II1 factors with exactly two
crossed product decompositions
NCGOA 2016
–
Lecture 1
Hausdorff Institute for Mathematics, Bonn, May 17-25, 2016
Stefaan Vaes∗
∗
Supported by ERC Consolidator Grant 614195
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Von Neumann algebras
Recall : two big families of von Neumann algebras.
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Group von Neumann algebras L(G ).
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Crossed product von Neumann algebras P o G .
In particular with P = L∞ (X ).
Questions :
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Flexibility : unexpected isomorphisms, e.g. for G amenable.
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Rigidity :
recover information on G or on G y P out of L(G ) or P o G .
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Cartan subalgebras
Some reminders :
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If (P, τ ) is tracial and G y (P, τ ) is trace preserving, then P o G is
tracial.
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L∞ (X ) ⊂ L∞ (X ) o G is a MASA iff G y (X , µ) is (essentially) free :
for all g 6= e, the set {x ∈ X | g · x = x} has measure zero.
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Thus : L∞ (X ) o G is a II1 factor when G y (X , µ) is a free ergodic
pmp action.
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In that case : L∞ (X ) ⊂ L∞ (X ) o G is a Cartan subalgebra.
Definition
A Cartan subalgebra A in a II1 factor M is a MASA such that
NM (A) = {u ∈ U(M) | uAu ∗ = A}
generates M.
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W∗ -superrigidity
W∗ -equivalence
We say that free ergodic pmp actions G y (X , µ) and Λ y (Y , η) are
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W∗ -equivalent if L∞ (X ) o G ∼
= L∞ (Y ) o Λ,
conjugate if there exist ∆ : (X , µ) → (Y , η) and δ : G → Λ such that
∆(g · x) = δ(g ) · ∆(x).
W∗ -superrigidity
We say that a free ergodic pmp action G y (X , µ) is W∗ -superrigid if
every W∗ -equivalent action is actually conjugate.
This means :
G and G y (X , µ) can be recovered from L∞ (X ) o G.
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W∗ -superrigid actions
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Peterson (2009) : existence of (virtually) W∗ -superrigid actions.
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Popa – V (2009) : concrete W∗ -superrigid actions.
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Ioana (2010) : if G has property (T), then the Bernoulli action
G y (X0 , µ0 )G is W∗ -superrigid.
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Ioana – Popa – V (2010) : if G = Γ1 × Γ2 with Γ1 infinite and Γ2
nonamenable, then again the Bernoulli action G y (X0 , µ0 )G is
W∗ -superrigid.
..
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Gaboriau – Ioana – Tucker-Drob (2016) : the action
PSL(2, Z) × PSL(2, Z) y PSL(2, Zp ) by left and right translation is
virtually W∗ -superrigid.
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How to prove W∗ -superrigidity ...
... for a given G y (X , µ).
Put A = L∞ (X ) and M = A o G.
Assume that M = B o Λ for some other Λ y (Y , η).
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First step : find u ∈ U(M) such that uBu ∗ = A.
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By Popa’s theorem, it suffices that B ≺ A.
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Conclusion of the first step (Singer) : the actions are orbit
equivalent.
Recall : this means ∃∆ : X → Y with ∆(G · x) = Λ · ∆(x).
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Second step : prove OE-superrigidity. This means : every action
that is OE to G y (X , µ) is actually conjugate.
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How to prove OE-superrigidity ...
... for a given G y (X , µ).
Assume that ∆ : X → Y is an orbit equivalence with Λ y (Y , η).
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Define Zimmer’s rearrangement 1-cocycle ω : G × X → Λ by
∆(g · x) = ω(g , x) · ∆(x).
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Prove that ω is cohomologous to a homomorphism δ : G → Λ.
This means : there exists ϕ : X → Λ such that
ω(g , x) = ϕ(g · x)−1 δ(g ) ϕ(x).
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e : X → Y : ∆(x)
e
Then, the map ∆
= ϕ(x) · ∆(x) satisfies
e · x) = δ(g ) · ∆(x).
e
∆(g
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Popa’s cocycle superrigidity theorem
Popa’s cocycle superrigidity theorem (2005-2006)
When G has property (T), or when G is a product group, or when ...,
every 1-cocycle for the Bernoulli action G y (X0 , µ0 )G with values in a
countable group Λ is cohomologous to a group homomorphism.
Target group Λ may be any closed subgroup of U(M) with M a II1 factor.
(2)
Conjecture : cocycle superrigidity holds if and only if β1 (G ) = 0.
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Focus on the first step
Definition
Let M be a II1 factor. We call B ⊂ M a group measure space (gms)
Cartan subalgebra if B ⊂ M is a Cartan subalgebra and there exists Λ
such that M = B o Λ.
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For the Bourbakists : existence of a subgroup Λ < NM (B) such that
(B ∪ Λ)00 = M and EB (v ) = 0 for every v ∈ Λ \ {1}.
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There may exist Cartan subalgebras that are not gms.
The first step in the approach to W∗ -superrigidity then becomes :
Does M have a unique gms Cartan subalgebra, up to unitary conjugacy.
More natural question : does M have a unique Cartan subalgebra, up to
unitary conjugacy.
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Uniqueness of Cartan subalgebras
Theorem (Ozawa – Popa, 2007)
Let G = Fn with 2 ≤ n ≤ ∞ and let G y (X , µ) be a free ergodic
profinite action.
This means : G y lim G/Gn .
←
−
∞
Then L (X ) is the unique Cartan subalgebra of L∞ (X ) o G up to unitary
conjugacy.
Chifan-Sinclair (2011) : the same for non elementary hyperbolic G.
Theorem (Popa – V, 2011-2012)
Let G = Fn or any non elementary hyperbolic group. For arbitrary free
ergodic pmp actions G y (X , µ), we have that L∞ (X ) is the unique
Cartan subalgebra of L∞ (X ) o G up to unitary conjugacy.
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A general dichotomy theorem
Theorem (Popa – V, 2011-2012)
Let G = Fn or any non elementary hyperbolic group.
Let M = P o G be an arbitrary tracial crossed product.
If Q ⊂ M is amenable relative to P, then
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either Q ≺ P,
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or NM (Q)00 stays amenable relative to P.
Intuition : read Q ≺ P as saying that “Q is finite relative to P”.
Then guess what relative amenability may be.
For groups : when Γ, Λ < G , we say that Γ is amenable relative to Λ if
G /Λ admits a Γ-invariant mean.
Precise definition : ...
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Consequences of the general dichotomy theorem
Definition
We say that G is Cartan-rigid if for every free ergodic pmp action
G y (X , µ), we have that L∞ (X ) is the unique Cartan subalgebra of
L∞ (X ) o G up to unitary conjugacy.
Theorem (Ioana, 2012)
All free products G = Γ1 ∗ Γ2 with |Γ1 | ≥ 2 and |Γ2 | ≥ 3 are Cartan-rigid.
Method : a family of embeddings θt : L∞ (X ) o (Γ1 ∗ Γ2 ) → P o F2 .
A stability result
Let C be the smallest class of groups
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containing all “nontrivial” free products and all non elementary
hyperbolic groups,
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that is stable under extensions (in particular, direct products).
Then all groups in this class are Cartan-rigid.
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How to prove uniqueness of Cartan subalgebras ?
Ozawa – Popa (2007) : M = L∞ (X ) o Fn with Fn y (X , µ) free
ergodic profinite.
Then, M has the complete metric approximation property (CMAP) :
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There exist ϕn : M → M, such that
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each ϕn is completely bounded and lim supn kϕn kcb = 1,
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each ϕn has finite rank,
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and for every x ∈ M, we have kϕn (x) − xk2 → 0.
Start of the proof : given a Cartan subalgebra B ⊂ M (or any amenable
von Neumann subalgebra), we get a sequence of normal functionals µn on
B ⊗ B op given by µn (a ⊗ b op ) = τ (ϕn (a)b).
This sequence has nice asymptotic invariance properties, etc, etc, ...
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Remarks
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To prove the dichotomy theorem for P o Fn :
work “relative” to P and build the “good” von Neumann algebra on
which we have the functionals µn .
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Every known uniqueness theorem for Cartan subalgebras
ultimately relies on this “miracle” to construct special
functionals µn using CMAP (or weak amenability).
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Conceptually, weak amenability has very little to do with
(non)uniqueness of Cartan subalgebras.
Indeed: weak amenability of a group G has nothing to do with G
having or not having abelian normal subgroups!
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Conclusion : uniqueness of Cartan subalgebras is largely non
understood !
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Back to uniqueness of gms Cartan subalgebras
Recall : enough to prove W∗ -superrigidity (in combination with
OE-superrigidity).
More conceptual approach (Popa – V, 2009 and Ioana, 2010) :
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any crossed product decomposition M = B o Λ gives rise to
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the dual coaction ∆ : B o Λ → (B o Λ) ⊗ L(Λ) given by
∆(bvs ) = bvs ⊗ vs for all b ∈ B, s ∈ Λ.
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We view ∆ : M → M ⊗ M.
Strategy : if M = A o G for a specific group G and a specific action
G y A (for instance, Bernoulli), we might be able to determine “all”
embeddings ∆ : M → M ⊗ M.
And thus, determine all possible group measure space decompositions
M = B o Λ.
That’s our program for the coming days.
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