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We describe the $C^*$-algebra of an $E$-unitary or strongly 0-$E$-unitaryinverse semigroup as the partial crossed product of a commutative $C^*$-algebraby the maximal group image of the inverse semigroup. We give a similar resultfor the $C^*$-algebra of the tight groupoid of an inverse semigroup. We alsostudy conditions on a groupoid $C^*$-algebra to be Morita equivalent to a fullcrossed product of a commutative $C^*$-algebra with an inverse semigroup,generalizing results of Khoshkam and Skandalis for crossed products withgroups.

On inverse semigroup $C^*$-algebras and crossed products