## Notes of Andreas Aaserud’s Cambridge lecture 20-04-2017

Property (T) and approximate conjugacy of actions

Joint work with Sorin Popa.

We let countable groups act on probability spaces. We focus on free and ergodic actions.

Example. Bernoulli action on ${X^\Gamma}$.

1. Actions on von Neumann algebras

Von Neumann algebras have states (linear functionals which are nonnegative with valu

Example. ${L^\infty(X,\mu)}$ has a state, defined by measure ${\mu}$.

Actions on standard probability spaces are in 1-1 correspondence with state preservong actions on separable abelian von Neumann algebras.

Ergodicity (no invariants but multiples of 1) and freeness (the largest projection fixed by all automorphisms fixes group elements) generalize to actions on von Neumann algebras.

There is a distance on automorphisms: sup of difference on elements of ${L^\infty}$-norm less than one, in ${L^2}$-norm.

Definition 1 Say two state-preserving actions are approximately conjugate if there exists a sequence of state-preserving ${\star}$-isomorphisms ${\theta_n}$ such that the ${\theta_n}$-conjugate of one action tends to the second action pointwise.

Ornstein-Weiss-Popa: all free ergodic measure-preserving actions of an amenable group are mutually approx. conjugate.

This is reminiscent of orbit equivalence, although there is no logical relation between aprox. conjugacy and orbit equivalence.

Proposition 2 Every group with an infinite amenable quotient has infinitely many non conjugate actions which are approx. conjugate.

We use Bernoulli actions, and the fact they are classified by entropy.

Theorem 3 For Kazhdan groups, appr. conjugacy implies conjugacy.

2. Property (T)

Theorem 3 follows from

Lemma 4 Let ${\Gamma}$ have property (T), let ${S}$ be a finite generating set. There exists ${\delta>0}$ such that if two actions are ${\delta}$-close, then they are conjugate.

The proof of the Lemma goes as follows.

1. Construct a von Neumann algebra ${M}$, generated by ${A}$ and one operator for each group element, which encodes the action.
2. To the inclusion ${A\subset M}$, Vaughan Jones associates the basic construction ${\mathcal{M}}$, generated by ${M}$ and by a projection ${e_A}$ of ${M}$ onto ${A}$. In admits a trace, hence a Hilbert space ${H}$ of elements of ${\mathcal{H}}$ with finite Hilbert-Schmidt norm.
3. From the two given actions of ${\Gamma}$ on ${A}$, cook up an action of ${\Gamma}$ on ${H}$ with ${e_A}$ almost fixed (it uses one action on the left and the other on the right). A fixed vector close to ${e_A}$ provides a projection ${e}$ in ${\mathcal{M}}$.
4. Show that ${e}$ is of the form ${e=ve_A v^*}$ where ${v}$ normalizes ${A}$. Then ${Ad(v^*)}$ is the desired automorphism.

2.1. Standard representation

The trace gives rise to a Hilbert space ${L^2(M,\tau)}$, wth a left-regular representation.

2.2. Cartan inclusions

It is ${A\subset M}$ where ${A}$ is maximal abelian in ${M}$ and ${M}$ is generated by the normalizer of ${A}$.

Let ${L^2(A)}$ be the closure of ${A}$ in ${L^2(M)}$. It comes with an orthogonal projection ${e_A:L^2(M)\rightarrow L^2(A)}$. The right-regular version of ${L^2(A)}$ and ${L^2(A)}$ generate a subalgebra ${\mathcal{A}}$.

Elements of the form ${xe_A y}$ generate a dense subalgebra of ${\mathcal{M}}$.

2.3. Feldman-Moore

This a construction in ergodic theory. The full group of an action is made of maps which coincide piecewise with group elements. This notion generalizes to actions on von Neumann algebras.

The von Neumann algebra ${M}$ is the cross-product constructed from the full group of the action. It comes with a trace, and ${A\subset M}$ is a Cartan inclusion. Going to the full group is necessary in order that ${A}$ be maximal abelian in ${M}$.

3. Open problems

Do non-Kazhdan groups have approx. conjugate non-conjugate actions ?

We can prove that non-amenable groups admit at least 2 non aprox. conjugate actions. Do all of them have infinitely many ?