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The Thompson group F has a natural unitary representation on $H=L^2[0,1]$.With some projections, we construct a family of projective unitaryrepresentations on a Fermionic Fock space associated with $H$. It comes fromthe representation of the associated CAR algebra. After $H^2(F;S^1)$ isobtained, we mainly study whether any of these projective unitaryrepresentations can be lifted to an ordinary one. We will discuss the liftingproblem of these projective representations.

A Family of Projective Representations of the Thompson Group and Lifting Problems