Inverse semigroups are a useful tool to generalize group actions on topological spaces. Compared to a group action, an inverse semigroup action allows for a more general construction of a transformation groupoid and its associated operator algebras. In this generality, every étale groupoid is canonically isomorphic to a transformation groupoid incorporating topological spaces on one end and groups on the other end. An immediate structural subtlety is to unify the ideas of a C_c(-) functor that exists on both ends. A continuous map between topological spaces induces a homomorphism between the corresponding C_c-function spaces in a contravariant way, while groups behave covariantly. The category of étale groupoid actors solves this problem. There are a covariant functor to the category of C_c-convolution algebras over the groupoids and, if 1 ≤ p < ∞ and if restricted to free actors, the induced homomorphisms extend to all reduced groupoid Lp-operator algebra completions F_\lambda^p(-).

When p=2, then the category of free actors between effective Hausdorff etale groupoids is equivalent to the category of Cartan pairs with Cartan maps as morphisms. When p ≠ 2 and with a suitable adaptation of Cartan theory and normalizers in place, this equivalence paves the way for a characterization of unital isometric homomorphism between F_\lampda^p-algebras of effective Hausdorff etale groupoids. The main observation is that such a unital isometric homomorphism induces a free actor between the underlying groupoids whose induced homomorphism, in turn, agrees with the initial one. In this talk, I will lay out the path towards this result and focus on the structural interplay between inverse semigroups, groupoid actors and induced homomorphisms by free actors.

This is joint work with Eusebio Gardella.

First lecture

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Second lecture

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Fifth lecture

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