有限AF-系统的叉积和不动点代数

Crossed Products and Fixed Point Algebras of Finite AF-Dynamical Systems

AF-代数A在有限群G作用下的叉积A×_αG和不动点代数A~α是否仍为AF-代数,这是一个迄今尚未解决的问题。本文在(A,G,α)具局部G-不变的条件下给出了这个问题的肯定回答.作者还证明了:若两有限AF-系统(A,Zn,p),(A,Zn,β)满足(?)<2/(n-1)·10^(-9),且A~α是AF的,则A~β亦然.本文最后还利用算子代数的扰动理论对此问题作了一些探讨。

It is a well-known open problem whether the crossed product A×_αG and the fixed point algebra A~α of an AF— algebra Aby a finite group Care AF.Wc obtain an affirmative answer under the condition that (A, G,α)Z_n is locally G-invariant.Moreover, it is shown that A~β is an AF -algebra if (A, α), (A, Z_n,β) are two finite AF-dynamical systems with ||α_1-β_1||<2/n-1·10^(-9) and A~α is an AF-algebra.Finally, wc have some discussions on the problem by the perturbation theory for operator algebras.