  We construct a weak conditional expectation from the section C*-algebra of a
Fell bundle over a unital inverse semigroup to its unit fibre. We use this to
define the reduced C*-algebra of the Fell bundle. We study when the reduced
C*-algebra for an inverse semigroup action on a groupoid by partial
equivalences coincides with the reduced groupoid C*-algebra of the
transformation groupoid, giving both positive results and counterexamples.
