... abelian∗-subalgebra A of a finite von Neumann factor M is called semiregular if N (A) generates a factor, equivalently, if N (A) ∩ M = C. Also, while maximalabelian ∗-subalgebras A with N (A) = ... pseudogroupaction (N.B.: v ≡ 1 is always a 2-cocycle, ∀R), there exists a type II1fac-tor with a Cartan subalgebra (A ⊂ M ) associated with it, via a group-measure space construction “ `a la” Murray-von ... maximal abelian∗-subalgebra A ⊂ M is a Cartan subalgebra if andonly if A ⊂ M is discrete, i.e., if and only if A ∩M ,A is generated byprojections that are finite in M ,A .(ii) Let A 1 ,A 2⊂...