Alain Connes (pdf)

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We shall come back later to the natural moduli space for the noncommutative tori and to the metric aspect when we have the correct general tools. But we first need to dispell the impression that the noncommutative torus, because of its ubiquity, is the only example of noncommutative space. A very large class of examples is provided first of all by the duals of discrete groups. We saw already the dual of the dihedral group which was truly very simple, but as soon as one considers more complicated discrete groups one finds that their duals are exactly like leaf spaces of foliations. It is important to understand at least on one example why the dual of a discrete non abelian group is in essence a quotient space, like the one we analysed above for foliations. Thus, let Γ be the solvable group Γ = Z 2 >✁ α Z obtained as the semi-direct product of the additive group Z 2 by the action of Z given by the matrix [ ] 1 1 T = ∈ SL(2, Z) . (31) 1 2 By Mackey’s theory of induced representations each element x of the 2-torus Ẑ2 which is the Pontrjagin dual of the group Z 2 , determines by induction an irreducible representation of Γ, π x ∈ Irrep(Γ) . (32) Moreover the two representations π x and π y associated to x, y ∈ Ẑ2 , are equivalent iff x and y are on the same orbit of the (transposed) transformation T of Ẑ2 . One does not get all irreducible representations of Γ by this procedure but the quotient space X = Ẑ2 / ∼ , x ∼ y iff y = T n x for some n (33) does correspond in the above sense to the smooth group ring of Γ. Here is a more complete list of basic examples, Space of leaves of foliations Space of irreducible representations of discrete groups Space of Penrose tilings of the plane Brillouin zone in the quantum Hall effect Phase space in quantum mechanics Space time Another rich class of examples arises from deformation theory such as deformation of Poisson manifolds, quantum groups and their homogeneous spaces. Let me finally mention the space of Adele classes, a noncommutative space whose understanding is intimately related to the location of the zeros of Hecke L-functions in the number field case. Thus there is no shortage of examples of noncommutative spaces that beg our understanding but which are very difficult to comprehend. The reason why I started working in noncommutative geometry is that I found that even at the very coarse level of measure theory, the general noncommutative case was becoming highly non trivial. When you look at an ordinary space and you do measure theory, you use the Lebesgue theory which is a beautiful theory, but all spaces are the same, there is nothing really happening as far as classification is concerned. This is not at all the case in noncommutative measure theory. What happens there is very surprising. It is an absolutely fascinating fact that when you take a non commutative algebra M from the measure theory point of view, such an algebra evolves with time ! 8
What I mean is that it admits a god-given time evolution given by a canonical group homomorphism, ([6]) δ : R → Out(M) = Aut(M)/Int(M) (34) from the additive group R to the center of the group of automorphism classes of M modulo inner automorphisms. This homomorphism is provided by the uniqueness of the, a priori state dependent, modular automorphism group of a state. Together with the earlier work of Powers, Araki-Woods and Krieger it was the beginning of a long story which eventually led to the complete classification ([6], [31], [26], [21], [7], [22]) of approximately finite dimensional factors (also called hyperfinite). They are classified by their module, Mod(M) ⊂ ∼ R ∗ + , (35) which is a virtual closed subgroup of R ∗ + in the sense of G. Mackey, i.e. an ergodic action of R ∗ + . One can interpret this classification as a non-trivial Brauer theory for Archimedian local fields. Local class field theory is interesting and non trivial for local fields which are non archimedian, like p-adic numbers because such fields K are very far from being algebraically closed. But there is nothing comparable when you look at class field over complex numbers, because the field of complex numbers is algebraically closed, so there is apparently nothing going on. Now it turns out that the theory of factors gives a non-trivial replacement for the missing Brauer theory at archimedian places. This can be seen as follows. To say that a field is algebraically closed can be formulated in terms of representation theory. It is equivalent to say that when we look at representations of semisimple objects (groups, algebras...) over this field any such representation can be decomposed in direct sums of multiples of representations whose commutant is the scalars. Of course this does hold when we take finite dimensional representations over C. But as soon as we take infinite dimensional representations a new phenomenon occurs and it is in perfect analogy with the unramified extensions of p-adic fields. Non trivial factors occur, they are commutants of representations in Hilbert space and though they have trivial center, they are not Morita equivalent to C. As we saw these factors have an invariant, which is called their module, very analogous to the module of central simple algebras over local fields; this module is not really a subgroup of R ∗ + , it’s a virtual subgroup of R ∗ + in the sense that it is a flow. And for factors which are approximately finite dimensional this flow turns out to give exactly a complete invariant of factors and all flows occur. This analogy goes quite far, noncommutative manifolds give examples of construction of the hyperfinite II 1 factor as the crossed product of the field K q of elliptic functions by a subgroup of its Galois group in perfect analogy with the Brauer theory. The occurence of the type III 1 factor from simple adelic constructions yields a spectral interpretation of the zeros of L-functions in number theory ([13]). So, we see that noncommutative measure theory is already highly non trivial, and thus we have all reasons to believe that if one goes further in the natural hierarchy of features of a space, one will discover really interesting new phenomenas. Measure theory is indeed a very coarse way of looking at a space, and a finer and finer picture is obtained by going up in the following hierarchy of points of view: 9
Page 1 and 2: NONCOMMUTATIVE GEOMETRY Alain Conne
Page 3 and 4: topological spaces. They are given
Page 5 and 6: i.e. X is the quotient of S 1 by th
Page 7: and the δ j are still derivations
Page 11 and 12: Lie groups. He also gave the first
Page 13 and 14: Of course such a dictionary that tr
Page 15 and 16: In practice the choice of the limit
Page 17 and 18: and is given by where D is the Dira
Page 19 and 20: There is a very important test of o
Page 21 and 22: in [18], encoded by a spectral trip
Page 23 and 24: [11] A. Connes: Cyclic cohomology a

Alain Connes (pdf)

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