Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Semigroups isometries với dự đoán phạm vi đi lại. | jj. OPERATOR THEORY 3409 5 311-346 ọ Copyright by INCREST 1985 SEMIGROUPS OF ISOMETRIES WITH COMMUTING RANGE PROJECTIONS HÉCTOR N. SALAS 1. INTRODUCTION Let H denote an infinite dimensional complex Hilbert space and let jS J denote the algebra of all bounded linear operators on H. Let K H denote the algebra of compact operators on H. An isometry u on H is a linear operator satisfying ji .v li-vll for .X in H. Let m be a natural number and let V n -. n e Z be a semigroup of isometries acting on . assume that LỢ satisfies F 4- K j E for all 77 I e Z . We will restrict our attention to those semigroups of isometries that have commuting range projections . V n V ri n e Z is a commuting family in this paper we will investigate some of the properties of the c -algebra c k ỉ generated by L i n e z . Examples of isometries with commuting range projections are the following. Let be a subset of Z that satisfies E 4-Z we call such a set a module over Z7 . Let ep p e be an orthonormal basis for 2 where ep is defined by ep q ôp q for q e E. Let u n n e Z acting onZ 2 be defined by the formula U n e ep n for p 6 E and n e Z l . Then U rt n 6 Z7 is a semigroup of isometries with commuting range projections. Let c be the c -algebra generated by n e Z . Our basic structure theorem Theorem asserts that from the point of view of c -algebras these are the only examples. Specifically we shall prove that if 7 e z is an irreducible semigroup of isometries on the Hilbert space H with commuting range projections n e Z and if is different from a multiple of the identity for all 77 e Z with 1 l t 1 l 0 then there is a ZỊ-module such that C V n is isomorphic to c . An essential ingredient in the proof of this theorem is the theory of grou-poid c -algebras. The condition on the commutativity of the range projections is provided precisely to make use of this theory. The idea of using groupoids to analyse the structure of concretely given c -al-gebras is not new. In 7 Cuntz studied c .
