Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Từ đơn giản C *- đại số tiếp tuyến cho các nhóm automorphism nhỏ gọn. | Copyright by INCREST 1985 J. OPERATOR THEORY 13 1985 189 -200 DERIVATIONS OF SIMPLE C -ALGEBRAS TANGENTIAL TO COMPACT AUTOMORPHISM GROUPS DEREK w. ROBINSON ERLING STƠRMER and MASAMICHI TAKESAKI 0. INTRODUCTION Recently 1 Kishimoto and the first named author showed that an asymptotic abelian automorphism r of a simple c -algebra it with identity gives rise to a multiple tensor product structure in the limit which allows one to prove that if an action a of a compact abelian group G as -automorphisms of 91 commutes with T hen any derivation Ỗ of the algebra 21F of z-finite elements into 21 commuting with T is automatically tangential to ac in the sense that a 5 is closable and b the closure 5 of 5 generates a one parameter subgroup of ac. Based on this limit tensor product structure we prove that the above result remains true without the commutativity assumption on the group G. Since every derivation given by a one parameter subgroup of ac commutes with T and is defined on 9IF our result characterizes the Lie algebra of ac as those derivations of 91F into 91 commuting with Ĩ. The proof is interesting in its own right. It consists of two parts. First we prove that any automorphism p commuting with T and leaving the fixed point algebra 9Í pointwise fixed is indeed an element of ac. This Galois type result will be shown by adapting the Tannaka duality arguments developed in a work of Araki-Haag-Kastler-Takesaki 2 We then prove that the derivation Ỏ leaves any a-invariant finite dimensional subspace globally invariant and this allows US to exponentiate 5 and obtain a one parameter group Pt of automorphisms of 9tF. In this proof we use the exterior product based on the limit multiple tensor product. Finally we prove that p extends to 91 by positivity arguments adapted from 3 and then apply the first result to conclude that P c aG. Before proceeding we recall a few essential definitions. 190 DEREK w. ROBINSON ERLING STORMER and MASAMICHI TAKESAKI First the c -algebra 91 is
