process

C -algebras associated with the ax C b-semigroup over N 211From our analysis of ˇk and from Lemma 6.6 we conclude that there are unitaries W kin M.A 0 / such that the following diagram commuteskn˛k A knA nA 0 nAd W kˇkA 0 knwhere the vertical arrows denote the natural inclusions M n .C.S 1 // ! K ˝ C.S 1 /.We conclude that these natural inclusions induce an isomorphism from K ˝ F Dlim!K ˝ A n to C 0 .R/ Ì Q D lim!A 0 n .Proof of Theorem 6.5. Consider the commutative diagram and the injection F ,!C 0 .R/ Ì Q constructed at the end of the proof of Lemma 6.7. Note also that theˇk are -unital and therefore extend to the multiplier algebra. This shows that theinjection transforms ˛p into ˇp times an approximately inner automorphism, for eachprime p. Therefore the injection transforms t , which is the dual action on the crossedproduct for the character n 7! n it of N , to the restriction of the corresponding dualaction on the crossed product .C 0 .R/ Ì Q/ Ì N .Finally, note that the endomorphisms ˇn of C 0 .R/ Ì Q are in fact automorphismsand that therefore ˇ extends from a semigroup action to an action of the group Q C .This shows that .C 0 .R/ Ì Q/ Ì N D .C 0 .R/ Ì Q/ Ì Q C D C 0.R/ Ì P C Q .Remark 6.8. If we consider the full space of adeles Q A D A f R rather than the finiteadeles A f or the completion at the infinite place R, we obtain the following situation.By [9], IV, §2, Lemma 2, Q is discrete in Q A and the quotient is the compact space. OZ R/=Z. Thus, the crossed product C 0 .Q A / Ì Q which would be analogous tothe Bunce–Deddens algebra is Morita equivalent to C.. OZ R/=Z/. Now,. OZ R/=Zhas a measure which is invariant under the action of the multiplicative semigroup N .Therefore the crossed product C 0 .Q A / Ì P C Q has a trace and is not isomorphic to Q N .7 The case of the multiplicative semigroup Z We consider now the analogous C -algebras where we replace the multiplicative semigroupN by the semigroup Z . This case is also important in view of generalizationsfrom Q to more general number fields (in fact the construction of Q Z and much of theanalysis of its structure carries over to the ring of integers in an arbitrary algebraic numberfield, at least if the class number is 1). Thus, on `2.Z/ we consider the isometriess n ;n2 Z and the unitaries u m ;m2 Z defined by s n . k / D nk and u m . k / D kCm .As above, these operators satisfy the relationss n s m D s nm ; s n u m D u nm s n ;Xn 1u i e n u i D 1: (1)iD0