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We introduce a new notion of twisted actions of inverse semigroups and show that they correspond bijectively to certain  regular  Fell bundles over inverse semigroups, yielding in this way a structure classification of such bundles. These include as special cases all the  stable  Fell bundles. Our definition of twisted actions properly generalizes a previous one introduced by Sieben and corresponds to Busby–Smith twisted actions in the group case. As an application we describe twisted étale groupoid  C *‐algebras in terms of crossed products by twisted actions of inverse semigroups and show that Sieben's twisted actions essentially correspond to twisted étale groupoids with topologically trivial twists.

Twisted actions and regular Fell bundles over inverse semigroups