A DIFFERENTIABLE STRUCTURE FOR A BUNDLE OF C-ASTERISK-ALGEBRAS ASSOCIATED WITH A DYNAMICAL SYSTEM

Let (M,G) be a differentiable dynamical system, and a be a transverse action for (M,G). We have a differentiable bundle (B, pi, M, C) of C- algebras with respect to a flat family F-sigma of local coordinate sys tems and we have a flat connection del in B. If G is connected, the bu ndle B is a disjoint union of rho(x)(C-r(G)) (x is an element of M), where G is the groupoid associated with (M, G) and rho(x) is the regul ar representation of C-r (G). We show that, for f is an element of C- c(infinity) (G), a cross section cs(f) : x --> rho(x)(f) is differenti able with respect to the norm topology, and calculate a covariant deri vative del(cs(f)). Though B is homeomorphic to the trivial bundle, the differentiable structure for B is not trivial in general. Let B-sigma be a subbundle of B generated by elements f with the property del(cs( f)) = 0. We show the triviality of the differentiable structure for B- sigma induced from that for B when C-r(G) is simple. We have a bundle RM (B) of right multiplier algebras and it contains B as a subbundle. Let (M,G) be a Kronecker dynamical system and a be a flow whose slope is rational. In this case, we have a subbundle D of RM (B) whose fibe rs are - isomorphic to C(T). The flat connection del(r) in D is not t rivial and the bundle B decomposes into the trivial bundle B-sigma and the non-trivial bundle D. Moreover, for a a-invariant closed connecte d submanifold N of M with dim N = 1, we show that C-r(G\N) is *-isomo rphic to C-r(D-x, Phi(x)), where Phi(x) is the holonomy group of del( r) with reference point x. If G is not connected, we also have suffici ently many differentiable cross sections of B and calculate their cova riant derivatives.