Graph c*-algebras (try googling it)

[marcus]

Given a directed graph there is a conventional well-studied way to make a C* algebra out of it. There may be other interesting ways to do this--but there is at least this one clear accepted way to do it. Try googling "graph c*-algebras"

That's already kind of interesting because graphs (and higher analogs such as foams) are used to describe geometry. All we can know about a geometry (based on a finite number of observations, which incidentally are of discrete-spectrum type) is naturally described by such objects. So they are the natural way to represent geometries.

On the other hand quantum field theory is naturally represented in terms of C* algebras---the axiomatic generalization of von Neumann algebras of observables. So an obvious way to go if you want a general covariant QFT, in other words a GCQFT is to inject the underlying geometry ALSO into the C* algebra, as well as the fields.

This suggests we should be looking at a mathematical object which is simultaneously a spin network, or spin foam, AND a pair (M,ω) where M is a C* algebra and ω is a state defined on M. Essentially M is all the measurements LIVING on the particular graph/complex and ω is what we think their expectation/correlation values are.

It means that it might be rewarding to look at the conventional way people have worked out to build a C* algebra corresponding to any given directed graph.

I'll assemble the links. Anyone who has already studied this area please contribute links to any sources you found helpful.

 

[marcus]

Remember there are probably several ways to construct a star-algebra based on a graph. this is just ONE possible way---but it looks like it is at least worth learning about. When you google you get this:
http://toknotes.mimuw.edu.pl/sem1/files/Rainer_kgca.pdf
Graph C*-algebras
lecture notes by Rainer Matthes, Wojciech Szymanski

I think pages 2-6 are the most interesting and what seems of particular interest is the example #10 on page 6. Here is what they say about it:
The example (10) can be treated as the C*-algebra of the quantum sphere S_q³.

The 3D hypersphere S³ is of considerable interest in cosmology because an outstanding possibility is that the universe is spatially S³ ---- the 2D "balloon model" analog in one higher dimension. Here they seem to be talking about a quantum or fuzzy S³.

 

[marcus]

Since that first google hit seemed so interesting, I looked up all Rainer Matthes papers:

http://arxiv.org/find/grp_math,grp_physics/1/au:+Matthes_R/0/1/0/all/0/1

and all those of Wojciech Szymanski :
http://arxiv.org/find/grp_math,grp_physics/1/au:+Szymanski_W/0/1/0/all/0/1

Wow! he has a lot of papers about the C* algebras of various (quantum) spheres
and he has a lot of papers where he co-authors with Roberto Conti.

I remember Conti (and Bertozzini) in connection with papers about Tomita flow, i.e. about Tomita modular theory on C*algebras.

What I would really like to see is a C*algebra that in some sense contains the 3D hypersphere S³ and which is extensive enough to have a non-trivial Tomita flow time. That would be a fun thing to see. But I have no idea how near or far away that is.

Other links obtained by goggling "graph c*-algebras":
http://www.math.uh.edu/~tomforde/WorkshopNotes.pdf (Tomforde workshop notes)
http://wolfweb.unr.edu/homepage/alex/pub/survey.pdf (Alex Kumjian survey+biblio)

 

[marcus]

Banff workshops are a big deal. This year they are having one on graph c*-algebras, in April 2013.
http://www.birs.ca/events/2013/5-day-workshops/13w5049
The complete title is:
Graph algebras: Bridges between graph C*-algebras and Leavitt path algebras
It runs 21-26 April. The organizers are:
Gene Abrams (University of Colorado)
Jason Bell (Simon Fraser University)
Soren Eilers (University of Copenhagen)
George Elliott (University of Toronto)
Marcelo Laca (University of Victoria)
Mark Tomforde (University of Houston)
In their introductory discussion, the organizers mentioned something I found interesting: C* algebras that are the DIRECT LIMIT of finite dimensional ones. the question that occurs to me is whether these could have a rich enough structure so support a non-trivial Tomita time. I'm looking for a "toy model"---a fairly simple basic C* algebra which is just complicated enough to have a Tomita flow.
This might not be it, but the possibility did present itself. These "almost finite" C* algebras are called "AF".

Just to have it handy, here is the google search for "graph C*-algebras"
http://www.google.com/#hl=en&sugexp=les;&gs_rn=1&gs_ri=hp&cp=8&gs_id=x&xhr=t&q=graph+C*-algebras

Another intriguing idea is the "row-finite" graph in which each vertex can emit at most finitely many edges. Here's a relevant paper:
http://nyjm.albany.edu/j/2000/6-14.pdf
The C∗-Algebras of Row-Finite Graphs
Teresa Bates, David Pask, Iain Raeburn, and Wojciech Szymanski

I'm being pretty unselective here, so just googling the one thing and recording whatever seems at all interesting in what comes up. I'd love a tutorial that introduces the topic from scratch in the simplest possible terms, but haven't come across one yet. I'll stop for now and try to digest my first impressions. Feel free to comment, anybody!

 

[member 11137]

Given a directed graph there is a conventional well-studied way to make a C* algebra out of it. There may be other interesting ways to do this--but there is at least this one clear accepted way to do it. Try googling "graph c*-algebras"

That's already kind of interesting because graphs (and higher analogs such as foams) are used to describe geometry. All we can know about a geometry (based on a finite number of observations, which incidentally are of discrete-spectrum type) is naturally described by such objects. So they are the natural way to represent geometries.

On the other hand quantum field theory is naturally represented in terms of C* algebras---the axiomatic generalization of von Neumann algebras of observables. So an obvious way to go if you want a general covariant QFT, in other words a GCQFT is to inject the underlying geometry ALSO into the C* algebra, as well as the fields.

This suggests we should be looking at a mathematical object which is simultaneously a spin network, or spin foam, AND a pair (M,ω) where M is a C* algebra and ω is a state defined on M. Essentially M is all the measurements LIVING on the particular graph/complex and ω is what we think their expectation/correlation values are.

It means that it might be rewarding to look at the conventional way people have worked out to build a C* algebra corresponding to any given directed graph.

I'll assemble the links. Anyone who has already studied this area please contribute links to any sources you found helpful.

I am not certain to understand everything you state here but I did start studying for my self the notion of C* algebra a few years ago. My first attempt was concerning terms like the one we can have in GTR: A_ab^c. u^a. u^b where the A_ab^c represent the local connection (usual e.g.: the Levi-Civita one) and where the u^a are the contravariant coordinates of the speed. Since the previous products define a kind of inner product in the space of the speeds (let us call it the u-space), the initial question was: does this sort of product gives us the possibility to build a C*algebra on the u-space? The answer seems to be yes when some conditions are realized.

So my question to you: "How can I go the inverse way than the one you are describing?" More clearly: how can I go the way "C*-algebra ----> graphs and probabilities?" Is it meaningful to look in that direction. If yes would it not open a new road in direction of a quantum gravity theory in 4D? In opposite case (it is a stupid proposition), sorry for having disturbing the forums.

 

[marcus]

...
So my question to you: "How can I go the inverse way than the one you are describing?" More clearly: how can I go the way "C*-algebra ----> graphs and probabilities?" Is it meaningful to look in that direction. If yes would it not open a new road in direction of a quantum gravity theory in 4D? In opposite case (it is a stupid proposition), sorry for having disturbing the forums.

Either way the answer turns out that is a really intriguing question! I think in fact that a construction "algebra→graph" could play a role in classifying C*-algebras and would be studied by mathematicians for their own abstract reasons. Graphs may turn out to be a tool by which to analyze *-algebras.

But I'm not good enough to answer your question. All I can say is it is not disruptive (!) it is on the contrary constructive whichever way the answer is. But what I want to emphasize is that THIS PARTICULAR CONSTRUCTION "graph→algebra" that they already have, and are studying could possibly not be the RIGHT one for the quantum relativist who is working in QG and who wants to implement geometry in a C* algebra.

This idea of graph C* algebra might be good to understand and it might inspire other ways of making C* algebra out of graph----including making out of spin networks and spin foams!

But this procedure that they already have might not be what is needed, but only an inspiration or starting point. That is why I was saying this:

Given a directed graph there is a conventional well-studied way to make a C* algebra out of it. There may be other interesting ways to do this--but there is at least this one clear accepted way to do it...

This is new to me and I'm at a low level of knowledge so I want to be cautious.

 

[marcus]

BF, another thing that is kind of interesting: there is a group that is currently studying "thermal equilibrium" states on graph C* algebras.

A name of one of the senior researchers is Iain Raeburn (as a handle to find the 2012 papers by this group). I think he is in New Zealand, if I remember right.

You know that the full picture of the world is not just the C* algebra M but the pair (M,ω) where the "state" ω is a linear functional defined on the algebra which tells us expectation values. And since it can give expectation values of products like XY it automatically gives us correlations (since it is defined on an algebra, which has multiplication).

So you can think of a directed graph as a shorthand description of a PROCESS where the system goes from state A to state B and then maybe back to A, or on to C, and from C back to A, and so on. And it can go from here to there by several different paths corresponding to possibly multiple edges to choose from. So what it does can be described by "words" consisting of vertices and edges of the graph, or maybe just of edges--moves that the system makes.

The C* algebra has become a one of the preferred mathematical tools to describe a dynamical system. So they have this concept of KMS state which is a state ω defined on the C* algebra that corresponds to a sort of thermal equilibrium state of the dynamical system.

That is really exciting to me because it relates to the idea of thermal time i.e. Tomita flow which arises from a system described by a pair (M, ω). This is just an intriguing footnote to the topic at this point, and the Raeburn papers are too technical and out of reach for me at this point but I'll get a link to have for later.
http://arxiv.org/find/grp_math,grp_physics/1/au:+Raeburn_I/0/1/0/all/0/1

 

[member 11137]

Thanks for your answer.

BF, another thing that is kind of interesting: there is a group that is currently studying "thermal equilibrium" states on graph C* algebras.

A name of one of the senior researchers is Iain Raeburn (as a handle to find the 2012 papers by this group). I think he is in New Zealand, if I remember right.

You know that the full picture of the world is not just the C* algebra M but the pair (M,ω) where the "state" ω is a linear functional defined on the algebra which tells us expectation values. And since it can give expectation values of products like XY it automatically gives us correlations (since it is defined on an algebra, which has multiplication).

So you can think of a directed graph as a shorthand description of a PROCESS where the system goes from state A to state B and then maybe back to A, or on to C, and from C back to A, and so on. And it can go from here to there by several different paths corresponding to possibly multiple edges to choose from. So what it does can be described by "words" consisting of vertices and edges of the graph, or maybe just of edges--moves that the system makes.

The C* algebra has become a one of the preferred mathematical tools to describe a dynamical system. So they have this concept of KMS state which is a state ω defined on the C* algebra that corresponds to a sort of thermal equilibrium state of the dynamical system.

That is really exciting to me because it relates to the idea of thermal time i.e. Tomita flow which arises from a system described by a pair (M, ω). This is just an intriguing footnote to the topic at this point, and the Raeburn papers are too technical and out of reach for me at this point but I'll get a link to have for later.
http://arxiv.org/find/grp_math,grp_physics/1/au:+Raeburn_I/0/1/0/all/0/1

Post 1:
This suggests we should be looking at a mathematical object which is simultaneously a spin network, or spin foam, AND a pair (M,ω) where M is a C* algebra and ω is a state defined on M. Essentially M is all the measurements LIVING on the particular graph/complex and ω is what we think their expectation/correlation values are.

Constructive proposition:
If you consider any trajectory as series of edges and if you remember that the speed of any particle in GTR (Einstein's version) is
(a) a spinor (Cartan's definition; i.e.: u² = 0 (4D formulation) but also
(b) a 3D vector the normalized norm of which is obligatorily contained in [0, 1] - suggesting a complicated link with the notion of probability; in extenso: a given and measured speed realizes more or less the vacuum state...
(c) transforming conformally with the help of Lorentz transformations, [L]

...would't it be an interesting and first step to prove that products like <u, [L]. u>_A are equiping the 4D space vector with a C*algebra ...? the "state" ω should be constrained to tell us where |³u| lies between 0 and 1...

Thanks also for the interesting links. And pleasedon't think that I am better than you in physics