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This paper studies injective envelopes of groupoid dynamical systems and thecorresponding boundaries. Analogue to the group case, we associate a bundle ofcompact Hausdorff spaces to any (discrete) groupoid (the Hamana boundary of thegroupoid). We study bundles of compact topological spaces equipped with anaction of a groupoid. We show that any groupoid has a minimal boundary (theFurstenberg boundary of the groupoid). We prove that the Hamana and Furstenbergboundaries are the same, for (discrete) groupoids. We find the relation betweenthe reduced crossed product of the $\G$-injective envelop of a groupoiddynamical system and the injective envelope of the reduced crossed product ofthe original system.

$C^*$-simplicity and Ozawa conjecture for groupoid $C^*$-algebras, part I: injective envelopes

$C^$-simplicity and Ozawa conjecture for groupoid $C^$-algebras, part I: injective envelopes