| We have a functor from Hilbert spaces to C*-algebras, which sends | every Hilbert space to the C*-algebra of its bounded operators. | | The GNS construction realizes every C*-algebra as the algebra of | bounded operators on some Hilbert space. | | Does that provide us with a pair of adjoint functors between Hilbert | spaces and C*-algebras?
Shouldn't you first ask the simpler question if the GNS construction provides a functor at all? I don't see that it does, given its dependence on a choice of state(s).
-- * Harald Hanche-Olsen <URL:http://www.math.ntnu.no/~hanche/> - It is undesirable to believe a proposition when there is no ground whatsoever for supposing it is true. -- Bertrand Russell
> | We have a functor from Hilbert spaces to C*-algebras, which sends > | every Hilbert space to the C*-algebra of its bounded operators. > | > | The GNS construction realizes every C*-algebra as the algebra of > | bounded operators on some Hilbert space. > | > | Does that provide us with a pair of adjoint functors between Hilbert > | spaces and C*-algebras?
> Shouldn't you first ask the simpler question if the GNS construction > provides a functor at all? I don't see that it does, given its > dependence on a choice of state(s).
Yes, right. Simpler or not, I should in any case have asked a more meaningful question! I did not properly appreciate the dependence of the GNS construction on the chosen state.
Please let me try again:
Presumably, then, what I am after is a category of pairs (A,f), with A a C*-algebra and f a state on it. Morphisms would be pairs consisting of algebra isomorphisms and the respective action on that state.
On that category of pairs the GNS construction should hopefully entend to a functor to Hilb. Does it?
Then, next, I would be after a functor going the other way around, which sends a Hilbert space to the pair consisting of its algebra of bounded operators and a state on that, such that this functor is weakly inverse, or at least adjoint to the former one.
Does that make sense now? Do functors like that exist?
In article <1173014137.629077.282...@c51g2000cwc.googlegroups.com>,
Urs Schreiber <urs.schrei...@googlemail.com> wrote: >Presumably, then, what I am after is a category of pairs (A,f), with A >a C*-algebra and f a state on it. Morphisms would be pairs consisting >of algebra isomorphisms and the respective action on that state.
You only want isomorphisms, eh? So this category is a groupoid. Let's call it X.
>On that category of pairs the GNS construction should hopefully entend >to a functor to Hilb. Does it?
Right, in fact we get a functor from X to the groupoid of Hilbert spaces equipped with unit vector, where the morphisms are unitary operators preserving the unit vector. Let's call this groupoid Y.
So, you've got a functor
GNS: X -> Y
>Then, next, I would be after a functor going the other way around, >which sends a Hilbert space to the pair consisting of its algebra of >bounded operators and a state on that,
The Hilbert space H needs to be equipped with a unit vector to get a state on the algebra L(H) of bounded linear operators on H. You seem to have forgotten the need for the unit vector.
But, if you include that, you get a functor going the other way around. We can call it
L: Y -> X
>such that this functor is >weakly inverse, or at least adjoint to the former one.
L is certainly not weakly inverse to GNS, because it's not essentially surjective: there are lots of C*-algebras that aren't isomorphic to one of the form L(H).
Furthermore, L can't even be adjoint to GNS, because adjunctions between groupoids are automatically equivalences! (The unit and counit of the adjunction are automatically invertible.)
You might try replacing X and Y by categories that aren't groupoids. I'm feeling too lazy right now to see what happens... I've done enough work for one post. :-)
John Baez wrote: > The Hilbert space H needs to be equipped with a unit vector to get a > state on the algebra L(H) of bounded linear operators on H. You seem > to have forgotten the need for the unit vector.
Yes, right, thanks -- originally I didn't pay proper attention to the role played by the "vacuum" vectors and their corresponding states.
Over at the "Noncommutative Geometry" blog by Alain Connes and others
The sensible statement that we finally seemed to have agreed on is:
There is an equivalence between the category
Hilb_vac
whose objects are pairs consisting of a Hilbert space and a cyclic vector on it, and whose morphisms are isomorphisms of Hilbert spaces,
and the category
C*_stat
whose objects are pairs, consisting of a C* algebra and a pure normal state on it, with morphisms again just being isomorphisms.
Masoud indicated that he didn't consider this a particularly exciting statement, and I guess he is right. But, while not exciting, I believe this is a _nice_ way to state a simple fact in quantum theory, namely the passage between the Schroedinger to the Heisenberg picture.
Again, for ordinary quantum mechanics this is not very exciting. But having a nice formulation is supposed to be helpful as we go on to extended n-dimensional quantum field theory, where it might help understand the relation between Segla-like functorial definitions (Schroedinger picture) and Haag-Kastler-like algebraic formulations (Heisenberg picture).
I talk about the above equivalence in this context here:
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