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I need to be looking again into the subject of the Gelfand-Naimark theorem for noncommutative $C^*$-algebras $A$ regarded as commutative $C^*$-algebras in the copresheaf topos on the poset of commutative subalgebras of $A$, as described in
Heunen, Landsman, Spitters, A topos for algebraic quantum theory.
While it seems clear that something relevant is going on in these constructions, I am still trying to connect all this better to other topos-theoretic descriptions of physics that I know of.
Here is just one little observation in this direction. Not sure how far it carries.
If I understand correctly, we have in particular the following construction: for $\mathcal{H}$ a Hilbert space and $B(\mathcal{H})$ its algebra of bounded operators, let $A : \mathcal{O}(X) \to CStar$ be a local net of algebras on some Minkowski space $X$. landing (without restriction of generality) in subalgebras of $B(\mathcal{H})$.
By the internal/noncommutative Gelfand-Naimark theorem we have that each noncommutative $C^*$-algebra that $A$ assigns to an open subset corresponds bijectively to a locale internal to the topos $\mathcal{T}_{B(\mathcal{H})}$ of copresheaves on the commutative subalgebras of $B(\mathcal{H})$.
So using this, our Haag-Kastler local net becomes an internal-locale-valued presheaf
$A : \mathcal{O}(X)^{op} \to Loc(\mathcal{T}_{B(\mathcal{H})}) \,.$So over the base topos $B(\mathcal{H})$ this is a “space-valued presheaf”. we could think about generalizing this to $\infty$-presheaves, probably (though I’d need to think about if we really get there given that the locales need not come from actual spaces). The we could think about if this generalization dually corresponds indeed to the “higher order local nets” such as factorization algebras.
Just a very vague thought. Have to run now.
The article is 69 pages so I need some help. You say that nc C star algebras are in correspondence with commutative C star algebras internally in some topos. I see that there is a construction of cosheaf of commutative C star algebras in this context from nc C star algebra, but is there a proof that this correspondence is a faithful functor (or even an equivalence of categories) ? By the way, star operation and properties make C star algebras smaller than the typical abstract noncommutative algebras, so it is not surprising to me that they are somehow commutative.
The article is 69 pages so I need some help.
I’ll try to write a summary a little later.
I was being distracted and now didn’t have muc time for writing a genuine summary, but a little bit is now at semilattice of commutative subalgebras.
But to reply to your question, Zoran:
in the article there seems to be just the construction of the internal locale $\Sigma(A)$ in the topos over $comSub(A)^{op}$ for a given non-commutative $C^*$-algebra $A$, but not a discussion of whether and how it produces an equivalence of categories.
But are you sure that they do claim explicitly somewhere that it is actually an equivalence (or at least a faithful functor) ? I mean proof more or less, but the statement…
Thank you for semilattice of commutative subalgebras. In the idea section you say that the same philosophy is thought even for abstract associative algebras, not necessarily operator algebras. This I have much harder time to believe. I mean the C-star algebras are more akin to associative algebras close to commutative ones, e.g. to the algebras of finite Gelfand-Kirillov dimension and some similar classes. The C-star envelope of very noncommutative (close to free associative) algebras are pretty tame in comparison to the latter. So for C-star (and in particular von Neumann) I could accept that noncommutative may somehow be interpreted via commutative. But not for guys close to free associative algebra.
you say that the same philosophy is thought even for abstract associative algebras, not necessarily operator algebras. This I have much harder time to believe.
You are right, the statement about commutative geometry applies to the $C^*$-case.
I have added now the qualifier “$C^*$-algebra” to the entry. That is needed in order that the internal algebra has a chance of being an internal commutative $C^*$-algebra, so that the internal Gelfand duality can be applied.
But are you sure that they do claim explicitly somewhere
As I said in my last message, I think they do not claim anything like this. The internal Gelfand duality is an equivalence of categories I guess, and their construction of the commutative $C^*$-algebra presheaf from a non-comutative $C^*$-algebra looks as natural as can be, but I haven’t seen any statement beyond that.
Very interesting.
I can not find the parallel thread where we went further (I wish once we have automatic backlinks from nLab :)) . I think my last question which I think would be useful to go further is about the colimit over all commutative subalgebras, to obtain the subalgebra of normal operators. It is not sufficient to know the commutative subalgebras, one really needs to know the connecting maps and how to compute the things related to a colimit in noncommutative algebras. Thus to have a use from an internal Gelfand duality, one would like to express the connecting morphisms and taking the colimit on the dual side. So one has dual side for each internal commutative subalgebra and one wants to connect them in a way they were connected in the original C-star algebra, for start. Then one wants to intepret this other side in such a way that the “extended duality” picture preserves the colimit, at least for various uses one can imagine. (like computing the observables from knowing some other combination of observables). Is there a hint how to systematically attack this ?
Hi Zoran,
I think we discussed this in the context of Bohrification.
That statement about the colimits is in
vdBerg-Heunen, Noncommutativity as a colimit (http://arxiv.org/abs/1003.3618)
They consider just the bare bones statement that I mentioned. I agree with you that it would be useful to see how much further this can be pushed.
Thanks for the reference. It is surprisingly recent!
I have currently a Bachelor student who is looking into seeing how this internal Gelfand duality can be used in the understanding of AQFT. So apart from the fact that I am still busy with finalizing my own thesis, I am looking into this stuff here. I’d be quite interested in talking about whatever aspect of this intersts you.
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