process

210 J. CuntzProof. The spectral projections p i .t/ for the different k th roots of e 2it 1, given byv 1 .t/ and v 2 .t/, have to be continuous functions of t. This implies that all the p i .t/ areone-dimensional and that, after possibly relabelling, we must have the situation wherep i .t C 1/ D p iC1 .t/:This means that the p i combine to define a line bundle on the k-fold covering of S 1by S 1 . However any two such line bundles are unitarily equivalent.We now determine the crossed product C 0 .R/ Ì Q where Q acts by translation.Lemma 6.7. The crossed product C 0 .R/ Ì Q is isomorphic to the stabilized Bunce–Deddens algebra K ˝ F .Proof. The algebra C 0 .R/ Ì Q is an inductive limit of algebras of the form C 0 .R/ Ì Zwith respect to the mapsˇk W C 0 .R/ Ì Z! C 0 .R/ Ì Zobtained from the embeddings Z Š Z 1 k,! Q. It is well known that C 0.R/ Ì Z isisomorphic to K ˝ C.S 1 /. An explicit isomorphism is obtained from the mapXf n u n 7 !X k .f n /e k;kCnn2Zn2Z;k2Zwhere k denotes translation by k, u k denotes the unitary in the crossed product implementingthis automorphism and e ij denote the matrix units in K Š K.`2Z/.This map sends C 0 .R/ Ì Z to the mapping torus algebraff 2 C.R; K/ j f.t C 1/ D Uf .t/U gŠK ˝ C.S 1 /where U is the multiplier of K D K.`2.Z// given by the bilateral shift on `2.Z/.A projection p corresponding, under this isomorphism, to e ˝ 1 in K ˝ C.S 1 /with e a projection of rank 1 can be represented in the form p D ug C f C gu with appropriate positive functions f and g with compact support on R. Underthe map C 0 .R/ Ì Z Š C 0 .R/ Ì kZ ! C 0 .R/ Ì Z, the projection p is mapped top 0 D u k g k C f k C g k u k where g k .t/ WD g.t=k/, f k .t/ WD f.t=k/. Now, p 0corresponds to a projection of rank k in K ˝ C.S 1 /.Let z be the unitary generator of C.S 1 /. Then the element e ˝ z corresponds thefunction e 2it .ug C f C gu / which is mapped to v D e 2it=k .u k g k C f k C g k u k /.Thus v k D p 0 and p 0 corresponds to a projection of rank k in K ˝ C.S 1 /.On the other hand F D lim A n where A n D M n .C.S 1 // and the inductive limit!is taken relative to the maps ˛k W M n .C.S 1 // ! M kn .C.S 1 // which map the unitarygenerator z of C.S 1 / to an element v such that v k D 1 ˝ z. Compare this now to theinductive system A 0 n D C 0.R/ Ì Z 1 n with respect to the maps ˇk considered above.