A differentiable structure for a bundle ofC∗-algebras associated with a dynamical system

Moto O'uchi
1995 Pacific Journal of Mathematics  
Let (M,G) be a difFerentiable dynamical system, and σ be a transverse action for (M,G). We have a differentiable bundle (B, π, M, C) of C*-algebras with respect to a flat family T σ of local coordinate systems and we have a flat connection V in B. If G is connected, the bundle B is a disjoint union of p x (C*(Q)) (x € M), where Q is the groupoid associated with (M, G) and p x is the regular representation of C*{9). We show that, for /EC c°°( 5),a cross section cs(f) : x H-> ρ x (f) is
more » ... x (f) is difFerentiable with respect to the norm topology, and calculate a covariant derivative V(cs(/)). Though B is homeomorphic to the trivial bundle, the difFerentiable structure for B is not trivial in general. Let B σ be a subbundle of B generated by elements / with the property V(cs(/)) = 0. We show the triviality of the difFerentiable structure for B σ induced from that for B when C*{G) is simple. We have a bundle RM(B) of right multiplier algebras and it contains ΰasa subbundle. Let (M, G) be a Kronecker dynamical system and σ be a flow whose slope is rational. In this case, we have a subbundle D of RM{B) whose fibers are *-isomorphic to C(T). The flat connection V r in D is not trivial and the bundle B decomposes into the trivial bundle B σ and the non-trivial bundle D. Moreover, for a σ-invariant closed connected submanifold N of M with dimiV = 1, we show that C*(G\N) is *-isomorphic to C*(D X ,Φ X )^ where Φ x is the holonomy group of V r with reference point x. If G is not connected, we also have sufficiently many difFerentiable cross sections of B and calculate their covariant derivatives.
doi:10.2140/pjm.1995.168.291 fatcat:vas7dlfdizb2vpc2fta4vt2som