Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Wiener-Hopf các nhà khai thác với biểu tượng gần như định kỳ piecewise liên tục. | Copyright by INCREST 1985 J. OPERATOR THEORY 14 1985 147-171 WIENER-HOPF OPERATORS WITH PIECEWISE CONTINUOUS ALMOST PERIODIC SYMBOL JINGBO XIA INTRODUCTION Several authors have studied the index theories of different types of singular integral operators with piecewise continuous symbol see 11 12 14 15 . Basically their results are generalizations of corresponding theory for the same type of operators with continuous symbol. What makes significant difference is that in the noncontinuous case the discontinuities of the symbol represent a part of the essential spectrum for the corresponding operator and become a part of the curve in the index counting. In the context of von Neumann algebra these results may be categorized as type Ico theory. In the present paper we shall develop a type IIoo analogue. It is well known that there is an index theory for Wiener-Hopf operators with continuous almost periodic symbol where the analytical index is obtained via a faithful representation in a type Iloo factor Jf see 6 7 4 . We are interested in adding discontinuities to these almost periodic symbols. We know that every continuous almost periodic function is identified with its Gelfand transform on RB the Bohr compactification of R. Also there is an obvious way to identify each piecewise continuous periodic function with an element in L RB . We simply mix these two types of functions to produce a c -algebra ỉf which is also considered as the uniform closure of all the piecewise continuous almost periodic functions. The main object of the study is the c -algebra 32 5 generated by Wiener-Hopf operators on 7 2 R with symbols in if. For each IF 6 32 5 the symbol p is regarded as a function on RB. Intuitively since the von Neumann algebra Jf comes from the group measure space construction on R X Rd it should be large enough to accommodate a faithful representation of 32 5 . Indeed it does. What we shall do is to extend the representation p used in 7 to 32 5 . Then the Breuer index .
