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 ℱσ of
local coordinate systems and we have a flat connection ∇ in B. If G is connected, the
bundle B is a disjoint union of ρx(Cr∗(𝒢))(x ∈ M), where 𝒢 is the groupoid
associated with (M,G) and ρx is the regular representation of Cr∗(𝒢). We show that,
for f ∈ Cc∞(𝒢), a cross section cs(f) : x↦ρx(f) is differentiable with respect to the
norm topology, and calculate a covariant derivative ∇(cs(f)). Though B is
homeomorphic to the trivial bundle, the differentiable structure for B is not trivial in
general. Let Bσ be a subbundle of B generated by elements f with the property
∇(cs(f)) = 0. We show the triviality of the differentiable structure for Bσ induced
from that for B when Cr∗(𝒢) is simple. We have a bundle RM(B) of right
multiplier algebras and it contains B a s a 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 ∇r in D is not trivial and the bundle B decomposes
into the trivial bundle Bσ and the non-trivial bundle D. Moreover, for a
σ-invariant closed connected submanifold N of M with dimN = 1, we show
that Cr∗(𝒢|N) is ∗-isomorphic to Cr∗(Dx,Φx), where Φx is the holonomy
group of ∇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.
