On spectral measures for certain unitary representations of R. Thompson's group F

The Hilbert space H of backward renormalisation of an anyonic quantum spin chain affords a unitary representation of Thompson's group F via local scale transformations. The group F is discrete and mysterious in many ways so the obvious questions of irreducibility and distinctness of these representations appear difficult and in a first step towards solving them we calculate the spectral measures of group elements in the representation. Given a vector in the canonical dense subspace of H we calculate the corresponding spectral measure and illustrate with some examples. To do this calculation we introduce the “essential part” (intimately related to the conjugacy class) of an element. The spectral measure for any vector in H is, apart from possibly finitely many eigenvalues, absolutely continuous with respect to Lebesgue measure. The same considerations and results hold for the Brown-Thompson groups Fn (for which F=F2).