We introduce the notion of a Cartan envelope for a regular inclusion (C,D).
When a Cartan envelope exists, it is the unique, minimal Cartan pair into which
(C,D) regularly embeds. We prove a Cartan envelope exists if and only if (C,D)
has the unique faithful pseudo-expectation property and also give a
characterization of the Cartan envelope using the ideal intersection property.
For any covering inclusion, we construct a Hausdorff twisted groupoid using
appropriate linear functionals and we give a description of the Cartan envelope
for (C,D) in terms of a twist whose unit space is a set of states on C
constructed using the unique pseudo-expectation. For a regular MASA inclusion,
this twist differs from the Weyl twist; in this setting, we show that the Weyl
twist is Hausdorff precisely when there exists a conditional expectation of C
onto D.
We show that a regular inclusion with the unique pseudo-expectation property
is a covering inclusion and give other consequences of the unique
pseudo-expectation property.Comment: 47 page
