 = \mathbb{Q}$ with discrete topology, $\Gamma_1 = \mathbb{R}$ and $\alpha$ is identity map. In that case, $\overline{E} = E$ and the image is not closed in $\mathbb{R}$.

Edit: Since nobody proved the statement or pointed out where my example is wrong I'm going to assume the lemma is indeed incorrect.

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    $\begingroup$ What locally compact group $G_2$ has its dual group isomorphic to $\Gamma_2 = \mathbb{Q}$ with the discrete topology? $\endgroup$ Feb 28, 2016 at 19:24
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    $\begingroup$ @NateEldredge $G_2$ is the dual of $\Gamma_2$, by the Pontrjagin duality. $\endgroup$ Feb 28, 2016 at 19:33
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    $\begingroup$ @NateEldredge Is this question rhetorical? Every discrete abelian group has a compact Pontrjagin dual. Without thinking much, I imagine that the dual of ${\mathbb Q}_d$ has some kind of solenoidal flavour (it certainly isn't anything like compact Lie, of course) $\endgroup$
    – Yemon Choi
    Feb 28, 2016 at 19:44
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    $\begingroup$ @YemonChoi: No, it wasn't rhetorical, just naive :-) $\endgroup$ Feb 28, 2016 at 19:52
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    $\begingroup$ The dual of $\mathbb Q$ is indeed a solenoid. You might call it "the" solenoid because it is the biggest, but that name is usually taken by the dual of $\mathbb Z[\frac12]$. Another name for the dual of $\mathbb Q$ is the adele quotient: $\mathbb A/\mathbb Q$. Similarly, the binary solenoid is $(\mathbb R\times \mathbb Q_2)/\mathbb Z[\frac12]$. $\endgroup$ Feb 28, 2016 at 20:47

1 Answer 1


[This is just moving my comment to an answer so that the question does not keep showing up as unanswered.]

The discussions above seem to demonstrate that the OP is correct, and that this part of the lemma is incorrect. According to my Canadian sources, it seems that this error has been noticed tacitly by other people: see the remarks before and after Lemma 1.3 in:

M. Ilie, N. Spronk. Completely bounded homomorphisms of the Fourier algebras, J. Funct. Anal. 225 (2005) no. 2, 480–499


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