## Notes of Claus Koestler’s Cambridge lecture 26-01-2017

An elementary approach to unitary representations of Thompson’s group ${F}$

${F=\langle X_n \,|\,X_k X_n=X_{n+1}X_k,\,0\leq k. Unknown wether amenable or not. Does not contain free subgroups.

1. Sources for unitary representations of ${F}$

Characters!

Theorem 1 (Gohm-Kostler, Dudko-Madzrodski) Extreme points of the set of characters and in one to one correspondance with points of the 2-torus (characters of the abelianisation ${{\mathbb Z}^2}$) plus ${\{0,0\}}$ (left regular representation).

1. ${F}$ is non-alenable.
2. ${C^*_r(F)}$ has a unique normalized trace.

Not so productive.

1.2. Subfactor approach

Theorem 3 (Jones 2014) TFAE Any subfactor yields a unitary representation of ${F}$.

1.3. Probabilistic approach

Theorem 4 (Gohm, Evans, Bhat, Wills, C. Jones) Every non-commutative stationary Markov chain yields a unitary representation of ${F}$.

1.4. Graphical picture

for the monoid ${F^+}$ (Belk 2004).

Represent generators ${X_0}$ and ${X_1}$ as diagrams: collection of edges joining an ${n+1}$-point set to an ${n}$-point set, with two edges joining at 0 (resp. at 1). Such diagram can be composed. Get a category whose objects are finite sets ${([n])_{n\in{\mathbb N}}}$ and morphisms are finite binary forests.

In the semicosimplicial category ${\Delta_S}$, same objects, morphisms are increasing functions. They satisfy again the relations of ${F}$.

There is a covariant functor from ${\Delta_S}$ to the category of NonCommutativePS. With Evans et de Finetti, we related coface identities to spreadability and non-commutative Bernoulli shift.

2. From Markov chains to representations

Start with construction on infinite sets, kind of limit of the previous one: partial shifts.

Pass to Hilbert spaces: partial shifts on sequences of vectors in a fixed Hilbert space ${D}$. Let ${U}$ and ${V}$ be unitaries ${D\rightarrow D\oplus D}$. Inserting them in partial shifts ${S_k}$, get unitaries ${U_k}$.

Theorem 5 The unitaries ${T_k:=U_{k+1}S_{k+1}}$ satisfy Thompson ${F}$‘s relations.

We call this the standard form representation of ${F}$.

Theorem 6 Let ${\pi:F\rightarrow\mathcal{H}}$ be a unitary representation such that

1. No fixed vectors.
2. ${\mathcal{H}}$ is generated by fixed vectors of generators ${X_n}$.

Then ${\pi}$ has standard form up to modifications of ${S_k}$‘s.

2.1. Outlook

Does the V. Jones’ example fit into this framework ?

Study the ${C^*}$ and von Neumann algebras of ${F}$.

2.2. Question

What if you embed ${F}$ into diffeos of the circle and compose with unitary representations coming from conformal field theory ?