For regular inclusions with abelian subalgebra, having a Cartan envelope is equivalent to having a faithful unique pseudo-expectation, now proven without the unital hypothesis.
Akemann and Frederic W
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Pseudo-Cartan Inclusions
For regular inclusions with abelian subalgebra, having a Cartan envelope is equivalent to having a faithful unique pseudo-expectation, now proven without the unital hypothesis.