II1 factors and equivalence relations with distinct fundamental groups
Abstract
We construct a group measure space II1 factor that has two non-conjugate Cartan subalgebras. We show that the fundamental group of the II1 factor is trivial, while the fundamental group of the equivalence relation associated with the second Cartan subalgebra is non-trivial. This is not absurd as the second Cartan inclusion is twisted by a 2-cocycle.
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