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

Abstract

The Hilbert space $\mathcal{H}$ of backward renormalisation of an anyonic quantum spin chain affords a unitary representation of Thompson's group $F$ via local scale transformations. Given a vector in the canonical dense subspace of $\mathcal{H}$ we show how to calculate the corresponding spectral measure for any element of $F$ and illustrate with some examples. Introducing the "essential part" of an element we show that the spectral measure of any vector in $\mathcal{H}$ 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 $F_n$ (for which $F=F_2$).