Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: On O_{n+1} . | J. OPERATOR THEORY 12 1984 247-264 Copyright by INCREST 1984 ON ơ 1 H. ARAKI A. L. CAREY and D. E. EVANS 1. INTRODUCTION The c -algebra o generated by 77-isometries with orthogonal ranges was introduced by Cuntz in 6 and shown to be simple and independent of the choice of generators. These CiS-algebras for 2 77 oo are closely associated with the full n-shift in topological Markov chain theory 8 9 and are c -analogues of factors of type Illi . They became important initially through providing counterexamples to various questions but subsequently have become interesting c -algebras in their own right. In 9 the third author gave a construction of 0 from annihilation and creation operators on a full Fock space and began an investigation of certain states and automorphisms which are natural analogues of quasifree states and quasifree automorphisms on the CAR algebra. These states and automorphisms on 0 are also termed quasifree. Mainly type I states were analysed in 9 while here we proceed to investigate further properties in the non-type I case. The properties of quasifree states on On are more complex than in the CAR case for example they need not be factor states and we concentrate on those which are compatible in a sense made precise below with the structure of the stable algebra V On as a C -crossed product C F z of an AF-algebra F and the shift. We investigate firstly in Section 3 the question of when the shift as an automorphism of F is extendable to an automorphism of the weak closure in the representation of Fn obtained by restricting a quasifree state of When the shift is implementable the von Neumann algebra generated by in a quasifree state is identified with a jy -crossed product. This identification enables US to give sufficient conditions for a quasifree state to be a factor state. The criterion of primariness in the general situation Theorem is shown by a different method. We then go on in 4 to show that the quasifree automorphisms defined in 9 are .
