Commutative .................................................Noncommutative

functions f: X \to C .................operators on Hilbert space; elements of an algebra

pointwise multiplication fg.....................................ab (composition)

range of a function................................spectrum of an operator

Complex variable................Operator on Hilbert space

Real variable..........................Self-adjoint operator

The very first entry of this dictionary however should be about the idea of a

**noncommutative space**. So what is a `noncommutative space', really? Let me quote here an excerpt from Alain's interview with George Skandalis to appear soon in

the EMS (European math society) journal:

"Question: What is noncommutative geometry? In your opinion, is "noncommutative geometry" simply a better name for operator algebras or is it a close but distinct field?

answer: Yes, it’s important to be more precise. First, noncommutative geometry for me is this duality between geometry and algebra, with a striking coincidence between the algebraic rules and the linguistic ones. Ordinary language never uses parentheses inside the words. This means that associativity is taken into account, but not commutativity, which would permit permuting the letters freely. With the commutative rules my name appears 4 times in the cryptic message a friend sent me recently: « Je suis alenconnais, et non alsacien. Si t’as besoin d’un conseil nana, je t’attends au coin annales. Qui suis-je ? »

Somehow commutativity blurs things. In the noncommutative world, which shows up in physics at the level of microscopic systems, the simplifications coming from commutativity are no longer allowed. This is the difference between noncommutative geometry and ordinary geometry, in which coordinates commute. There is something intriguing in the fact that the rules for writing words coincide with the natural rules of algebraic manipulation, namely associativity but not commutativity. Secondly, for me, the passage to noncommutative is exactly the passage from a completely static space in which points do not talk to each other, to a noncommutative space, in which points start being related to each other, as isomorphic objects of a category. When some points are related to each other, they will be represented by matrices on the algebraic side, exactly in the same way as Heisenberg discovered the matrix mechanics of microscopic systems. One does not go very far if one remains at this strictly algebraiclevel, with letter manipulations... and the real point of departure of noncommutative geometry is von Neumann algebras. What really convinced me that operator algebras is a very fertile field is when I realized –because of the 2 by 2 matrix trick – that a noncommutative operator algebra evolves with time! It admits a canonical flow of outer automorphisms and in particular it has “periods”! Once you understand this, you realize that the noncommutative world instead of being only a pale reflection, a meaningless generalization of the commutative case, admits totally new and unexpected features, such as this generation of the flow of time from noncommutativity. However, I don’t identify noncommutative geometry with operator algebras; this field has a life of its own. New phenomena are discovered and it is very important to study operator algebras per se -I have spent a large part of my life doing that. But on the other hand, operator algebras only capture certain aspects of a noncommutative space, and the “only” commutative von Neumann algebra is L∞[0; 1]! To be more specific, von Neumann algebras only capture the measure theory, and Gelfand’s C*-algebras the topology. And there are many more aspects in a geometric space: the differential structure and crucially the metric. Noncommutative geometry can be organized according to what qualitative feature you look at when you analyze a space. But, of course, as a living body you cannot isolate any of these aspects from the others without destroying its integrity. One aspect on which I worked with greatest intensity in recent times is a shift of paradigm which is almost forced on you by noncommutativity: it bears on the metric aspect, the measurement of distances. This is where the Dirac operator plays a key role. Instead of measuring distances effectively by taking the shortest path from one point to another, you are led to a dual point of view, forced upon you when you are doing non-commutative geometry: the only way of measuring distances in the noncommutative world is spectral. It simply consists of sending a wave from a point a to a point b and then measuring the phase shift of the wave. Amusingly this shift of paradigm already took place in the metric system, when in the sixties the definition of the unit of length, which used to be a concrete metal bar, was replaced by the wavelength of an atomic spectral line. So the shift which is forced upon you by noncommutative geometry already happened in physics. This is a typical example where the noncommutative generalization corresponds to an abrupt change even in the commutative case."

I was planing to continue with a detailed analysis of the question, but I think it is important to stop right now and answer some questions. I would particularly encourage students and others to come online and pose their questions, comments and remarks about issues discussed so far.

## 15 comments:

If one converts states |a), to density matrices |a)(a|, one converts states to operators. The necessity of the Hilbert space then goes away and one can describe quantum mechanics entirely in the operator algebra, except for the symmetry groups.

The large number of elementary particles suggests that they are composite. One can therefore hope that the symmetry groups can appear from the bound states. In fact, this is how symmetry groups were originally applied in physics to describe the state of molecules.

With this sort of program, one could hope to unify the elementary particles entirely within the operator algebra alone. The Hilbert space and the symmetry groups would be derived things.

Here're my questions. It's very nice of You that You devote Your time here also to laymen. Thanks!

1) About real/complex <---> self-adjoint/general operators

I'm not sure I understood AC's comment: Complex is simpler than real because although complex contains real, we restrict our attention to functions which have particularly good properties in complex domain (like, say, maximum principle). Now, general operators are easier than self-adjoint operators because... what? Because we're interested in this action of functions AC described and it's simpler in case of all operators?

2) a) Unfortunately, I understood next to nothing from David Goss article. Maybe some simpler version with additional explanations? :-)

b) Did NCG solve any problem in classical number theory? Maybe there are classicaly solved problems in number theory that can be solved also, for educational purposes, by NCG methods?

c) What does it mean that index of Fredholm operator is a model for integers? or that a such that [a,a*]=1 is a model for naturals? I'm sure there is something more to it then just that index is integer and spec consists of natural numbers?

3) Here're some quotes from above which I _totally_ don't understand (but which are very intriguing, even "magical" ;-). I would appreciate some explanations.

"the passage to noncommutative is exactly the passage from a completely static space in which points do not talk to each other, to a noncommutative space, in which points start being related to each other, as isomorphic objects of a category. When some points are related to each other, they will be represented by matrices on the algebraic side, exactly in the same way as Heisenberg discovered the matrix mechanics of microscopic systems."

In particular, is there some geometric picture one has in mind when one thinks of NGC?

"a noncommutative operator algebra evolves with time! It admits a canonical flow of outer automorphisms and in particular it has “periods”!"

"Instead of measuring distances effectively by taking the shortest path from one point to another, you are led to a dual point of view, forced upon you when you are doing non-commutative geometry: the only way of measuring distances in the noncommutative world is spectral. It simply consists of sending a wave from a point a to a point b and then measuring the phase shift of the wave."

But there are no points in NGC, are there? So what we want to measure?

Thought on language: noncommutative things were always much more diffcult for me than commutative ones. Actually, in my first algebra course I just ignored concept of noncommutative group (I thought that something like this could never be of any use :-). Now I wonder: maybe it's because polish, which I'm native speaker of, has _much_ more flexible syntax than french and one can toss the words around a whole sentence quite freely? :-)

I forgot one more: Throughout this blog word "integrality" appears here and there.

I don't understand what it refers to.

What is algebraic non-commutative geometry (or maybe non-commutative algebraic geometry)?

Hi Masoud

Before saying anything else let me say "thanks" for organizing this blog. I have enjoyed Lieven's blog the past several years and it is good to now have a blog that focuses on the operator algebra approach to non-commutative geometry.

Hi Alain,

"Thanks" to you too, Alain, for taking the time to educate us in this informal way and for encouraging us to send in questions.

In your March 16 post "What is a non-commutative space?" you said "because of the 2 by 2 matrix trick [...] a noncommutative operator algebra evolves with time! It admits a canonical flow of outer automorphisms and in particular it has “periods”!"

I have heard you mention this time evolution before but have not made any effort to understand what you have meant. Can you say a little more. For example, I don't know "the 2-by-2 matrix trick". Or perhaps I do, but without that label attached to it.

Are you using the outer automorphisms to get this time evolution? If so, how?

Should there be some shadow of this time evolution within the realm of non-commutative algebraic geometry, or is it a purely operator algebra phenomenon?

Paul Smith

Alain,

I have a question and a comment regarding "points"

in noncommutative geometry.

What I ask/say is related to part (3) of Sirix's comment.

You say "passage to noncommutative is exactly the passage from a completely static space in which points do not talk to each other, to a noncommutative space, in which points start being related to each other, as isomorphic objects of a category. When some points are related to each other, they will be represented by matrices on the algebraic side, exactly in the same way as Heisenberg discovered the matrix mechanics of microscopic systems. "

I agree that "points talk[ing] to each other" is a fundamental aspect of non-commutativity.

To me, coming from non-commutative algebraic geometry rather than operator algebras, the simplest "ring of functions on a non-commutative space" is the ring R of upper triangular 2-by-2 matrices over a field k. This ring has two maximal two-sided ideals, and correspondingly two simple left modules up to isomorphism. Let's call them L and N. Then, with the correct labelling, Ext^1(N,L) is non-zero, i.e., there is a non-split extension

0-->L-->M-->N-->0.

I think of the two maximal ideals as two different "points" but they "talk to each other" in the sense that there is this non-split extension.

In commutative algebraic geometry if one has two

distinct closed points p and q on a scheme X with corresponding structure sheaves (or skyscraper sheaves) O_p and O_q, then Ext_X^i(O_p,O_q)=0

for all i. More generally, if F and G are coherent sheaves with disjoint supports then Ext^i_X(F,G)=0

for all i. In this sense, F and G do not talk to each other.

And of course this fits in perfectly with our everyday geometric ideas: if Y and Z are disjoint closed subspaces of a space X the algebraic and geometric properties of Y are unaffected by the algebraic and geometric properties of Z; the ambient space X can not suddenly create some interaction between Y and Z.

The idea of a "point" in non-commutative geometry still remains somewhat mysterious to me. There are often some things that deserve to be called points, i.e., that will be points (maybe I should say closed points) under any reasonable definition, but the precise boundary of the definition of "closed point" is not clear to me.

For example, if I have a finitely generated algebra R over a field k, a simple R-module having finite dimension over k should, I think, be a closed point (or perhaps some fraction of a point).

Similarly, if, in the above situation, there is a two-sided ideal I in R such that R/I is commutative, we should think of Spec(R/I) being a closed subspace of the nc-space (nc=non-commutative) corresponding to R. And, closed points of Spec(R/I) should then be considered as points on the nc-space. In this way, a nc-space can have lots of points.

Let me return again to the ring R of upper triangular matrices and for brevity I will

write Sp(R) for the nc-space related to it. Why do I think Sp(R) is the simplest nc-space rather than, say, the space related the ring M_2(k) of all 2-by-2 matrices?

Well, the category of modules over M_2(k) is equivalent to the category of modules over k. So from the module-theoretic point of view M_2(k)

is commutative. But Mod(R) is not equivalent to the module category of any commutative ring.

Despite all the above I wonder whether I am missing something important because the ring R of upper triangular 2-by-2 matrices over the complex numbers is not a C^*-algebra. Is there

any spectral triple that behaves like R?

Finally, I don't think I appreciate what you are saying Alain when you say "related to each other, as isomorphic objects of a category". Are you saying more than this: in a non-commutative ring R one can have different maximal left ideals I and J such that the modules R/I and R/J are isomorphic?

Paul Smith

Dear Paul,

Thanks for your comments and good to be in touch again after the Newton Institute NCG conference! Just wanted to make a small comment about your last comment (though it is posed to Alain, but I venture an answer anyhow!). I guess what Alain means here is that this `categorification' (I can't believe myself using this word!) of the idea of an `equivalence relation' to a `groupoid' and the resulting groupoid algebra construction leads directly to noncommutative algebras of interest in NCG. One of course should use topological groupoids to get really interesting examples. Would be nice to see a relation with your example also.

A comment about the possible meaning of non-commutativity. It is often thought that non-commutativity could reflect something emerging at Planck length scale. Inclusions of hyper-finite factors of type II_1, which I am personally fond of, would suggest an alternative interpretation in terms of quantum measurement theory with finite measurement resolution: the notion of resolution is indeed lacking from standard quantum measurement theory.

For hyper-finite factors of type II_1 the probability of reduction to complex ray vanishes so that one is forced to introduce projections to infinite-D subspaces and the notion of measurement resolution in terms of the sub-factor N for the inclusion N subset M. Quantum measurement would project to N-ray instead of complex ray and quantum operator space M/N and corresponding quantum state space would define finite-D state space of reduced degrees of freedom whose rays are N-rays.

One can assign to Jones inclusions quantum counterparts of SU(2) fundamental representations and according to Wenzl's thesis SU(k) allow quantum counterparts of all representations of these groups as inclusions defined by Hecke algebras. I believe that this generalizes to all compact Lie groups. The choice of quantum symmetry group and its representation would thus also characterize quantum measurement. For instance, the flavor groups SU(3) and SU(4) of early particle physics do not correspond to standard model symmetries and might be interpreted in terms of measurement resolution. It might be also possible to understand the dualities of M-theory in this picture.

Matti Pitkanen

Paul: Thanks, I was away in Rome and just came back. The stuff about "points talking to each other" in NCG is exactly as you explain and, as Masoud says, this is the way arrows in a groupoid give rise to an algebra. The algebra R of triangular matrices is precisely what is used in the two by two matrix trick where one uses the nilpotent element of R to relate two different states of the algebra. I believe it is better if I try to make a "post" just on that point, but it will take me some time to write it properly.

Sirix: Thanks, for c) the meaning

of "that index of Fredholm operator is a model for integers"

is that quite often there is a payoff when you are handed an integer invariant (think of the Euler characteristic of a polyhedron as an example) to find a natural operator of which this integer is the index. All your other questions are relevant but take time to answer. Note that the nuance between real and complex in the quote of Weil is not about simplicity but "the complex world is beautiful, the real world is dirty"....

Carl: Yes it is a good point that the Hilbert space is a "derived" thing in quantum mechanics where the more fundamental datum is the algebra of observables. This was from the start the point of view of Heisenberg.

hi anonymous: to your question: "what is noncommutative algebraic geometry?" I can say this. NCAG is an attempt to define and find NC analogues of algebraic varieties and schemes in general. These noncommutative analogues are usually encoded in terms of an abelian, or just triangulated, category with some specific extra structure. The first main observation is that an algebraic variety or scheme can be reconstructed fully in terms of the abelian category of coherent sheaves on the scheme. This is a kind of duality theorem similar in its spirit to Gelfand's theorem for commutative C* algebras. I think instead of going any further I better ask Paul to comment and answer this question more fully please!

AC: In this particular case of polyhedrons, what could be the relevant operator? It would be operator on what space?

Sirix: in the case of polyhedra or in fact of any finite CW complex the Euler characteristic is the index of any operator going from the space of even dimensional cochains to odd dimensional cochains. Here an n-dimensional cochain assigns a real number to any n-cell. In the case of operators acting in finite dimensional vector spaces the index is independent of the choice of the operator and if the operator T goes from E to F its index is just the dimension of E minus the dimension of F. What is really important then is what happens when you pass to a situation where the operator acts on infinite dimensional spaces. For the Euler characteristic of a manifold M the corresponding operator is the sum of the de-Rham differential d (on the space of differntial forms) with its adjoint d* (for a metric on M).

Alain Connes was quoted as saying, in small part:

"What really convinced me that operator algebras is a very fertile field is when I realized [...] that a noncommutative operator algebra evolves with time! It admits a canonical flow of outer automorphisms."

Now, given a symplectic space and the (noncommutative) Weyl algebra A obtained from it, can one associate a "canonical flow of outer automorphisms" to it? And what is it like?

As far as I understand, the "intrinsic time evolution" is (nontrivially) defined precisely for von Neumann algebra type III factors, where it is induced by the adjoint action of imaginary powers of the positive operator "Delta", which is provided by Tomita-Takesaki theory.

Type III factors appear as algebras of observables in particular in 2-dimensional conformal field theory. In these theories, there is also a "standard" time evolution represented on these algebras.

How is this "standard" time evolution related to the "intrinsic time evolution"?

Or more generally: what are concrete examples that show that it is sensible to address the canonical 1-parameter action by outer automorphisms that exists for type III factors as "time evolution".

(I am in particular interested in this, because over at the n-Cafe we are currently thinking about how and when canonical quantum dynamics can be associated to a given setup #. For a moment I was hoping that this might have a relation to Alain Connes' "intrisic time evolution". But maybe not.)

Thank you for responding Masoud. Perhaps I should refocus my question a little. When is a non-commutative space algebraic? Cheers.

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