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Stream: deprecated: analysis

Topic: Generalizations of C* algebras

Owen Lynch (Mar 03 2023 at 19:44):

I've come across pro C* algebras, which are (it seems to me) essentially C* algebras where the underlying space is only a locally convex space instead of a Banach space: https://ncatlab.org/nlab/show/pro-C-star-algebra (that is of course not the definition the nlab gives)

Does anyone know if there is a category of topological spaces that is equivalent to the dual of the category of pro C* algebras?

John Baez (Mar 03 2023 at 19:51):

You mean commutative pro C*-algebras, of course.

John Baez (Mar 03 2023 at 19:51):

I don't know! But we can figure out something pretty easily.

John Baez (Mar 03 2023 at 19:52):

The category of commutative C*-algebras is equivalent to the opposite of the category of locally compact Hausdorff spaces.

John Baez (Mar 03 2023 at 19:54):

[[pro-objects]] in any category correspond to [[ind-objects]] in the opposite category - this is just formal nonsense, like how limits in a category are colimits in the opposite category. Pro-objects are "imaginary, ideal limits" of cofiltered diagrams in a category, and ind-objects are "imaginary, ideal colimits" of filtered diagrams in a category.

John Baez (Mar 03 2023 at 19:56):

So, commutative pro-C ${}^\ast$ -algebras, or more precisely pro-(commutative C*-algebras) (which may well be the same thing) correspond to ind-(locally compact Hausdorff spaces).

John Baez (Mar 03 2023 at 19:59):

So my answer is this question: what are ind-objects in the category locally compact Hausdorff spaces like?

John Baez (Mar 03 2023 at 20:00):

More hand-wavily: what sort of big spaces could we do by taking formal cofiltered colimits, or "increasing unions", of locally compact Hausdorff spaces?

John Baez (Mar 03 2023 at 20:01):

For example how about

$\mathbb{R} \hookrightarrow \mathbb{R}^2 \hookrightarrow \mathbb{R}^3 \hookrightarrow \cdots$ ?

John Baez (Mar 03 2023 at 20:02):

In Top the colimit of this is called $\mathbb{R}^\infty$ and it's not locally compact.

John Baez (Mar 03 2023 at 20:02):

But it might be a fine ind-object in the locally compact Hausdorff spaces!

John Baez (Mar 03 2023 at 20:03):

One has to be careful because the morphisms of these ind-objects might be quite different from morphisms in Top (= continuous maps).

Owen Lynch (Mar 03 2023 at 20:06):

John Baez said:

The category of commutative C*-algebras is equivalent to the opposite of the category of locally compact Hausdorff spaces.

I learned today that this is not true in the way I thought it was true!! https://math.stackexchange.com/questions/170984/are-commutative-c-algebras-really-dual-to-locally-compact-hausdorff-spaces

Owen Lynch (Mar 03 2023 at 20:07):

But OK, this makes sense. I'll have to think more about what this means and learn more about pro and ind objects.

John Baez (Mar 03 2023 at 20:07):

Oh, okay! I just naively extrapolated from the unital case.

Owen Lynch (Mar 03 2023 at 20:07):

Are there other examples of pro and ind objects that you know of?

Owen Lynch (Mar 03 2023 at 20:07):

I only learned about that today too.

John Baez (Mar 03 2023 at 20:08):

Profinite groups are super important in Galois theory.

Owen Lynch (Mar 03 2023 at 20:08):

Now I seem to remember David Jaz Myers going on about profinite sets in homotopy type theory...

John Baez (Mar 03 2023 at 20:10):

There are many uses of pro-objects, but if you want something "established" think about how things like the absolute Galois group of $\mathbb{Q}$ is best treated as a profinite group, not just a bare group.

Owen Lynch (Mar 03 2023 at 20:10):

What is the absolute Galois group?

John Baez (Mar 03 2023 at 20:11):

Forget it. :upside_down:

Owen Lynch (Mar 03 2023 at 20:11):

OK :)

John Baez (Mar 03 2023 at 20:12):

It's just like the most mysterious object in math, the group that has driven people insane... it's the Galois group of the algebraic numbers over the rational numbers.

Owen Lynch (Mar 03 2023 at 20:12):

Oh, because each extension by adding an extra root is a finite extension

John Baez (Mar 03 2023 at 20:13):

Right.

Owen Lynch (Mar 03 2023 at 20:14):

So the whole group is a filtered colimit of all of those Galois groups which come from adding a finite number of roots.

Owen Lynch (Mar 03 2023 at 20:14):

Neat!

John Baez (Mar 03 2023 at 20:16):

I get the problem now:

It makes sense that to prove Gelfand-Naimark duality for nonunital C*-algebras people would adjoin a unit and reduce it to the unital case. But then one is treating locally compact Hausdorff spaces as pointed compact Hausdorff spaces with the specified point (called "infinity") removed, and the maps are really basepoint-preserving maps between those pointed spaces! Which are, alas, not the same as arbitrary continous maps between locally compact Hausdorff spaces. They're maps that do something rather simple at infinity.

Owen Lynch (Mar 03 2023 at 20:18):

Right!

Owen Lynch (Mar 03 2023 at 20:18):

I think those are called "proper" maps or something

John Baez (Mar 03 2023 at 20:19):

Owen Lynch said:

So the whole group is a filtered colimit of all of those Galois groups which come from adding a finite number of roots.

I agreed with this at first but it's not right. It's a filtered limit of those finite Galois groups.

Owen Lynch (Mar 03 2023 at 20:19):

OK, I mixed up pro and ind; pro is for limits?

Owen Lynch (Mar 03 2023 at 20:20):

Ah, right, because $\mathbb{R}^{\infty}$ is a colimit

Owen Lynch (Mar 03 2023 at 20:20):

OK, that's how I'll remember this

John Baez (Mar 03 2023 at 20:20):

Yes.

If you have field extensions F $\subset$ F' $\subset$ F'' then the Galois group of F' over F is a quotient of the Galois group of F'' over F' not a subgroup.

John Baez (Mar 03 2023 at 20:21):

Actually I guess this is just true for Galois extensions.

John Baez (Mar 03 2023 at 20:22):

Anyway, the example of $\mathbb{R}^\infty$ is easier for us geometrically minded folks.

Owen Lynch (Mar 03 2023 at 20:23):

Huh, I learned Galois theory before I learned algebraic geometry, so I didn't think about it this way at the time, but I wonder if you can think of Galois theory as another algebra/geometry duality, where the groups are the geometry half

John Baez (Mar 03 2023 at 20:24):

It's a duality right but I don't want to answer your "wonder".

Owen Lynch (Mar 03 2023 at 20:24):

Wise :)

John Baez (Mar 03 2023 at 20:24):

Here's a readable short explanation of Galois groups as profinite groups.

John Baez (Mar 03 2023 at 20:26):

But it's actually confusing where it says

For example, the field extensions ${\displaystyle \mathbb {Q} ({\sqrt {a}})/\mathbb {Q} }$ for a square-free element ${\displaystyle a\in \mathbb {Q} }$ each have a unique degree 2 automorphism, inducing an automorphism in ${\displaystyle \operatorname {Aut} (\mathbb {C} /\mathbb {Q} )}$

because this makes it sound like the big Galois group ${\displaystyle \operatorname {Aut} (\mathbb {C} /\mathbb {Q} )}$ is a union, or colimit, of the smaller Galois groups!

Owen Lynch (Mar 03 2023 at 20:27):

Right, that seems like an injection!

Owen Lynch (Mar 03 2023 at 20:29):

OK, neat; now it's back to the functional analysis grindstone for me :)

John Baez (Mar 03 2023 at 20:31):

You should be studying algebraic geometry and number theory now that I'm thinking about that. :goofy:

Owen Lynch (Mar 03 2023 at 20:48):

When you come to Topos, I'll try and nerd snipe you back into analysis :)

John Baez (Mar 03 2023 at 20:52):

I still like analysis. I'm not sure I call pro-objects in the category of C* algebras "analysis", it reminds me of a category theorist's idea of what analysis should be about. :upside_down: But I like that too.

Reid Barton (Mar 04 2023 at 19:41):

It occurs to me that $\mathbb{R}^\infty$ is not just an ind-(locally compact Hausdorff space), but also an ind-(compact Hausdorff space). For example, it is the union of the subsets $[-N, N]^N$.

Reid Barton (Mar 04 2023 at 19:42):

When you say "pro-C*-algebra", is it implicitly assumed that the maps in the pro-system are of some specific kind, say quotients by closed ideals?

Reid Barton (Mar 04 2023 at 19:52):

It looks to me like section 2 of the first paper linked on the nlab page on pro-C*-algebras answers your question

John Baez (Mar 04 2023 at 21:23):

I'm trying to find what's you're talking about in Amini's paper and I found this typo, clearly the result of injudicious global search and replace:

There are dif and only iferent topologies on...

John Baez (Mar 04 2023 at 21:29):

I didn't find the answer to the question "what sort of spaces correspond to pro-C*-algebras?" in Amini's paper. Maybe you meant some other paper?

John Baez (Mar 04 2023 at 21:29):

What's the answer?

Reid Barton (Mar 04 2023 at 21:40):

Oh sorry, I meant the paper by N. C. Phillips in the first bullet point

Reid Barton (Mar 04 2023 at 21:41):

The link in that paragraph could be easy to overlook.

Reid Barton (Mar 04 2023 at 21:42):

The answer is, if I'm not mistaken, the category of "compactological spaces", which today are maybe better known as quasiseparated condensed sets. They are also the filtered colimits of compact Hausdorff spaces with injective transition maps.

John Baez (Mar 04 2023 at 22:11):

Oh, thanks! I didn't see there was a link in there.

I think you just satisfied my curiosity, at least for now.