Structure for regular inclusions. II: Cartan envelopes, pseudo-expectations and twists

We introduce the notion of a Cartan envelope for regular inclusion ( C , D ) . When exists, it is unique, minimal pair into which regularly embeds. prove exists if and only has unique faithful pseudo-expectation property also give characterization using ideal intersection property. For any covering inclusion, we construct Hausdorff twisted groupoid appropriate linear functionals description in terms twist whose unit space set states on constructed pseudo-expectation. MASA this differs from Weyl twist; setting, show that precisely when there conditional expectation onto with other consequences