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created an entry Bohrification
Are you getting closer now to my question how far is the thing from faithful ? Unless one has an equivalence of categories (what I doubt) I would not call a construction of non-faithful functor a “generalization” of Gel’fand-Neimark theorem. (Generalization includes a full statement in the special case.)
The entry looks neat and interesting but the above questions bother me. There are lots of non-faithful spectra-like functors in the history of noncommutative geometry, and the fact that one looses significant information most of the time is a source of frustration of practioners, including me.
Hi Zoran,
I was planning to add this anyway, now I have:
what is currently proposition 3 in the entry states a result of Bas Spitters’, that if instead of the presheaf topos over commutative subalgebras one uses the sheaf topos with respect to th double negation topology, then Bohrification coincides with Gelfand duality on commutative $C^\ast$-algebras.
Urs, it seems you misunderstood me.
For usual C-star algebras, noncommutative, there is a Gelfand transform which nevertheless gives a commutative space. But it can not be called duality as it is not an equivalence of categories. The fact that in commutative case one has a Gelfand-Neimark theorem is not generalizing to the Gelfand transform in general. Generalizing the map is in that classical situation easy, generalizing the duality/theorem is difficult. You talk generalizing the duality but you just quote a result on generalizing the map, just in a different way. When we specialize the theorem that there is such a construction we do not get a statement that in fact we have a duality – for this one needs also to generalize the inverse, as the duality has two sides. I understand that internalization gives more, but I suspect it does not give a generalization of Gelfand-Neimark duality but only of a map in one direction. It is confusing saying that one has a generalization of a duality if one generalizes just the functor (for what there are so many ways to do), not the whole duality/theorem.
It would be good to say “constructive Gelfand transform”, not constructive Gelfand duality (unnless I am wrong). In operator algebras term Gelfand transform is used widely in noncommutative case, while duality is used in commutatiive as there is a true duality/antiequivalence of categories. Otherwise the terminology is misleading.
Hi Zoran,
okay, I have changed the first sentence of the entry to
The construction called Bohrification by some authors is a generalization of the Gelfand spectrum for commutative C-star algebras to a context of noncommutative $C^*$-algebras.
Is that better?
On the other hand, where I say “constructive Gelfand duality” I really mean the duality between commutative $C^\ast$-algebras and suitable locales, only that in the discussion it appears internally in some topos other than $Set$.
Aha, then constructive is really duality, unlike the Bohrification in general. Thanks.
I was suspecting optimistically that if one would look at many different internal commutative shadows of a single noncommutative operator algebra, and having a commutative duality for each of them, then all the shadows together could maybe give a faithful information. This was my first impression of this whole work. That is why I search for precise statement, to see if this is eventually possible or targetted.
Right, so I think the story is to be thought of like this:
in a first step, the information in the noncommutative algebra is reduced to that which can be seen by the collection of all its “classical contexts” aka commutative subalgebras. This loses a bit of information. But not too much. It seems to remember precisely the “observable” information in the sense of quantum mechanics: the partial subalgebra of normal elements.
But then, in a second step, in the topos over these commutative subalgebras, we do have internal commutative Gelfand duality, and this is now a true duality, internally.
Can one make the words “observable information” into a mathematical statement logically precedeng and independent from this theory ?
One precise statement is this:
the Bohrification of the nc $C^\ast$-algebra $A$ depends only on its partial $C^\ast$-subalgebra $N(A)$ of normal elements;
in the category of partial $C^\ast$-algebras $N(A)$ is the colimit over its commutative $C^\ast$-subalgebras.
Oh, thanks, this is very helpful mathematically.
I still need to educate myself in the foundations of QM to understand the precise meaning of the $N(A)$ from foundational QM point of view.
The standard lore is that it is the self-adjoint operators that are observable. These are contained in the normals and for $a$ normal, $a^* a$ is self-adjoint and positive.
The other result is that iso-classes of posets of commutative subalgebras of $A$ correspond to isos of the Jordan algebra $J(A)$. Again, the Jordan algebra structure is supposed to capture the observable content.
This is Harding-Döring, the other statement was vdBerg-Heunen, all referenced in the entry.
Finally the Spitters-Landsman-Heunen result says that the internal integrals over the Bohrified phase space give the quantum states. I need to write this out in more detail in the entry, eventually.
What the abstract passage from commutative subalgebra in some abstract category of (noncommutative) “partial C-star algebras” would correspond from the point of view of measurement ? The noncommutativity is about the impossibility of simulteneous measurement. Now you have things which are commutative, and for which you can say some geometry in some internal point of view. Now the colimit fact tells you that the full thing is in some sense determined from the pieces. But maybe this abstract colimit in somewhat arbitrary category has some interpretation from the measurement point of view ?
I have added something like the following parapgraph to the Idea-section at Bohrification
One thing that is nice about Bohrification is that it makes the following statement true: “quantum states on a quantum algebra $A$ are precisely classical states internal to the Bohrified ringed topos corresponding to $A$”.
This is essentially a direct re-interpretation of Gleason’s theorem: this theorem says that quantum states on $A$ are already fixed by demanding them to be maps on $A$ that are (positive, normed) linear functions on all commutative subalgebras of $A$. Now, the immediate formalization of a map $A \to \mathbb{C}$ that is required to preserve certain structure on all commutative subalgebras is a fully structure-preserving function, but internal to the presheaf topos over the comutative subalgebras. That presheaf topos is the “Bohrification” of $A$, since Bohr said things that can be interpreted as being formalized by this process.
I am re-organizing and expanding the entry. And have renamed it to Bohr topos .
At Bohr topos I have considerably expanded the previous Idea-section. After polishing a bit more I am planning to make this a post to the $n$Café. (So all comments and criticism on the exposition are most welcome!)To account for the length of the section now, I have split it into subsections “Brief” and “More detailed”.
I have considerably expanded the section The Bohr topos at Bohrification during the last few days.
I see that you’re again writing about a full and faithful functor from $Top$ to $Loc$. This functor is neither full nor faithful: on maps to a non-$T_0$ space, it’s not faithful; on maps to a space without enough points (a non-sober space whose $T_0$ reflection is still non-sober), it’s not full.
As I see it, you have three options:
Just use a functor and don’t worry about its surjectivity properties.
Remark that the spaces that you need are all sober and use $Sob \hookrightarrow Loc$ instead.
Skip $Top$ entirely, and use locales from the very beginning.
(Skipping $Loc$ and going straight to $Topos$ will not help.)
I see that you’re again writing about a full and faithful functor from $Top$ to $Loc$.
Sorry. That’s not “again” but is a remnant that I have overlooked. That’s embarrasing. But now I have removed it.
Remark that the spaces that you need are all sober
No, they are not. $Alex \mathcal{C}(A)$ is almost never sober, I think.
Let me try to clarify: there are two aspects that are needed for the discussion of the decent = locality thing.
The faithfulness of $Poset \to Topo_{ess}$. This is now discussed in detail.
The fact that $Alex : Poset \to TopSpace$ (but not $Alex : Poset \to Locale$) preserves limits. (And then the descent objects are formed in the category of ringed topological spaces).
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