When we started this blog we promised to gradually build a dictionary (see here and here ) of concepts in use in NCG (= noncommutative geometry). We started with the following list
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.