Variations on Pontryagin Duality

November 8, 2008

Posted by John Baez

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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!

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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 $f : A \to U (1)$ . $A^{*}$ 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

$A^{* *} ≅ A$

in a natural way!

For example, we have

$ℤ^{*} ≅ U (1)$

and

$U (1)^{*} ≅ ℤ$

$ℝ$ 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

$A' = hom (A, ℂ^{*})$

be the set of continuous homomorphisms from $A$ to the group $ℂ^{*}$ 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 $σ$ -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 $A^{*}$ .

Posted at November 8, 2008 7:32 AM UTC

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Re: Variations on Pontryagin Duality

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I can tell you that the answer to the first question is “in general, no”. Let

A = \sum_{ℕ} ℤ,

a coproduct of countably many copies of $ℤ$ with the discrete topology. Then

hom (\sum_{ℕ} ℤ, ℂ^{*}) ≅ \prod_{ℕ} hom (ℤ, ℝ \times S^{1}) ≅ ℝ^{ℕ} \times (S^{1})^{ℕ}

but $ℝ^{ℕ}$ 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 $ℂ^{*} ≅ ℝ \times S^{1}$ , we really just have to worry about the factor $ℝ$ . If $A$ is connected, then we should have an injection

hom (A, p) : hom (A, ℝ) \to hom (A, S^{1})

where $p : ℝ \to S^{1}$ is the usual covering projection, realizing $hom (A, ℝ)$ as a closed subgroup of its Pontryagin dual.

Posted by: Todd Trimble on November 8, 2008 1:53 PM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

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Thanks! I knew that $ℂ^{*} ≅ U (1) \times ℝ$ let me reduce my puzzles to one question about ordinary Pontryagin duality:

Question 2a: If $A$ is a second-countable locally compact Hausdorff topological group, is $A^{*}$ also second-countable?

together with two more:

Question 1a: If $A$ is a locally compact Hausdorff topological group, is $hom (A, ℝ)$ also locally compact Hausdorff?

Question 1b: If $A$ is a second-countable locally compact Hausdorff topological group, is $hom (A, ℝ)$ also second-countable?

Planetmath assures me the answer to question 2a is yes — but without a reference to a proof. But I didn’t think of looking for a counterexample to part 1a. So thanks: ithe answer is no.

(It’s a bit sad that locally compact Hausdorff abelian groups don’t form a closed category, given that $U (1)$ seems to want to be a dualizing object.)

If $A$ is connected, then we should have an injection

$hom (A, p) : hom (A, ℝ) \to hom (A, S^{1})$

where $p : ℝ \to S^{1}$ is the usual covering projection, realizing $hom (A, ℝ)$ as a closed subgroup of its Pontryagin dual.

Nice! I guess $hom (A, p)$ should be an injection because a continuous homomorphism from a connected topological group to a topological group should be determined by its values on an open neighborhood of the identity. This seems to be true without any conditions: when we know a continuous homomorphism on an open neighborhood of the identity, we know it on the open subgroup it generates, and an open subgroup is also closed.

(I’m stating my reasoning in hopes that someone will point out flaws if there are any.)

Posted by: John Baez on November 8, 2008 4:25 PM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

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I think the answer to (2a) is yes. The argument I concocted may be unnecessarily complicated, but here it is:

A regular second countable space $A$ is metrizable (Urysohn metrization theorem), and if it is also locally compact, then it is $σ$ -compact (easy exercise).
A locally compact Hausdorff abelian group $A$ is metrizable iff $A^{*}$ is $σ$ -compact (source: Sidney Morris’s Lectures on Pontryagin Duality, theorem 29).
By combining 1. and 2., $A^{*}$ is also $σ$ -compact and metrizable, i.e., is a countable union of compact metric subspaces. Since a compact metric space is second countable [it admits finite coverings by $(1 / n)$ -balls for each $n$ ], so is a $σ$ -compact metric space.

Posted by: Todd Trimble on November 8, 2008 5:04 PM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

Nice! Thanks again, Todd.

Posted by: John Baez on November 9, 2008 1:23 AM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

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It turns out my hunch was right: Pontryagin’s original work on duality only considered locally compact Hausdorff abelian groups $A$ that were second-countable. He proved that if $A$ satisfies these conditions, so does its dual $A^{*}$ . It’s Theorem 31 on page 128 of the 1946 edition of his book Topological Groups.

His proof is not particularly elegant: it even uses the interval $[- \frac{1}{10}, \frac{1}{10}]$ .

But, he’s the man with the original vision! So, I’ll cite him for this result.

Posted by: John Baez on November 11, 2008 7:28 PM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

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Here’s a more direct proof of 2a: the Fourier transform is a continuous homomorphism from the convolution algebra

L^{1} (A)

to the algebra

C_{0} (A^{*})

of continuous functions vanishing at infinity, and the image of this homomorphism is dense. If

A

is second-countable, then

L^{1} (A)

is separable, and hence so is

C_{0} (A^{*})

. But separability of

C_{0} (A^{*})

clearly implies secound-countability of

A^{*}

, since inverse images of rational balls under a countable dense subset of

C_{0} (A^{*})

form a basis.

Posted by: Eric on November 14, 2008 3:32 AM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

Clever!

Posted by: Todd Trimble on November 14, 2008 2:05 PM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

Since Todd mentions countable sums of copies of the integers, readers might find the following paper of interest:
15. ( R. BROWN. P.J. HIGGINS and S.A. MORRIS), “Countable products of lines and circles: their closed subgroups, quotients and duality properties”, {\em Math. Proc. Camb. Phil. Soc.} 78 (1975) 19-32.

In general duality is not inherited by subgroup/quotient group pairs. This paper also contains a notion of strong duality which is inherited.

Ronnie

Posted by: Ronnie Brown on November 11, 2008 3:35 PM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

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I’m beginning to want to compile a table of theorems of this form:

If the locally compact Hausdorff topological group $A$ has property $P$ , then its Pontryagin dual $A^{*}$ has property $P^{*}$ .

I mentioned the most famous ones:

If $A$ is compact, $A^{*}$ is discrete.

If $A$ is discrete, $A^{*}$ is compact.

and of course there’s this:

If $A$ is finite, $A^{*}$ is finite.

I’ve bumped into a few more:

If $A$ is torsion-free and discrete, $A^{*}$ is connected and compact.

If $A$ is connected and compact, $A^{*}$ is torsion-free and discrete.

If $A$ is a Lie group, $A^{*}$ has finite rank.

If $A$ has finite rank, $A^{*}$ is a Lie group.

If $A$ is second countable, $A^{*}$ is second countable.

If $A$ is separable, $A^{*}$ is metrizable.

To add to my collection, I should get ahold of these books:

Lev Semenovich Pontryagin, Topological Groups, Gordon & Breach, 1966.
Sidney A. Morris, Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Math. Soc. Lecture Notes 29, Cambridge U. Press, 1977.

Posted by: John Baez on November 8, 2008 4:44 PM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

By the way, as I mentioned here Ulrich Bunke has the notion and applications for Pontryagin duality of higher categorical groups. He says that as far as the gerbes involved are concerned, T-duality is just Pontryagin duality of higher groups.

See

Bunke, Schick, Spitzweck, Thom, Duality for topological abelian group stacks and T-duality

Abstract: We extend Pontrjagin duality from topological abelian groups to certain locally compact group stacks. […] As an application of the theory we interpret topological T-duality of principal Tⁿ-bundles in terms of Pontrjagin duality of abelian group stacks.

Posted by: Urs Schreiber on November 8, 2008 9:38 PM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

So, are any equivalents of John’s list known in this higher setting?

Posted by: David Corfield on November 10, 2008 8:47 AM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

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You say Bunke uses $Σ^{2} ℤ$ as duality object. But isn’t that just categorifying relative to the first part of the product:

$hom (A, ℂ^{*}) = hom (A, U (1)) \times (A, ℝ) ?$

Posted by: David Corfield on November 10, 2008 8:53 AM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

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So, are any equivalents of John’s list known in this higher setting?

I have to admit that I did not study their article in detail, so I can’t tell off the top of my head.

But isn’t that just categorifying relative to the first part of the product:

That may be. In their application one of the main points was that the higher Pontryagin duality also induces duality on categorical degrees.

For instance

$Hom (B^{2} ℤ, B^{2} ℤ) ≃ ℤ$

$Hom (B ℤ, B^{2} ℤ) ≃ B ℤ$

if I am not mistaken. (They switch between $B ℤ$ and $U (1)$ a bit, which makes the pattern less obvious.)

Posted by: Urs Schreiber on November 10, 2008 1:55 PM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

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Todd showed that even if the abelian group $A$ is locally compact, the abelian group

$A' = hom (A, ℂ^{*})$

may not be. His example was an infinite coproduct of copies of $ℤ$ :

$A = ⨁_{n \in ℕ} ℤ$

Yves de Cornulier has allowed me to share an email in which he argues that in a certain nice sense, this group $⨁_{ℕ} ℤ$ is the ‘only obstacle’ to $A'$ being locally compact. There are aspects of his argument that I don’t understand, but it seems promising.

Here is his email after a bit of LaTeXing:

$A'$ is not necessarily locally compact. E.g, this fails with $⨁_{ℕ} ℤ$ , an infinite direct sum of copies of $ℤ$ (see below). I think that this is essentially the only counterexample: if a locally compact abelian group does not have a discrete quotient of infinite rank (i.e. containing $⨁_{ℕ} ℤ$ as a subgroup), then $A'$ is locally compact.

Here is how I see this.

As a topological group, $ℂ^{*}$ is nothing else than $ℝ \times U (1)$ , $ℝ$ being the reals. So

$hom (A, ℂ^{*}) = hom (A, U (1)) \times hom (A, ℝ),$

and the discussion boils down to the study of $hom (A, ℝ)$ .

Now $A$ has a compact subgroup $K$ such that $A / K$ is a Lie group (maybe with infinitely many components). We can pick $K$ so that the unit component of $A / K$ is $ℝ^{d}$ . As

$hom (A, ℝ) = hom (A / K, ℝ),$

we can reduce to the study of $A / K$ , so I suppose $A = A / K$ .

As $ℝ^{d}$ is a divisible abelian group, it has a direct factor $B$ in $A$ (viewed as a discrete group), but as $B \cap ℝ^{d}$ is trivial $B$ is actually discrete in the topological group $A$ . Again, since morphisms to $ℝ$ vanish on the torsion, we can suppose that $B$ is torsion-free.

Now suppose that $B$ has finite rank: it contains a subgroup isomorphic to $ℤ^{k}$ such that the natural restriction morphism

$hom (B, ℝ) \to hom (ℤ^{k}, ℝ)$

is an isomorphism (actually of topological groups). Now $hom (ℤ^{k}, ℝ)$ is locally compact and second countable, so $hom (A, R)$ is so as well.

On the other hand, suppose that $B$ has infinite rank. Let $V$ be a neighbourhood of 0 in $hom (B, ℝ)$ : it contains, for some compact(=finite) subset $F$ of $B$ and some $r > 0$ , the set $V (K, B (r))$ of homomorphisms $B \to ℝ$ mapping $F$ into the $r$ -ball $B (r)$ . I can find, in this $V (K, B (r))$ , a sequence with no cluster point. Namely, since $B$ has infinite rank, there exists $y$ in $B$ such that the cyclic subgroup generated by $y$ has trivial intersection with the subgroup generated by $K$ . So there exists a homomorphism $u_{n} : B \to ℤ$ mapping $K$ to zero and $y$ to $n$ . Clearly, this has no cluster point, but it lies in $V (K, B (r))$ .

So we get the equivalences:

$hom (A, ℂ^{*})$ is locally compact $\Leftrightarrow$

$hom (A, ℝ)$ is locally compact $\Leftrightarrow$

$hom (A, ℤ)$ is locally compact $\Leftrightarrow$

$⨁_{ℕ} ℤ$ is not a subgroup of a discrete quotient of A $\Leftrightarrow$

$⨁_{ℕ} ℤ$ is not a discrete subgroup of A.

In these cases, $hom (A, ℝ)$ and $hom (A, ℤ)$ are also 2nd-countable.

The other question should reduce to the study of whether $hom (⨁_{ℕ} ℤ, R)$ is 2nd-countable, which shouldn’t be a big deal.

In response to a question, he added:

Note that by Lie group I don’t assume connectedness. So any discrete group is a Lie group, for instance.

It’s well known (and not trivial) that any locally compact abelian group $A$ has a compact subgroup $K$ such that $A / K$ is a Lie group. Clearly, any homomorphism from $A$ to $ℝ$ has to vanish on $K$ , so factors thru $A / K$ .

Posted by: John Baez on November 10, 2008 12:12 AM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

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I was able to follow most of the nice argumentation by Yves, but there was one small sticking point for me: it’s in the paragraph just before the various equivalences, where he says

So there exists a homomorphism $u_{n} : B \to ℤ$ mapping $K$ to zero and $y$ to $n$ .

The trouble is that I don’t see why $B$ need admit any nonzero homomorphisms to $ℤ$ . Consider for example the case where the discrete group $B$ is a vector space over $ℚ$ .

On the other hand, this is a minor complaint, and the argument is easily adjusted: we can extend any homomorphism $⟨ K, y ⟩ \to ℝ$ to a homomorphism $B \to ℝ$ because $ℝ$ is an injective $ℤ$ -module, i.e., a divisible abelian group. So we need only replace $ℤ$ by $ℝ$ and strike out reference to $hom (A, ℤ)$ to make the argument work.

[And just to add a few more words to make the argument clear to myself: the point is that the basis element $V (K, B (r))$ of the compact-open topology cannot have compact closure, because the sequence of maps $u_{n} : B \to ℝ$ considered by Yves has no accumulation point. In other words, the closure is not sequentially compact, and therefore not compact. So $hom (B, ℝ)$ cannot be locally compact.]

Regarding

The other question should reduce to the study of whether $hom (\oplus_{ℕ} ℤ, ℝ)$ is 2nd-countable, which shouldn’t be a big deal.

I think both parts are correct: that we can indeed reduce to consideration that space, and indeed that space is second countable. For the first part: if the direct factor $B$ has uncountable rank, then as a discrete space it is obviously not second countable, so for the purposes of John’s question we can ignore that case. If it has countable rank, then by what Yves said earlier we have isomorphisms

hom (B, ℝ) ≅ hom (\oplus_{ℕ} ℤ, ℝ) ≅ ℝ^{ℕ}

and we indeed have a countable basis for the last space named, consisting of products

\prod_{n \in ℕ} U_{n}

where $U_{n} = ℝ$ for all $n$ except for finitely many $k$ where $U_{k}$ is an open interval with rational endpoints.

So, summarizing that last bit: if $A$ is a locally compact Hausdorff abelian group and is second countable, then $hom (A, ℂ^{*})$ is also second countable.

Posted by: Todd Trimble on November 10, 2008 3:32 PM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

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Todd wrote:

I was able to follow most of the nice argumentation by Yves…

Thanks for clearing up that sticky point. This stuff will show up in an appendix to that paper on reps of 2-groups, with big kudos to you and Yves.

By the way, do you know a reference for this fact he mentioned?

It’s well known (and not trivial) that any locally compact abelian group $A$ has a compact subgroup $K$ such that $A / K$ is a Lie group.

I’ll get ahold of those Pontryagin duality books the day after tomorrow. Maybe this result is in one of those. I also have a yen to peruse Loomis’ Abstract Harmonic Analysis.

(I usually go in to teach on Tuesday, but tomorrow is Veteran’s Day. I thought it was always on Monday! Maybe it shifts to Tuesday when we’re in more than one war?)

Posted by: John Baez on November 10, 2008 10:12 PM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

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By the way, do you know a reference for this fact he mentioned?

I think it’s also in that book by Sidney Morris book you mentioned back here. Annoyingly, I must have misplaced my copy, but I’ll email you chapter and verse when I locate it.

I like that book: it’s solidly written, unpretentious, and covers all the basics in a reasonable number of pages. By perusing that book, you’ll probably find some more items to that list you started!

The reason I happened to have the counterexample at the ready in my first comment is that I once daydreamed: wouldn’t it be nice if locally compact Hausdorff abelian groups formed a *-autonomous category? I knew the chances of that were minuscule (I would have heard of that one!), but I had to think a while before hitting upon a reason they don’t. But that’s okay: even if they don’t, that’s what Chu space categories are for! That is, one can use the dualizing object $S^{1}$ to cook up a *-autonomous category in which this category fully embeds in a nice way.

Posted by: Todd Trimble on November 11, 2008 12:04 AM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

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Okay. I’ve drafted a version of the whole argument! You can see it in Appendix A.3 near the end of this paper — around page 82 in the current version. If anyone (say, for example, Todd) can spot mistakes, I’d love to hear about them.

I want to get the pesky remaining details cleared up soon, so I’m gonna run over to the library and get ahold of that book by Armacost! I’m sort of excited about all the fine nuances of locally compact abelian groups now, and I should take advantage of that mood before it fades.

Posted by: John Baez on November 11, 2008 1:26 AM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

Hmm, this looks like one reference to that ‘well-known and not trivial result’ Yves mentioned, which seems to be called the ‘fundamental structure theorem for locally compact abelian groups’:

David A. Armacost, The Structure of Locally Compact Abelian Groups, Dekker, New York, 1981.

There’s an informative review in the AMS Bulletin.

Amusingly, it says: ‘It would be unreasonable to expect mathematicians to buy this book in droves; the price is enough to inhibit people.’ The price? Back then it was $23.75!

Posted by: John Baez on November 10, 2008 10:38 PM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

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Another question for you, Todd. When books you read talk about ‘locally compact abelian groups’, do they secretly mean locally compact Hausdorff abelian groups? Some people sneak in Hausdorffness without mentioning it. This is a bit unnerving.

For example: Mackey, in his book Unitary Group Representations in Physics, Probability and Number Theory, says “We remind the reader that a topological group is a group that is at the same time a $T_{1}$ topological space in such a fashion that $(x, y) \mapsto x y^{- 1}$ is a continuous function”.

This is initially even more unnerving, since being $T_{1}$ is weaker than being Hausdorff, also known as $T_{2}$ …. and if different folks start sneaking fine print of differing strength into their definitions, what’s a poor fellow like me to do?

(Even in my nerdiest undergraduate days, I didn’t groove to the subtleties of separation axioms. So how am I supposed to remember all their intricate logical relations now?)

Luckily, I think Mackey is just showing off his ability to split hairs, since Wikipedia says that a topological group that’s just $T_{0}$ is already $T_{3 \frac{1}{2}}$ ! So, his topological groups are implicitly Hausdorff.

Anyway, I’ll soon get ahold of a bunch of books on locally compact abelian groups, and read the fine print to see what they really mean by this term.

Posted by: John Baez on November 11, 2008 12:26 AM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

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That $T_{1}$ and $T_{2}$ coincide for topological groups is a kind of “ $ε / 2$ -argument”, and probably carries over to general spaces defined by a uniform structure. Namely, suppose $G$ is $T_{1}$ and suppose $x \neq e$ in $G$ . Choose an open set $V$ containing $e$ but not $x$ . Then because we’re dealing with a topological group, there exists an open neighborhood $U$ of $e$ such that $U \cdot U \subseteq V$ , and by replacing $U$ by $U \cap U^{- 1}$ if necessary, we may assume $U = U^{- 1}$ . Then I’ll leave it to you to check that $x U \cap U = \emptyset$ , so that $x$ and $e$ are separated by open sets, and therefore $G$ is Hausdorff.

I think by massaging this type of argument, it’s not hard to show $T_{0} = T_{1}$ and $T_{2} = T_{3}$ (the latter is listed as an exercise in Morris’s book). The $T_{3 \frac{1}{2}}$ I didn’t know about, but I find it sort of believable.

I figure some of these older writers feel that $T_{0}$ is such a mild hypothesis that no one in their right mind would deny it [:-)], so they just leave it off. But almost everyone means Hausdorff even if they forget to say so.

Posted by: Todd Trimble on November 11, 2008 12:56 AM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

That T₁ and T₂ coincide for topological groups is a kind of “ε/2-argument”, and probably carries over to general spaces defined by a uniform structure. … The T_3½ I didn’t know about, but I find it sort of believable.

What you need to know is this: a topological space is T_3½ if and only if it is both T₀ and capable of supporting a uniform structure. (Or this: it’s T_3½ if and only if it is capable of supporting a separated uniform structure. Also related: it’s completely regular iff capable of supporting a uniform structure, where of course T_3½ is defined to mean both T₀ and completely regular.)

Posted by: Toby Bartels on November 11, 2008 1:14 AM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

Nice bit of wisdom there (and nice to be hearing from you again, Toby). As I say, I find it believable, but do you have a reference?

Posted by: Todd Trimble on November 11, 2008 1:36 AM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

You want a reference now, too, Todd? Don’t you trust me? ^_^

Most of what I know about separation axioms is from Eric Schechter’s magnificent Handbook of Analysis and its Foundations. At http://www.math.vanderbilt.edu/~schectex/ccc/excerpts/separat.html is a brief excerpt which (if you look carefully) contains the statement, and you can certainly find the proof in the body of that book.

If you understand Stone–Čech compactification, then you can probably come up with a proof on your own; it’s all the same basic idea of embedding things in cubes.

Posted by: Toby Bartels on November 12, 2008 4:35 AM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

Actually, I found a proof for the case of topological groups outlined in exercise 5 on page 237 of Munkres’s Topology. The proof for general uniform spaces ought to be easily extractable from the hints given.

Posted by: Todd Trimble on November 11, 2008 2:37 AM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

Suppose we adopt the point of view that the concept of topos is better than the concept of topological space, and (in particular?) that in abstract contexts it’s better not to refer to points at all. Then what kind of separation axioms can we still state?

I would love to be able say the ugliness of the separation axioms is an artifact of the formalism of topological spaces.

Posted by: James on November 11, 2008 6:32 AM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

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There is some literature on this; for example there is a notion of regular locale. (You probably already know this, but a frame is a lattice in which every subset has a sup and in which finite meets distribute over arbitrary sups, and a frame morphism is a map which preserves finite meets and arbitrary sups. The category of locales is by definition the opposite of the category of frames: frames generalize topologies of spaces $Open (X)$ and locales spaces $X$ , and continuous maps of locales $X \to Y$ are identified with morphisms $Open (Y) \to Open (X)$ of their frame counterparts.)

If you want to learn more about pointless separation axioms, you might start by looking at Johnstone’s Stone Spaces. It is apparently possible to define a notion of Hausdorff locale, but it’s unwieldy. Regular locales are much easier to deal with, and Johnstone also discusses completely regular and normal locales in his book.

Posted by: Todd Trimble on November 11, 2008 11:40 AM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

Similar in spirit perhaps to:

This situation, like so often already in the history of our science, simply reveals the almost insurmountable inertia of the mind, burdened by a heavy weight of conditioning, which makes it difficult to take a real look at a foundational question, thus at the context in which we live, breathe, work – accepting it, rather, as immutable data. It is certainly this inertia which explains why it took millennia before such childish ideas as that of zero, of a group, of a topological shape found their place in mathematics. It is this again which explains why the rigid framework of general topology is patiently dragged along by generation after generation of topologists for whom “wildness” is a fatal necessity, rooted in the nature of things.” (Grothendieck, Sketch of a Programme, 1984: 259)

Then again, if John needs this material to do analysis perhaps it’s necessary:

After some ten years, I would now say, with hindsight, that “general
topology” was developed (during the thirties and forties)by analysts and in
order to meet the needs of analysis, not for topology per se, i.e. the study
of the topological properties of the various geometrical shapes. (p. 258)

Posted by: David Corfield on November 11, 2008 11:44 AM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

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James wrote:

Suppose we adopt the point of view that the concept of topos is better than the concept of topological space, and (in particular?) that in abstract contexts it’s better not to refer to points at all. Then what kind of separation axioms can we still state?

Todd is the expert on this sort of thing, but he left out one buzzword which it’s handy to know: sober space. Sobriety is a condition that lets you recover a space from it lattice of open subsets. So, you can think of sober spaces as a step en route to locales, where you forget points and work directly with the lattice of opens.

I would love to be able say the ugliness of the separation axioms is an artifact of the formalism of topological spaces.

One could argue that, and topos theorists seem to prefer working with locales.

I would also argue that the problem with topological spaces is that they’re a kind of umbrella formalism that covers a wide range of different ideas.

For example, analysts love locally compact Hausdorff spaces since these are just another way of thinking about commutative $C^{*}$ -algebras. They form a very nice context for measure theory, too, since the dual of the commutative $C^{*}$ -algebra of bounded continuous functions on a locally compact Hausdorff space $X$ is the space of finite regular Borel measures on $X$ . (Non-analysts should take my word for it: this is a good thing.)

So, when people study Pontryagin duality for ‘locally compact abelian groups’, you can instantly guess they should mean ‘locally compact Hausdorff abelian groups’, because the beautiful approach to Pontryagin duality uses integration and operator algebras.

And, it turns out that every book I got from the library last night, which discusses ‘locally compact abelian groups’, secretly inserts the word ‘Hausdorff’ into the definition of this concept.

In short: in some contexts Hausdorffness is a given, because it’s part of a package: part of the definition of some well-behaved category of spaces.

On the other hand, algebraic geometers or people working with sheaves strongly avoid Hausdorffness in certain situations.

Algebraic topologists are doing a third completely different thing, so they have settled on compactly generated weak Hausdorff spaces as their ‘convenient category’ of spaces.

The concept of ‘topological space’ covers all three of these applications, but only loosely. The study of general topological spaces has little inner vitality — it’s various categories of special spaces that are interesting.

Posted by: John Baez on November 11, 2008 7:20 PM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

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Thanks Todd, David, and John.

I will have a look at Johnstone’s book next time I drop by the library. One minor objection I have, though, is that as it appears to me, a locale is an extra structure on a topos, not a property of a topos. So I would think a true topos purist would think that separation axioms expressed in terms of locales, or more generally sites, would be only a partial improvement over axioms expressed in terms of points. (And besides, my favorite site is not a locale!)

There is the notion of a quasi-separated object in a topos. (I think this has been renamed by some post-Grothendieck writers.) It means that the fiber product, over the given object, of any two compact objects is again compact. This is natural and important but rather weak. Also, I think you should be able to express in topos terms the requirement that the diagonal map $X \to X \times X$ be a closed embedding. (Here’s a stab: $X$ is isomorphic over $X \times X$ to the complement of an essential monomorphism $U \to X \times X$ .) That one seems very natural to me, though I don’t recall having seen it written down anywhere, so I wonder if I’m missing a subtlety.

Posted by: James on November 12, 2008 6:19 AM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

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Concerning your article in preparation on infinite dimensional 2-representations:

we talked about this a little bit before somewhere, but since you don’t mention it: it looks to me like the 2-category $Meas$ you describe should pretty closely be related to the 2-category ${Bimod}_{vN}$ whose

- objects are (commutative) von Neumann algebras

- morphisms are von Neumannn bimodules

- 2-morphisms are homomorphisms of these,

or rather the image in $Vect - Mod$ under ${Bimod}_{vN} \to Vect - Mod$ which sends

- algebras to their categories of modules;

- bimdodules to functors induced by tensoring with these

- bimodule homomorphisms to natural transformations of these.

Commutative vonNeumann algebras are precisely the algebras of measurable functions, I think. Indeed, as you mention on p. 24, with reference to Dixmier and Arveson, the “fields of Hilbert spaces” which you discuss are precisely the modules for commutative vonNeumann algebras.

Given that, a bimodule of commutative von Neumann algebras should be interpretable equivalently as a “field of Hilbert spaces” on a product space $X \times Y$ , as it indeed appears in the definition of morphisms on $Meas$ on p. 29.

I understand that you have a different angle here, but wouldn’t it be fun to relate this a bit for instance to the interesting 2-representation of the strict String-2-group on – in this language – “fields of Hilbert spaces” over non-commutative measurable spaces (i.e. using a non-commutative von Neumann algebra, namely that of a highest weight rep of the centrally extended loop group sitting in $String (n)$ , the way it is described for instance in the context of String 2-bundles in my article with Konrad?

Posted by: Urs Schreiber on November 11, 2008 8:58 PM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

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Hi, Urs!

Yes, I really want to rewrite this whole paper in the language of bimodules over commutative von Neumann algebras.

Unfortunately the paper is already very long and my younger coauthors (who must publish or perish) would rebel if I tried such a massive change of formalism at this late stage.

Luckily, the work we’ve done is far from wasted! A lot of the theorems we’ve proved will serve to make future papers easier. And best of all, the paper is almost done.

Commutative von Neumann algebras are precisely the algebras of measurable functions, I think.

Yes. Indeed, many things we do in this paper that look a bit ‘technical’ or ‘arbitrary’ have been carefully chosen to make the story very beautiful once it is retold in the language of von Neumann algebras.

And, to be brutally frank, I sort of hope you don’t figure out this story and tell everyone about it before I work out the details.

I understand that you have a different angle here, but wouldn’t it be fun to relate this a bit for instance to the interesting 2-representation of the strict String-2-group on – in this language – “fields of Hilbert spaces” over non-commutative measurable spaces (i.e. using a non-commutative von Neumann algebra, namely that of a highest weight rep of the centrally extended loop group sitting in String(n), the way it is described for instance in the context of String 2-bundles in my article with Konrad?

Yes, that would indeed be a wonderful thing to do. However, I’m enjoying a break from String(n)-related projects, which are sufficiently fashionable that I always feel competitors breathing down my neck when I’m working on them. What’s exciting me now is that a lot of classic but ‘dry’ and ‘technical’ ideas from analysis — standard Borel spaces, Polish groups, and so on — turn out to be exactly right for studying the important bicategory you’re calling ${Bimod}_{vN}$ .

Posted by: John Baez on November 12, 2008 1:59 AM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

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Hi John,

thanks for the reply!

Yes, I really want to rewrite this whole paper in the language of bimodules over commutative von Neumann algebras.

I didn’t mean to suggest that. I was just hoping I’d find a remark, pointing out that what you discuss fits into a bigger picture which relates to stuff other people have already thought about, such as Stolz-Teichner, for instance, whose construction, with slight modification, can nicely be re-interpret as a (essentially strict) 2-representation of strict $String (n)$ .

I just come from a discussion about the relevance of this point in axiomatic CFT, so it is close to my heart.

Indeed, many things we do in this paper that look a bit ‘technical’ or ‘arbitrary’’have been carefully chosen to make the story very beautiful once it is retold in the language of von Neumann algebras.

I am enjoying the geometrical interpretation you give. I had been vaguely aware, from general nonsense, that such must exist, but it’s great to see it worked out in such detail.

In fact, the geometrical picture is closer to my preferred internal way of thinking. In a way it is for “historical” reasons really that I started to look at 2Vect in terms of Bimod (the reason is, you might recall, that this point of view appeared as very fruitful in rational CFT – I talked to CFT people who found it strange to hear I had trouble promoting the idea that the fibers of 2-vector bundles generally ought to be objects in Bimod).

[…] $String (n)$ -related projects, which are sufficiently fashionable that I always feel competitors breathing down my neck when I’m working on them.

Is that really so in this case?

In any case, I am thinking that presenting a nice representation theory for strict 2-groups provides as a byproduct a nice incentive for “competitors” (which are incidentally those who might cite one’s articles – your younger coauthors might be interested in that aspect :-) to consider the strict version of the String-2-group. My impression is that the usefulness of having a (i.e.: our :-) strict model has not been fully appreciated yet.

Posted by: Urs Schreiber on November 12, 2008 3:03 PM | Permalink | Reply to this

Re: Variations on Pontryagin Duality

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Urs wrote:

John wrote:

I really want to rewrite this whole paper in the language of bimodules over commutative von Neumann algebras.

I didn’t mean to suggest that.

I know you didn’t. Nonetheless it’s true. There’s something sad about writing a 90-page paper and realizing halfway through that it could all be done more beautifully. I’ve resigned myself to that, but just barely.

I was just hoping I’d find a remark, pointing out that what you discuss fits into a bigger picture which relates to stuff other people have already thought about, such as Stolz-Teichner, for instance, whose construction, with slight modification, can nicely be re-interpret as a (essentially strict) 2-representation of strict $String (n)$ .

I can certainly stick in a little remark to that effect in the Conclusions. But if I start trying to explain it clearly, I run the risk of starting to write the paper I secretly want to write. I guess that’s why I’ve avoided mentioning this stuff.

John wrote:

[…] $String (n)$ -related projects […] are sufficiently fashionable that I always feel competitors breathing down my neck when I’m working on them.

Is that really so in this case?

I may just be imagining it. There aren’t dozens of people who like von Neumann algebras, bicategories and the string group. But there are some very smart people who do — for example you, Stolz, Teichner and Henriques.

My impression is that the usefulness of having a (i.e.: our :-) strict model has not been fully appreciated yet.

Take advantage of this situation while it lasts! It won’t last forever.

Posted by: John Baez on November 14, 2008 10:58 PM | Permalink | Reply to this

Read the post Locally Compact Hausdorff Abelian Groups
Weblog: The n-Category Café
Excerpt: Learn the Principal Structure Theorem for locally compact Hausdorff abelian groups --- the kind of groups that show up in Pontryagin duality.
Tracked: November 12, 2008 4:36 AM

Re: Variations on Pontryagin Duality

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Everyone interested in Pontryagin duality should read this:

Michael Barr, On duality of topological abelian groups.

He explains:

Did you know that there is a *-autonomous category of topological abelian groups that includes all the LCA groups and whose duality extends that of Pontrjagin? The groups are characterized by the property that among all topological groups on the same underlying abelian group and with the same set of continuous homomorphisms to the circle, these have the finest topology. It is not obvious that such a finest exists, but it does and that is the key.

Posted by: John Baez on December 31, 2008 10:05 PM | Permalink | Reply to this

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