Pull-backs of $C\sp \ast$-algebras and crossed products by certain diagonal actions

Abstract

Let G be a locally compact group and p: Q -T a principal G-bundle. If A is a C*-algebra with primitive ideal space T, the pull-back p*A of A along p is the balanced tensor product Co((Q) ?C(T) A. If ,3: G -Aut A consists of C(T)-module automorphisms, and -y: G -Aut Co (Q) is the natural action, then the automorphism group y 0X 3 of Co(Q) 0 A respects the balancing and induces the diagonal action p*,3 of G on p*A. We discuss some examples of such actions and study the crossed product p* A x p G. We suggest a substitute D for the fixed-point algebra, prove p*A x G is strongly Morita equivalent to D, and investigate the structure of D in various cases. In particular, we ask when D is strongly Morita equivalent to A-sometimes, but by no means always-and investigate the case where A has continuous trace. Let B be a C*-algebra and G a locally compact group acting on B as a strongly continuous automorphism group a. Our goal here is to study the crossed product C*-algebra B x , G for two classes of diagonal actions for which the induced action of G on B is free. The first class includes actions of the form -y 0X3 on B Co (Q) 0 A, where p: Q -* T is a principal G-bundle, -y is the dual action of G on Co (Q), and 3: G -* Aut A is an action of G on another C*-algebra A. We also consider diagonal actions on algebras which are the pull-backs of another algebra A along a principal bundle p: Q -* T: if A is a C*-algebra with primitive ideal space T, then the pull-back p*A is the balanced tensor product Co(Q) ?Cb(T) A. When 3: G -* Aut A consists of C(T)-module automorphisms, the product action ty 0 /3 preserves the balancing, and the diagonal action p*3 is, by definition, the induced action on p* A. In general, if f: X -* Y and q: PrimA -* Y are continuous, then Cb(Y) acts on Co(X) by composition with f, and on A by composition with q and the DaunsHofmann theorem. We can therefore define the pull-back f*A of A along f as the C*-algebraic tensor product Co(X) ?Cb(y) A. The reason for the name is that when A is the algebra of sections of some C*-bundle E over Y, there is a natural isomorphism of f*A onto the algebra of sections of the pull-back f*E. In ?1 we discuss this and other basic properties of pull-backs and give some evidence to show they are likely to be of interest. In particular, we show that if G is abelian and a: G -* Aut A is locally unitary in the sense of [18], then the crossed product Received by the editors October 20, 1983 and, in revised form, February 15, 1984. 1980 Mathematics Subject Classification. Primary 46L40, 46L55.