Almost $T$-invariant vectors and continuity of rotations for Pythagorean representations of Thompson's groups

Brothier and Jones (2019) constructed, from any Pythagorean tuple of operators satisfying $\sum_i A_i^*A_i=1$, unitary representations of Thompson's groups $F$, $T$, $V$ (and of the Cuntz algebra) on an inductive-limit Hilbert space, and proved that no such representation of $T$ has nonzero invariant vectors. They left two explicit analytic questions open: (i) whether some Pythagorean representation of $T$ admits almost $T$-invariant vectors, i.e. whether the trivial representation of $T$ can be weakly contained in a Pythagorean representation (Question 5.4 of the source); and (ii) for which Pythagorean tuples the dyadic rotations $\pi(r_n)$ converge to the identity, so that the representation extends to a strongly continuous unitary representation of the full circle-rotation group (Remark 6.3 of the source) -- a question the source settles only in the scalar case.

Background

Thompson's groups $F\subset T\subset V$ are the groups of piecewise-linear homeomorphisms with dyadic breakpoints and slopes that are powers of two: $F$ acts on the unit interval, $T$ on the circle $\mathbf S$ (adding dyadic rotations), and $V$ on the Cantor set (allowing discontinuous dyadic rearrangements). Whether $F$ is amenable is a famous open problem, and weak-containment questions about representations of these groups sit close to it.

A unitary representation of a discrete group $G$ is a homomorphism from $G$ into the group of unitary operators on a Hilbert space. Vaughan Jones discovered around 2014 that such representations of $F$ and $T$ can be built functorially from algebraic data carried by a Hilbert space, via the category of binary forests (a binary forest is a finite disjoint union of finite rooted binary trees, with morphisms given by grafting the roots of one forest onto the leaves of another). Brothier and Jones (2019) introduced the cleanest instance of this technology: a Pythagorean pair is a Hilbert space $H$ together with two operators $A,B$ satisfying the Pythagorean equality $A^*A+B^*B=1$ (equivalently, a linear isometry $H\to H\oplus H$); more generally an $n$-ary Pythagorean tuple $(A_1,\dots,A_n)$ with $\sum_i A_i^*A_i=1$ serves the $n$-ary Thompson groups. Such data define a functor from the category of binary forests, hence a unitary representation $\pi$ of $F$ -- canonically extended to $T$ and $V$ -- on an inductive-limit Hilbert space $\mathcal H=\overline{\bigcup_t H_t}$, where $t$ runs over finite binary trees and $H_t$ is the direct sum of one copy of $H$ per leaf of $t$. Every representation of the Cuntz algebra $\mathcal O_n$ (the universal C*-algebra generated by $n$ isometries $S_1,\dots,S_n$ with $\sum_i S_iS_i^*=1$) arises this way, so the class of Pythagorean representations is very large.

The archetype is the scalar pair $A=B=1/\sqrt2$ on $H=\mathbb C$, for which $\pi$ is the representation of $T$ on $L^2(\mathbf S)$ by half-densities, $f\mapsto\sqrt{g'}\,(f\circ g)$; this representation extends to all orientation-preserving diffeomorphisms of the circle. Here $r_n\in T$ denotes the rotation of the circle by the angle $2^{-n}$, and $\operatorname{Rot}(\mathbf S)$ the compact group of all rotations of $\mathbf S$; a representation of the dyadic rotations is called continuous when $\pi(r_n)\to\mathrm{id}$ (in the strong, or at least weak, operator topology) as $2^{-n}\to0$, which is exactly what permits extending $\pi$ to a strongly continuous unitary representation of $\operatorname{Rot}(\mathbf S)$, i.e. a representation for which, for every vector $\xi$, the orbit map $r\mapsto\pi(r)\xi$ is continuous in the Hilbert-space norm.

Two analytic facts from the source anchor the open questions. First, Proposition 5.3: no Pythagorean representation contains a nonzero $T$-invariant vector. Second, Proposition 6.4: for a scalar pair $(a,b)$ with $\omega=a\bar b$, the powers of the dyadic rotations converge in the weak operator topology to scalars, $\pi(r_n^j)\to x_j\,\mathrm{id}$, where $x_1=\bar\omega/(1-\omega)$ and $x_{2k}=x_k$, $x_{2k+1}=\bar\omega x_k+\omega x_{k+1}$. Consequently, for scalar tuples the rotations are continuous exactly when $\omega=1/2$ -- essentially only the standard half-density case -- while generically the limit of rotations is not the identity.

A unitary representation $\pi$ of a group $G$ is said to admit almost-invariant vectors if for every finite set $S\subset G$ and every $\varepsilon>0$ there is a unit vector $\xi$ with $\|\pi(g)\xi-\xi\|<\varepsilon$ for all $g\in S$; this is equivalent to the trivial representation of $G$ being weakly contained in $\pi$ (matrix coefficients of the trivial representation approximated uniformly on finite sets). Amenability of $G$ itself is equivalent to the regular representation admitting almost-invariant vectors, so existence questions for almost-invariant vectors carry an amenability flavor; for instance, the degenerate pair $(A,B)=(1,0)$ on $H=\mathbb C$ gives a representation of $T$ induced from the trivial representation of $F$, for which weak containment of the trivial representation of $T$ amounts to co-amenability of $F$ in $T$.

In parallel with these questions, the structure theory of Pythagorean representations was developed by Brothier and Wijesena (2022--2025): every such representation splits into a diffuse part, which is Ind-mixing (it contains no subrepresentation induced from a finite-dimensional representation of a subgroup), and an atomic part; for finite-dimensional defining data the atomic part is a finite direct sum of monomial representations (representations induced from a character of a subgroup), built from the parabolic subgroups $Y_p=\{g\in Y:g(p)=p\}$ with $p$ a point of the Cantor set. Irreducible representations of finite so-called Pythagorean dimension (in their terminology, the minimal Hilbert-space dimension of defining data producing the representation) were classified through smooth moduli manifolds, and a tensor product with computable fusion rules was added. None of this structure theory addresses the two questions below, which concern the analytic behavior of these representations rather than their algebraic decomposition.

Problem Statement

Settle the two analytic questions that Brothier and Jones explicitly posed and left open about Pythagorean representations of Thompson's group $T$, in the full generality of the construction (arbitrary Hilbert spaces, arbitrary Pythagorean tuples, including non-scalar operators and $n$-ary tuples):

(1) Almost-invariant vectors (Question 5.4). Does there exist a Pythagorean representation $\pi$ of $T$ that admits almost $T$-invariant vectors, i.e. such that for every finite subset $S\subset T$ and every $\varepsilon>0$ there is a unit vector $\xi$ in the representation space with $\|\pi(g)\xi-\xi\|<\varepsilon$ for all $g\in S$ -- equivalently, is the trivial representation of $T$ weakly contained in some Pythagorean representation of $T$ built from any Pythagorean tuple, in any Hilbert-space dimension?

(2) Continuity of rotations (Remark 6.3). For which Pythagorean tuples do the dyadic rotations $\pi(r_n)$ converge (in the strong, or at least weak, operator topology) to the identity as the rotation angle $2^{-n}\to0$, so that $\pi$ extends to a strongly continuous unitary representation of the full rotation group $\operatorname{Rot}(\mathbf S)$ of the circle? The scalar case is already settled in the source (continuity holds exactly when $\omega=a\bar b=1/2$); the open content concerns general tuples, in particular non-scalar operators acting on higher-dimensional Hilbert spaces and $n$-ary tuples.

Both questions probe the same boundary behavior -- how a representation assembled from the Pythagorean equality behaves under the rotation subgroup of $T$ in the limit of fine dyadic scales -- and were posed together as the explicit open problems of the source paper, in the full generality of the construction, with no restriction to finite dimension, scalar tuples, or a fixed arity (for an $n$-ary tuple the $n$-ary Thompson group $T_n$ and its rotations by $n$-adic angles take the place of $T$ and the dyadic rotations in both questions).

References

1. Arnaud Brothier and Vaughan F. R. Jones, 'Pythagorean representations of Thompson's groups', Journal of Functional Analysis 277(7) (2019) 2442-2469. DOI: 10.1016/j.jfa.2019.02.009, https://doi.org/10.1016/j.jfa.2019.02.009 (author's version: arXiv:1807.06215, https://arxiv.org/abs/1807.06215)
2. Arnaud Brothier and Dilshan Wijesena, 'Jones' representations of R. Thompson's groups not induced by finite-dimensional ones', Annales de l'Institut Fourier 75(6) (2025) 2677-2716. DOI: 10.5802/aif.3689, https://doi.org/10.5802/aif.3689 (arXiv:2211.08555, https://arxiv.org/abs/2211.08555)
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