Quick computation of the Pontryagin dual group of torus

Question

I'm looking for a quick way to compute the Pontryagin dual group of the n-dimensional torus $\mathbb{T}^n$ (with $\mathbb{T} := \mathbb{R} / \mathbb{Z}$). The only way I know is from "Dikran Dikranjan - Introduction to Topological Groups" and is very tedious.

I thought so: $\mathbb{T}^n$ is compact then $\widehat{\mathbb{T}^n}$ has the discrete topology. The set $E$ of all the functions

$\mathbb{T}^n \to \mathbb{C} : x \mapsto e^{2 \pi a \cdot x}$

with $a \in \mathbb{Z}^d$ is a subset of $\widehat{\mathbb{T}^d}$ which separates the points of $\mathbb{T}^d$ (I mean: for any $x,y \in \mathbb{T}^d$ with $x \neq y$ exists some $\chi \in E$ such that $\chi(x) \neq \chi(y)$. Now I was hoping that Stone-Weierstass theorem could help me to prove that $E$ is dense in $\widehat{\mathbb{T}^d}$ and so that $\widehat{\mathbb{T}^d} = E$, because $\widehat{\mathbb{T}^d}$ is discrete. However I don't know how to apply Stone-Weierstass theorem in this case.

I thank you for any ideas!

Answer 1

The key case is the 1-dimensional torus. We want to show every continuous homomorphism $\chi \colon {\mathbf R}/{\mathbf Z} \rightarrow {\mathbf T}$ has the form $x \bmod {\mathbf Z} \mapsto e^{2\pi inx}$ for some integer $n$.

Any character of ${\mathbf R}/{\mathbf Z}$ can be pulled back to a character of ${\mathbf R}$ by composing with the canonical map ${\mathbf R} \rightarrow {\mathbf R}/{\mathbf Z}$. Let $\chi' \colon {\mathbf R} \rightarrow {\mathbf T}$ be the resulting character of ${\mathbf R}$, i.e., $\chi'(x) = \chi(x \bmod {\mathbf Z})$. All continuous homomorphisms ${\mathbf R} \rightarrow {\mathbf T}$ have the form $x \mapsto e^{2\pi ixy}$ for some real number $y$. Accepting this for a moment, we'd have $\chi'(x) = e^{2\pi ixy}$ for some $y$. Thus $\chi(x \bmod {\mathbf Z}) = e^{2\pi ixy}$. Taking $x = 1$, we get $1 = e^{2\pi iy}$. Therefore $y$ is an integer. Writing $y$ as $n$, for psychological purposes, $\chi(x \bmod {\mathbf Z}) = e^{2\pi inx}$ for an integer $n$, which is what you wanted.

Our task now is to compute the characters of ${\mathbf R}$. For this I like the method from Conway's "A Course in Functional Analysis". For any continuous homomorphism $\gamma \colon {\mathbf R} \rightarrow {\mathbf T}$, we want to show $\gamma(x) = e^{2\pi ixy}$ for some real number $y$. We will use differential equations. Because $\gamma(0) = 1$, by continuity $\int_0^a \gamma(t)dt \approx a$ for small positive $a$, so $\int_0^a\gamma(t)dt \not= 0$ for an appropriate $a$. Fixing such $a$, for all real $x$ we have $$\int_x^{x+a} \gamma(t)dt = \int_0^a \gamma(t+x)dt = \int_0^a \gamma(t)\gamma(x)dt = \gamma(x)\int_0^a \gamma(t)dt.$$ Therefore $$ \gamma(x) = \frac{\int_x^{x+a} \gamma(t)dt}{\int_0^a\gamma(t)dt}. $$ The right side, by the fundamental theorem of calculus, is a differentiable function of $x$, and therefore $$ \gamma'(x) = \frac{\gamma(x+a) - \gamma(x)}{\int_0^a\gamma(t)dt} = \frac{\gamma(a)-1}{\int_0^a\gamma(t)dt}\gamma(x), $$ so $\gamma'(x) = s\gamma(x)$ for a complex number $s$. By the theory of ODE, $\gamma(x) = Ce^{sx}$. Since $\gamma(0) = 1$, we have $C = 1$, so $\gamma(x) = e^{sx}$. Since $|\gamma(x)| = 1$, we have $e^{{\rm Re}(s)x} = 1$ for all real $x$, so ${\rm Re}(s) = 0$. Write $s = 2\pi iy$ for a real number $y$ and we're done.

I don't think you're going to find a 2-line proof of the computation of the character group of ${\mathbf R}/{\mathbf Z}$ or ${\mathbf R}$ because you need to get exponential functions from somewhere. Above we get them from our knowledge of solutions to the most basic first-order ODE, and to bring in an ODE we needed a clever idea to see that the a priori continuous function $\gamma$ is in fact differentiable.

Answer 2

You can read off a simple proof from Section VII.2 of Bourbaki's General Topology Part 2. Roughly, the proof goes as follows. Let $\mathbb{T}:=\mathbb{R}/\mathbb{Z}$, then any continuous homomorphism $\mathbb{T}^m\to\mathbb{T}^n$ lifts to a continuous homomorphism $\mathbb{R}^m\to\mathbb R^n$ which then is linear and maps $\mathbb{Z}^m$ into $\mathbb{Z}^n$.

Answer 3

If your purpose of calculating the dual of the torus is to have the Fourier inversion formula, I just want to add the remark that you don't need to go through all the machinery of locally compact topological groups in order to prove the basic results of Fourier analysis.

For the circle, one needs only note that the distribution $u(x) = \sum_{n \in {\mathbb Z}} e^{2 \pi i n x}$ on the torus ${\mathbb R}/{\mathbb Z}$ is invariant under multiplication by $e^{2 \pi i x}$, and is therefore a multiple of the delta-function $\delta(x)$ -- integrating against the smooth function $1$, you see that, in fact, $u(x) = \delta(x)$. (This move is simply how you usually calculate the "sum of a geometric series".) Substituting into the general formula $f(x) = \int_{{\mathbb R}/{\mathbb Z}} f(y) \delta(x - y) dy$ gives the Fourier inversion formula.

The above is really just for a conceptual point of view, since it's unnecessary to quote distribution theory for this purpose. To give a direct proof, suppose $f(x)$ is smooth; after translating, it is enough to show that $f(0) = \sum_{n \in {\mathbb Z}} \hat{f}(n) = \sum_n \int f(x) e^{- 2 \pi i n x} dx$ (which converges absolutely because you can integrate by parts enough times to prove decay of $\hat{f}(n)$). We can assume without loss of generality that $f(0) = 0$ by subtracting the constant $f(0)$ (this is how we know "the constant" in the inversion formula is correct and corresponds to testing against $1$ in the previous argument).

In the case $f(0) = 0$, we can write $f(x) = (e^{2 \pi i x} - 1) g(x)$ for some smooth function $g(x)$ (which proves the previous claim that if $e^{2 \pi i x} u = u$ then $u(x) = C \delta(x)$). But then,

$\sum_{n \in {\mathbb Z}} \hat{f}(n) = \sum_{n \in {\mathbb Z}} (\hat{g}(n - 1) - \hat{g}(n))$

but this is $0$ because $\hat{g}$ is integrable. (In particular, assuming $f$ to be $C^\infty$ was certainly unnecessary, and the Fourier inversion formula holds pointwise for functions in a certain class.)

If you like to think in terms of Stone Weierstrass, you can view this argument is saying that the map $Tf$ which takes a Fourier transform and then performs the inverse Fourier transform is the identity. It relies on the fact that $f(x) = (e^{2 \pi i x} - e^{2 \pi i x_0}) g(x)$ whenever $f$ vanishes at $x_0$ which is a strong sense in which trigonometric polynomials separate points. The conclusion that $T$ is a multiple of the identity then follows from $T$ being linear not only over ${\mathbb C}$ but also over the algebra of trigonometric polynomials.

Another remark about distribution theory here is that for the argument of KConrad above, the equation $\gamma'(x + s) = \gamma'(s) \gamma(x)$'' already makes sense when $\gamma$ is a distribution (in particular, if $\gamma$ is continuous). The way it makes sense is by integrating by parts against a test function, which is related to the integration performed the accepted proof. One can use a very similar argument to KConrad's and show that the only distributions solving $\gamma(x + y) = \gamma(x)\gamma(y)$ are exponentials as I'll (almost) show below.

Finally, observe that the fact that the Fourier inversion formula holds for $L^2$ functions implies in particular that all the characters have been accounted for. After all, if you had any other character, it would be orthogonal to all the exponential functions and hence have $\hat{\gamma} = 0$. However, this argument requires not only that the Fourier transform is an isometry on smooth functions in $L^2$ (a consequence of the inversion formula), but one also has to argue that $L^2$ functions can be approximated by smooth functions, which is typically done by mollification $f(x) \approx \int f(x + h) \eta(h) dh$ for a smooth function $\eta$ of integral $\int \eta(h) dh = 1$ and small support. The same kinds of mollifications or similar calculations are used in proofs of Stone Weierstrass theorem; by a similar reasoning, once you have enough characters to separate points, you know you have a complete set.

But, as KConrad showed with $\eta$ the characteristic function of a short interval, when $f = \gamma$ is a character, mollification shows $\int \gamma(x + h) \eta(h) dh = \gamma(x) \int \gamma(h) \eta(h) dh$, which implies in particular that $\gamma(x)$ is smooth because $\int_{{\mathbb R}/{\mathbb Z}} \gamma(x +h) \eta(h) dh = \int_{{\mathbb R}/{\mathbb Z}} \gamma(h) \eta(h - x) dh$ is always smooth (even when $\gamma$ is just a distribution).

But then once you have established that $\gamma$ is smooth, the same equation $\gamma(x) \gamma(y) = \gamma(x+y)$ which proves the orthogonality of characters can also be differentiated classically and evaluated at $y = 0$, and then (as KConrad noted) you just need to solve the ODE that defines the exponential functions. So I'll end here now that we have "come full circle".

Answer 4

You can "almost" prove that rather abstractly. The dual group (this is a group because the torus is abelian) is discrete. Moreover, because the torus is connected, the dual group is torsion-free. Hence the dual group is isomorphic to a product of $\mathbb{Z}$.

The only problem here is to limit the rank of this group to be 1. Obviously, this has something to do with the geometry and the topology of $T$ (such as the fact the group is monothetic, or moving into infinitesimal equivalence as in Keith's answer, which can be thought as some Lie theoretic framework), otherwise, the exact argument will work for say $\mathbb{T}^{n}$ for any $n$.

Another possibility here (which one might call cheating) is to use Peter-Weyl theorem (because $\mathbb{T}$ is compact, you don't need the full proof of the Pontryagin duality here). Then out of the blue, recalling Poincaré's inequality and get that the eig. functions of the Laplacian will be a basis to $L^{2}(T)$, and then find them and see that "magically" they are happen to be characters. This gives you an easy differential eq. with boundary conditions which you can solve explicitly.