process

C -algebras associated with the ax C b-semigroup over N 209It follows from the previous proposition that we have a representation of P C Qin theunitary group of M. xQ N /. Denote by A f the locally compact space of finite adeles overQ, i.e.,A f Df.x p / p2P j x p 2 OQ p and x p 2 OZ p for almost all pgwhere P is the set of primes in N.The canonical commutative subalgebra D of Q N is by the Gelfand transform isomorphicto C.X/ where X is the compact space OZ D Q O p2PZ p . It is invariant underthe endomorphisms ' n and we obtain an inductive system of commutative algebras.D; ' n /. The inductive limit xD of this system is a canonical commutative subalgebraof xQ N . It is isomorphic to C 0 .A f /. In fact the spectrum of xD is the projective limit ofthe system (Spec D; O' n / and ' n corresponds to multiplication by n on OZ.Theorem 6.4. The algebra xQ N is isomorphic to the crossed product of C 0 .A f / by thenatural action of the ax C b-group P C Q .Proof. Denote by B the crossed product and consider the projection e 2 C 0 .A f / Bdefined by the characteristic function of the maximal compact subgroup Q O p2PZ p A f . Consider also the multipliers Ns n and Nu of B defined as the images of the elements100nand1101of PCQ , respectively. Then the elements s n D Ns n e and u D Nuesatisfy the relations defining Q N . Moreover, the C -algebra generated by the s n and ucontains the subalgebra eC 0 .A f / of B. This shows that this subalgebra equals eBeand, by simplicity of Q N , it follows that eBe Š Q N .It is a somewhat surprising fact that we get exactly the same C -algebra, eventogether with its canonical action of R if we replace the completion A f of Q at thefinite places by the completion R at the infinite place.Theorem 6.5. The algebra xQ N is isomorphic to the crossed product of C 0 .R/ by thenatural action of the ax+b-group P C Q . There is an isomorphism xQ N ! C 0 .R/ Ì P C Qwhich carries the natural one parameter group t to a one parameter group 0 t suchthat 0 t .f u g/ D a it fu g where u g denotes the unitary multiplier of C 0 .R/ Ì P C Qassociated with an element g 2 P C Qof the form 1 bg D0 a(the KMS-state is carried to the state which extends Lebesgue measure on C 0 .R/).In order to prove this we need a little bit of preparation concerning the representationof the Bunce–Deddens algebra F as an inductive limit. As noted above it is an inductivelimit of the inductive system .M n .C.S 1 /// with maps sending the unitary generator zof C.S 1 / to a unitary v in M k .C/ ˝ C.S 1 / satisfying v k D 1 ˝ z. We observe nowthat such a v is unique up to unitary equivalence.Lemma 6.6. Let v 1 ;v 2 be two unitaries in M k .C/˝C.S 1 / such that v k 1 D vk 2 D 1˝z.Then there is a unitary w in M k .C/ ˝ C.S 1 / such that v 2 D wv 1 w .