AN ANALOGUE OF THE THOM ISOMORPHISM EOR CROSS PRODUCTS OF A C ALGEBRA BV AN ACTION OF R ALAIN COMNES Institut des flautes Etudes Scientifiques route de Chartres, 9744o - Bures-sur-vette (France) 17cι6ιικ/Sani 0861 Aaenues

– 2 – Construction of the pairing K(A) ΘK(R) FK(AN RR) Let (A, R, a) be a C* dynamical system, and δ be the corresponding unbounded t (x) –x derivation of A: δ(x) - Iim(–– Let us first assune that A has a tO unit and construct for each pair (lel, lul) e K (A) x K (R) an element of K, (AMR), vhere e ê Proj M, (A) and u is a unitary in C, q) a c(s1) Replacing A by M (A) one can for the construction assume that n - 1, also one can perturb e slightly in norm and assume that it is smooth, in particular that e E Dom δ. Let 61 –6* Ad (le, 6(e))), then it is a nev derivation of A defining an action α! of R on A such that α (e) = e Vs ER, indeed le, 6(e)]7 -– 1e, 6(e)] so 6′ is selfadjoint, also: 61 (e) - 6(e) * IIe, 6(e)), el -0 (because eö (e) * 6(e)e - 6(e7) - 6(e) and eö (e)e -0). Let s 5 V, be the canonical uni tary representation of R in the multipliers M(A*, RR), then to the action αl, corresponds a neu representation Vt, with vt e ugV,, u being the canonical accocycle with u! - le, δ (e)]. Ne extend the repre¬ sentation v1 in a homomorphism π᾽ of cα) in (AN R). Note that e commutes with π' ((cλ)7) so that we define a unitary by the formula v = en! (u) 1 1–e vhere e EA is considered as a multiplier of AN, R, for this we need to assume, which is obvious ly not a restriction, that u - 1A, c ê Crq) so: v = ex“ (c) * 1 Let t(le), lul) be the class of v in K, (A * R). It is obvious that t((e), (u)) depends only on the class of u and that it is an additive function of this class. AIso if one changes e in e! with le) - le), one can assume, by connectedness of the equivalence classes of projections that seze is as smalI as needed and also that [6(e)-6(e1)] is as smal! as needed, it

– 5 – corolary. Let (A, R, a) be as above, with δ the basic derivation, let τ be an α invariant trace on A and τ the dual trace on A* R, then the image of K, (AX R) by the trace τ is: τ(6 (u)u1), u smooth element of GL(A)) In a fortheoming paper we shal! extend to arbitrary c dynami cal systens (A, G, a) with G a Lie group, the pseudo-differential calculus of (3) and show how the above formula follows from the index theorem for flows. Let us end this short note by a feu examples: Example L. Let, q be a minimal diffeomorphism of the 3 sphere 8° (cf. (4)) and A = C(8*) x Z, then by (8) A is simple, with unit, moreover by amenabi li ty of Z, it has a faithful trace τ, now the above corollary and the equality H1(s°, z) - (0) show that τ(K, (A)) -Z, thus A has no non trivial projections. (I was informed by J. Cuntz that B. Blackadar had al ready obtained an example of a simple C* algebra without projections (2)). Example 2. The above theorem allovs to compute the K theory of cross products apparent ly not related to flous, for instance one obtains that if F C SL(2, IR) is a cocompaet discrete subgroup if one lets it act on P, () one has for the cross product a K theory isomorphic to the usual K theory of the compact 3-mani fold SL(2, 1R)/r Exanple 3. The horocycle flow, i. e. the action by lest multiplications of 1c Ut 1J tER in SL(2, 1R)/r vith Y as above gives a foliation (V, 6) for which the homology class of the Ruelle-Sullivan current is O (for the unique invariant transverse measure A), thus the index theorem of (3) gives Ind (D) - O for any elliptic D vhile the index with values in K, (cr(v, 6)) is actually an isomorphism with this group e K1(sL(2, 1)/r). AIso cr(v, 6)

– 6 – is, in this example, a simple C algebra without non trivial projections, vhi Ie K 40. Bibliography (1) M. F. Atiyah and 1. Singer, The index of elliptic operators I, Ann. of Math. 87 (1968) p. 484–530. (2) B. BIackadar, A simple unital projectionless C* algebra, Preprint, University of Nevada, Reno. (3) A. Connes, Sur Ia theorie non commutative de l'integration, Springer Lecture Notes in Math. 725 (1979) p. 19–143. (4) A. Fathi et M. Herman, Existence de diffeomorphismes minimaux, Preprint Ecole Polytechnique (1976). (5) P. Green, The structure of imprimitivity algebras, Preprint, I. A. S., Prince ton. (6) J. MiInor, Introduction to algebraic K-theory, Annals of Math. studies, No 72 Princeton. (7) M. Pimsner and D. Voiculescu, Exact sequence for K groups and Ext groups of certain cross product, C* algebras. (8) J. L. Sauveageot, Ideaux primi tifs induits dans les produits croisens, J. Funct. Analysis. (9) 1. Takai, on a duality for cross products of C algebras, J. Funct. Analysis 19 (1975) p. 25–39. SU