Variations on Pontryagin Duality
Posted by John Baez
I’m slaving away on a paper about infinite-dimensional representations of 2-groups, which needs to be done in time for the thesis defense of Aristide Baratin — a coauthor who is due to finish grad school very soon.
This paper has forced me to brush up on my analysis — a nostalgic if slightly painful experience. I’ll probably explain some of what I’ve learned in This Week’s Finds. But to finish it, we still need the answers to some questions related to Pontryagin duality.
Help!
Pontryagin duality goes like this. Suppose $A$ is a locally compact Hausdorff topological abelian group. Let $A^*$ be the set of characters: that is, continuous homomorphisms $f : A \to \mathrm{U}(1)$. $A^*$ becomes an abelian group thanks to pointwise multiplication of characters. It becomes a topological group with the compact-open topology — that is, the topology of uniform convergence on compact sets. We call $A^*$ the Pontryagin dual of $A$.
Then, $A^*$ is again a locally compact Hausdorff topological abelian group, and
$A^{**} \cong A$
in a natural way!
For example, we have
$\mathbb{Z}^* \cong \mathrm{U}(1)$
and
$\mathrm{U}(1)^* \cong \mathbb{Z}$
$\mathbb{R}$ is its own dual! More generally, for any finite-dimensional real vector space $V$ with its usual topology, $V^*$ is the same as the dual vector space. So, Pontryagin duality generalizes vector space duality.
My questions concern a variation on this theme. Suppose $A$ is an abelian locally compact Hausdorff topological group, and let
$A' = hom(A,\mathbb{C}^*)$
be the set of continuous homomorphisms from $A$ to the group $\mathbb{C}^*$ consisting of invertible complex numbers. $A'$ is an abelian group, and it becomes a topological group with the compact-open topology.
Question 1: Is $A'$ again locally compact and Hausdorff?
Question 2: if $A$ is second countable, is $A'$?
(I don’t even know the answer to this second question for $A^*$, but I suspect it might be in that case, since Wikipedia says “The foundations for the theory of locally compact abelian groups and their duality were laid down by Lev Semenovich Pontryagin in 1934. His treatment relied on the group being second-countable and either compact or discrete.” It’s well-known that the Pontryagin dual of a compact group is discrete, and vice versa. Does duality preserve second countability? I really need this result for $A'$, but I’ll take what I can get.)
The point of the second question might be clearer after studying Mackey’s classic book:
- G. W. Mackey, Unitary Group Representations in Physics, Probability and Number Theory, Benjamin–Cummings, New York, 1978.
He makes a good case for working with second-countable locally compact Hausdorff groups when you’re studying unitary representations: for example, these groups have $\sigma$-finite Haar measures. So, it would be nice if Pontryagin duality mapped groups of this sort to other groups of this sort.
But since we’re thinking about non-unitary representations and their categorified kin, we need to think about $A'$ instead of $A^*$.
Re: Variations on Pontryagin Duality
I can tell you that the answer to the first question is “in general, no”. Let
$A = \sum_{\mathbb{N}} \mathbb{Z},$
a coproduct of countably many copies of $\mathbb{Z}$ with the discrete topology. Then
$\hom(\sum_{\mathbb{N}} \mathbb{Z}, \mathbb{C}^*) \cong \prod_{\mathbb{N}}\hom(\mathbb{Z}, \mathbb{R} \times S^1) \cong \mathbb{R}^{\mathbb{N}} \times (S^1)^{\mathbb{N}}$
but $\mathbb{R}^{\mathbb{N}}$ is not locally compact Hausdorff. (Notice also that $A$ in this case is countable as a set and therefore second countable.)
Unless I’m making a stupid mistake, I think the answer to question (1) is “yes” in the case where $A$ is connected. Since $\mathbb{C}^* \cong \mathbb{R} \times S^1$, we really just have to worry about the factor $\mathbb{R}$. If $A$ is connected, then we should have an injection
$\hom(A, p): \hom(A, \mathbb{R}) \to \hom(A, S^1)$
where $p: \mathbb{R} \to S^1$ is the usual covering projection, realizing $\hom(A, \mathbb{R})$ as a closed subgroup of its Pontryagin dual.