Equivalent Constructions of Cartan Pairs

Feldman and Moore [4] introduce Cartan subalgebra of the von Neumann algebra M on a separable Hilbert space H from the natural subalgebra of M(R, sigma), the twisted algebra of matrices over the relation R on a Borel space (X, B, muy). They show that if M has a Cartan subalgebra A, then M is isomorphic to M(R, sigma) where A is the twisted algebra onto the diagonal subalgebra L^inf (X, muy). The relation R is unique to isomorphism and the orbit of the two-cohomology class on R in the torus T, which is the automorphism group of R, is also unique. Three decades later, based on Feldman-Moore work and utilizing etale groupoids from C*(G, Sigma), Renault [9] constructs equivalent Cartan pairs. Nearly another decade later, using extensions of inverse semigroups from extensions of Cartan inverse monoids and Feldman-Moore work, Donsig, Fuller, and Pitts [2] construct other equivalent Cartan pairs. In this paper, we study all Cartan pairs of Feldman and Moore, Renault, and Donsig et al. Our objective is to show that these Cartan pairs are equivalent.