A C*-algebra is an algebra of quantum operators (with some functions like addition, multiplication, conjugation, norm, and identity).
Yes, like B(H) for some Hilbert space H, bounded linear operators
A commutative C*-algebra has a set of operators which all commute together
Yes
A non-commutative C*-algebra has a set of operators in which any pair does not commute
Yes
We are mostly concerned with commutative C*-algebras
At the point of measurement, yes. We do care about the non-commutative C* algebra and there are different ways of trying to characterize it, including through its commutative sub-algebras
This commutative C*-algebra/will be the analog for Andrés’ monads...?
I wasn’t trying to connect it to monads. I am not sure I understand the idea of monads. Perhaps I don’t believe in them. Nevertheless, non-commutative C* algebras have some aspects of “privacy” in that they cannot be exhaustively measured.
These form a meet-semilattice; a partial order of commutative C*-algebras and their intersection sets
Yes. At least finite dimensionally, a commutative C* algebra gives a PVM. Sub-algebras give coarser PVMs.
For every commutative C*-algebra there is a corresponding topological space
Yep, by Gelfand duality. For finite dimensional Hilbert space H, B(H) gives a discrete topology over a count of elements equal to dimension of H. It’s more interesting in the infinite dimension case.
For every non-commutative C*-algebra there is a corresponding non-commutative topological space.
There are some attempts at this in non-commutative geometry, but I’m not very familiar. It at least does not give a topological space in the normal sense.
So you take the meet-semilattice on the left; you can map this to a graph of topological spaces on the right. In my head there is a graph of classical systems on the left and an isomorphic graph of little toruses and spheres and whatever on the right. My intuition kinda fails here also; if you take a point on the surface of a torus, what do you see this corresponding to in the corresponding classical system/C*-algebra?
Let’s think of the easier case, where we start with a commutative C* algebra at root. It corresponds with some topological space. It has commutative sub-algebras. These correspond with quotient spaces. There is a category theoretic reason for this: Sub-objects in the category of commutative C* algebras and homomorphisms are specified by monomorphisms, and when reversing the arrows (contravariant duality), these give epimorphisms. Epimorphisms (specifically regular epimorphisms) give quotient objects in the category of compact Hausdorff spaces and continuous maps. So there are ways of “coarsening the opens” in a topological space to get a “quotient space” which glues together some points, making them less distinguishable by opens.
It of course is a bit complicated when the root C* algebra is non commutative. Here it is a bit like an abstract object which you can take quotients of, and some of those quotients lead to topological spaces, but it isn’t a topological space (properly at least; non-commutative topology might say something) prior to quotienting. Some of the motivation here is the idea that we might be able to better understand a non commutative C* algebra by looking at which topological spaces can be found within its commutative sub-algebras.
As for specific points: Given a commutative C* algebra A, the points in the topological space are the characters, which are C* homomorphisms from A into the complex numbers. Let’s start with the finite dimension case. Let be taken as a commutative C* algebra (element wise addition / multiplication). Now the topology for is a discrete topology over n elements. A character of is a C* algebra homomorphism from into . Since C* algebra homomorphisms are complex-linear, this is a “covector” of , often represented as a row vector. Moreover, it must map the unit of (which is copies of 1) to 1, and preserve multiplication. The only possible homomorphisms here are those that read a given component, e.g. .
Conceptually, you would use to represent a classical complex-valued operator on a system with possible states. A character would accordingly be a given state; interpreted as a character, it maps an operator to its value. This is of course not very interesting on its own, but it illustrates the general idea.
A more interesting test case would be to take some commutative sub-algebra of B(H) for infinite dimension H, for example, the sub-algebra for position of one particle. Here, a point in the topological space corresponds to giving a value to position-measuring operators. That is equivalent to specifying a position. Unlike with the finite dimension case, the topology is non-trivial.
Yes, like B(H) for some Hilbert space H, bounded linear operators
Yes
Yes
At the point of measurement, yes. We do care about the non-commutative C* algebra and there are different ways of trying to characterize it, including through its commutative sub-algebras
I wasn’t trying to connect it to monads. I am not sure I understand the idea of monads. Perhaps I don’t believe in them. Nevertheless, non-commutative C* algebras have some aspects of “privacy” in that they cannot be exhaustively measured.
Yes. At least finite dimensionally, a commutative C* algebra gives a PVM. Sub-algebras give coarser PVMs.
Yep, by Gelfand duality. For finite dimensional Hilbert space H, B(H) gives a discrete topology over a count of elements equal to dimension of H. It’s more interesting in the infinite dimension case.
There are some attempts at this in non-commutative geometry, but I’m not very familiar. It at least does not give a topological space in the normal sense.
Let’s think of the easier case, where we start with a commutative C* algebra at root. It corresponds with some topological space. It has commutative sub-algebras. These correspond with quotient spaces. There is a category theoretic reason for this: Sub-objects in the category of commutative C* algebras and homomorphisms are specified by monomorphisms, and when reversing the arrows (contravariant duality), these give epimorphisms. Epimorphisms (specifically regular epimorphisms) give quotient objects in the category of compact Hausdorff spaces and continuous maps. So there are ways of “coarsening the opens” in a topological space to get a “quotient space” which glues together some points, making them less distinguishable by opens.
It of course is a bit complicated when the root C* algebra is non commutative. Here it is a bit like an abstract object which you can take quotients of, and some of those quotients lead to topological spaces, but it isn’t a topological space (properly at least; non-commutative topology might say something) prior to quotienting. Some of the motivation here is the idea that we might be able to better understand a non commutative C* algebra by looking at which topological spaces can be found within its commutative sub-algebras.
As for specific points: Given a commutative C* algebra A, the points in the topological space are the characters, which are C* homomorphisms from A into the complex numbers. Let’s start with the finite dimension case. Let
Conceptually, you would use
A more interesting test case would be to take some commutative sub-algebra of B(H) for infinite dimension H, for example, the sub-algebra for position of one particle. Here, a point in the topological space corresponds to giving a value to position-measuring operators. That is equivalent to specifying a position. Unlike with the finite dimension case, the topology is non-trivial.