  We describe the $C^*$-algebra of an $E$-unitary or strongly 0-$E$-unitary
inverse semigroup as the partial crossed product of a commutative $C^*$-algebra
by the maximal group image of the inverse semigroup. We give a similar result
for the $C^*$-algebra of the tight groupoid of an inverse semigroup. We also
study conditions on a groupoid $C^*$-algebra to be Morita equivalent to a full
crossed product of a commutative $C^*$-algebra with an inverse semigroup,
generalizing results of Khoshkam and Skandalis for crossed products with
groups.
