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Báo cáo toán học: "Crossed products by locally unitary automorphism groups and principal bundles "
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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Vượt qua sản phẩm của các nhóm automorphism ở địa phương đơn nhất và bó chính. | g Copyright by INCREST 1984 J. OPERATOR THEORY 1 1 1984 215 241 CROSSED PRODUCTS BY LOCALLY UNITARY AUTOMORPHISM GROUPS AND PRINCIPAL BUNDLES JOHN PHILLIPS and IAIN RAEBURN Let A be a continuous trace c -algebra with spectrum T and let a z - - Aut t be a group consisting of C T -automorphisms with generator Ỉ ax. We proved in 13 Section 2 that there is a cohomology class in the Cech group H2 T Z which vanishes precisely when p is implemented by a unitary element of the multiplier algebra ểll í or equivalently when a is implemented by a homomorphism u z - Elements of H2 T Z are associated with isomorphism classes of principal -bundles over T and in fact the class Ii fi is constructed in 13 as the transition functions of such a bundle. The starting point of our present work was the observation that this principal bundle appears naturally as the spectrum of the crossed product c -algebra A z the action of s1 on A xa Z A comes from the dual action of s1 z on XxaZ and the bundle projection p A X Z - A is given by sending 7Ĩ X Us A xaZ A to the irreducible representation K of A. We shall show here that Theorem 2.1 of 13 is a special case of general theorems which relate a class of abelian automorphism groups of a type 1 c -algebra A to locally trivial principal bundles over A. Let A be a type I 7 -aIgebra G a locally compact abelian group and a G - - Aut A a strongly continuous automorphism group. We shall say that a is implemented by w G - SKẠA in the representation 7Ĩ of A if 7ĩ ag ữ 7i wgaw for g e G as A we call a locally unitary if there are maps of G into ẩR. 4 which implement a locally in A. A theorem of Russell 17 shows that singly generated groups of C l -auto-morphisms of continuous trace c -algebras are always locally unitary and we show in Section 1 that there are other circumstances in which automorphism groups are automatically locally unitary. Suppose a. G - Aut A is locally unitary. The dual action a of G on A xa G induces an action of G on XxaG A and it