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Báo cáo toán học: "Gauge actions on ${\rmO_A}$ "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Đánh giá hành động trên $ {\} $ rmO_A. | J. OPERATOR THEORY 7 1982 79-100 Copyright by INCREST. 1982 GAUGE ACTIONS ON A DAVID E. EVANS 1. INTRODUCTION Suppose A is a zero-one 77 X n matrix satisfying certain standard mild conditions as in 12 9 10 . Then A the Cti -algebra generated by n non-zero partial isometries Sj . s satisfying s s A i j SjS 7 1 2 . n was introduced in 12 and shown to be uniquely determined by these relations. Here we consider the natural gauge action of the circle group T on A where t 6 T sends a generator S of A to íSị. If G G p is the torsion subgroup of T determined by a generalized integer p we consider C A G pỴ the crossed product of A under this action of G p and compute the K-groups of this crossed product in particular taking G the c -algebra of the full n-shift this enables us to show that for fixed n p is a complete invariant for G p . In 18 we showed how to construct from shift operators on Fock space CO C . In 2 this construction is modified to get A from shift operators on a m . .0 subspace FA of Fock space. This enables one to give a canonical definition of A for any matrix A without restriction. Thus if A is a permutation matrix A is canonically isomorphic to B X C T where B is a finite dimensional c -aigebra. In the other extreme we consider the full 77-shift and give an elementary proof of simplicity of Gn. In 2 we consider the C i -algebra produced by considering weighted shifts on Fa of period p for some positive integer p. This algebra can be identified as GA p for the matrix A p A X Jp where Jp is an irreducible p xp permutation matrix- A p can also be identified as C A G pỴ 11 17 . This particularly concrete representation of 0A P naturally leads US to embedd 0Aịpp in GA l 2 if Pjlp and define 80 DAVID E. EVANS oo - u A Pị I if p is a generalized integer determined by a sequence of positive i l integers where P Pi- . Then x p can be identified with C A G p . If n 1 A --- 1 these algebras are precisely the weighted shift algebras of Bunce and Deddens 4 In 4 we .