We introduce the notion of a Cartan envelope for a regular inclusion $(\mathcal{C},\mathcal{D})$. When a Cartan envelope exists, it is the unique, minimal Cartan pair into which $(\mathcal{C},\mathcal{D})$ regularly embeds. We prove a Cartan envelope exists if and only if $(\mathcal{C},\mathcal{D})$ 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 $(\mathcal{C},\mathcal{D})$ in terms of a twist whose unit space is a set of states on $\mathcal{C}$ 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 $\mathcal{C}$ onto $\mathcal{D}$.

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.

我们介绍了常规包含的Cartan信封的概念 $（\mathcal{C}，\mathcal{d}）$。当存在Cartan信封时，它是唯一的，最小的Cartan对$（\mathcal{C}，\mathcal{d}）$定期嵌入。我们证明只有当且仅当存在Cartan信封时，$（\mathcal{C}，\mathcal{d}）$ 具有独特的忠实伪期望属性，并且还使用理想的相交属性对Cartan信封进行了表征。

对于任何覆盖物，我们使用适当的线性函数构造Hausdorff扭曲的组群，并给出了Cartan包络的描述 $（\mathcal{C}，\mathcal{d}）$ 根据单位空间是一组状态的扭曲 $\mathcal{C}$使用唯一的伪期望构造。对于常规的MASA包含，此扭曲不同于Weyl扭曲；在这种情况下，我们证明存在条件期望的情况下，Weyl扭曲恰好是Hausdorff$\mathcal{C}$ 到 $\mathcal{d}$。

我们显示具有唯一伪期望属性的常规包含是覆盖包含，并给出唯一伪期望属性的其他结果。