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

Valeriano Aiello; Vaughan F. R. Jones

doi:10.1016/j.jfa.2020.108777

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arXiv · 1905.05806

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

Valeriano Aiello·Vaughan F. R. Jones

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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 $F=F_2$).

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BibTeX RIS

Valeriano Aiello, Vaughan F. R. Jones. 2019-05-14. On spectral measures for certain unitary representations of R. Thompson's group F. https://doi.org/10.1016/j.jfa.2020.108777

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