This work explores some aspects of the deformation/stiffness theory of II1 factors, originally developed by Sorin Popa at the beginning of the previous decade [76]. The powerful techniques and strategies that allowed Popa to accomplish this shift in perspective form the basis of the II1 factor deformation/stiffness theory.
An outline of the dissertation
Somewhat surprisingly, we are able to do this by working with weak “deformations”, which do not preserve von Neumann algebraic structure, but remain controllable only in a dense C∗-subalgebra.
Some preliminaries
An actionσ is said to be ergodic (or weak mixing, mixing, etc.) if the Koopman representation of πσ is in the sense of the above definition. If M is a finite von Neumann algebra and B ⊂(A⊗M)oΓ is a subalgebra such that ζα converges uniformly to the identity on (B)1 as α →0, then αt converges uniformly to the identity on (B)1 as t → 0.
ON COCYCLE SUPERRIGIDITY FOR GAUSSIAN ACTIONS
Introduction
The interaction between representation theory and the ergodic theory of a group via the Gaussian action has been fruitfully exploited in the literature (cf. the seminal works of Connes and Weiss and of Schmidt inter alios). As a consequence, we obtain our first result, that the cohomology of the representation provides a barrier to the Ufin-cocycle superrigidity of the connected Gaussian action.
Preliminaries
- Gaussian actions
- Cocycles from representations and from actions
- Closable derivations
If (X, µ) is a one-point probability space, the construction above gives rise to the group von Neumann algebra, which we will denote by LΓ. The "representation theory" of a finite von Neumann algebra is captured in the structure of its bimodules, also called correspondences (cf. [5, 68]).
Deformations
- Popa’s deformation
- Ioana’s deformation
- Malleable deformations of Gaussian actions
Mee (A, τ) aljebraa von Neumann kan babal’ate dhuma qabu haa ta’u, v∈A⊗Abe A⊗1,1⊗A ⊂ A⊗A tiif yuunitarii Haar maddisiisa, akkaataa walduraa duubaan. Aljebraa Niwumaan dhuma qabu (B, τ) yoo kenname, ̃B = B ∗ LZ, bu’aa bilisaa aljebraa NiwumaanB fi LZ haa ta’u.
Cohomology of Gaussian actions
This is actually just a consequence of fully positive deformations becoming asymptotically A-bimodular. This will converge uniformly for γ ∈ Γ if and only if the cocycle b is bounded and so the result follows. The exponentiation map described above induces an injective homomorphism H1(Γ, π) →H1(Γ, σ,T)/χ(Γ), where χ(Γ) is the character group ofΓ.
The above statement, together with the fact that this map is a homomorphism, shows that this map is injective.
Derivations
- Derivations from s-malleable deformations
- Tensor products of derivations
- Derivations from generalized Bernoulli shifts
Denote by ˆNj the tensor product of Nis obtained by omitting the jth index, by specifying an arbitrary order in which successive tensor powers are resolved in N, we have a natural identification N = ˆNj⊗Nj for each j ∈I. By Lemma II.5.3 this formula is then extended to A0 and since ˜T acts on ˜Unitary and A0 is a kernel for δβ, σT(D(δβ)) =D(δT β) and this formula remains valid for a∈D (δβ ). If we denote by N = AoΓ the corresponding construction of the space of group measures, we can define the N-N Hilbert bimodule structure on K = ˜H ⊗`2Γ corresponding to Lemma II.5.3.
However, it is not difficult to check, using the fact that both derivatives arise as tensor product derivatives, that if ζα0 are the solvent maps corresponding to the infinitesimal generator ofα, then we have the inequality τ(ζα(a)a∗)≤2τ( ζα0(a) )a∗), for alla∈A.
L 2 -rigidity and U fin -cocycle superrigidity
In the case of SO(n,1) the restriction of the lattice subgroup Γ to the connected component of the identity SO(n,1)0 is always i.c.c. If P ⊂N is a von Neumann subalgebra, then His is left susceptible to P (in the sense of Theorem III.3.2) if and only if for every nonzero projection p∈ Z(P0∩N) and finite subset F ⊂ You (P), we did that. Now, sticking to the same notations, assume that the orthogonal representation b : Γ → O(K) is such that there exists a K > 0 such that π⊗K is weakly contained in the left regular representation.
On the other hand, the same calculations as in the proof of (IV.3.5) together with the inequality.
STRONG SOLIDITY FOR GROUP FACTORS FROM LATTICES IN SO(N, 1) AND
Introduction
In their paper [60], Ozawa and Popa brought new techniques to the study of free group factors which enabled them to show that these factors possess a strong structural property, what they called "strong solidity". These factors are already known from the work of Ozawa and Popa [61] to have no Cartan subalgebras. To circumvent this problem, we first note that Ozawa and Popa's techniques actually make it possible to derive a kind of relative inclusiveness of the normalizer subalgebra with respect to the bimodule, given in terms of an "invariant mean".
Since the property of having a nearly invariant sequence of vectors is stable under taking tensor powers, we are able to transfer relative malleability to a large tensor power of the bimodule to derive malleability from the normalizer algebra.
Preliminaries
- Representations, correspondences, and weak containment
- Cocycles and the Gaussian construction
- Weak compactness and the CMAP
In the theory of von Neumann algebras, correspondences play a role analogous to unitary representations in the theory of countable discrete groups. For the von Neumann algebras N and M, we must consider that an N-M correspondence is a ∗-representation π of the algebraic tensor N Mo in the bounded operators on a Hilbert space H, which is normal when restricted to both N and Mo. We refer the reader to the theory of tensor products of correspondences and the basic theory of correspondences in general.
It was discovered by Parthasarathy and Schmidt [62] that cocycles also fit well into the framework of the Gaussian construction, inducing one-parameter families of deformations (i.e. cocycles) of the action σ.
Amenable correspondences
A construction identical to the one we propose has already appeared in the work of Sauvageot [85] for an arbitrary factor M. So, if P⊂M satisfies one of the conditions in the above theorem, then P is relatively suitable for QinsideM (PlM Q ) in the sense of Theorem 2.1 in [60]. It is easy to see that the space of all such operators Tξ,η is a ∗-subalgebra of B(H) which we will denote by Nf.
Using another application of the generalized Powers-Størmer inequality, it is easy to check that ξn=ηn1/2.
Proofs of main theorems
Since ( ˜ξn) are vectors in a correspondence that is weakly contained in the coarse M-M correspondence, this holds for every finite subset F ⊂ U(Q). We say that a countable discrete group Γ is in the classQH if it admits an array q : Γ → Hπ for an inadmissible unitary representation π : Γ → U (Hπ). As in the case of the uniform Roe algebra, we will see that Cu∗(Γ yσ
Also using proposition IV.3.6 one can find a positive number tε>0 such that for all tε > t >0.
ON THE STRUCTURE OF II 1 FACTORS OF NEGATIVELY CURVED GROUPS
Introduction
Remarkable in its generality, Ozawa's argument relies on a subtle interplay between C∗-algebraic and von Neumann-algebraic techniques [7]. In their fundamental works on the rigidity of group actions [45, 46], Monod and Shalom proposed a more inclusive cohomological definition of negative curvature in group theory given in terms of non-vanishing of the second degree bounded cohomology for Γ with coefficients in the left-regular representation. Invoking Ozawa's proof of the weak adaptability of hyperbolic groups [56], Theorem B allows us to fully solve the strong solidity problem—hence the Cartan problem—in the positive for i.c.c.
Our method then borrows Ozawa's insight into using local reflexivity to move from C∗λ(Γ) to the entire von Neumann algebra.
Cohomological-type properties and negative curvature
In the previous definition, we could also have relaxed the strict anti-symmetry condition to the condition that kπγ(q(γ−1)) +q(γ)k is bounded. A map q : Γ→ It is called an aquasi-1-cocycle for the representationπ if we can find a constant K≥0 such that for allγ, λ∈Γ we have. In the following, we will drop the "1" and refer to (quasi-)1-cocycles as (quasi-)cocycles.
If the representation π can be chosen such that it is weakly `2, then we say that Γ belongs to the class QHreg.
Deformations of the uniform Roe algebra
- Schur multipliers and the uniform Roe algebra
- Construction of the extended Roe algebra C u ∗ (Γ y σ X)
- A path of automorphisms of the extended Roe algebra associated with the
Since arguments of Burger and Monod [8] show that Z2 oSL(2,Z) is not in the class Dreg (in fact admits no proper quasi-cocycle in any representation), as a corollary of Theorem IV.2.2 , we have that the class QHreg is strictly larger than the class Dreg. Then the extended Roe algebra Cu∗(Γyσ X) is defined as the C∗-algebra generated by L∞(X×Γ) and the unitarysuγinnerB(L2(X)⊗`2(Γ)). Furthermore, when restricted to the uniform Roe algebra one can recover from αt the multipliers introduced above: E ◦αt(x) = mt(x) for all x∈Cu∗(Γ).
Since the elements in Cλ∗(Γ) can be approximated in uniform norm by finitely supported elements, using the triangle inequality it suffices to show (IV.3.6) only forx=P.
Proofs of the Main Results
Furthermore, the Cauchy-Schwartz inequality together with (3) and (IV.4.9) allows us to see that the latter quantity is less than. Applying the triangle inequality successively and using (IV.4.9), which is unity, and (IV.4.10), we obtain that the previous quantity is less than. Also using statement IV.3.6, after contracting tε, if necessary, we can additionally assume that for all u∈F we have. IV.4.14) So if we use the triangle inequality together with (IV.4.13) and the Cauchy-Schwartz inequality, we get this.
Using (IV.4.12) together with the Cauchy-Schwartz inequality, (3) and (IV.4.14) we further see that the last quantity above is less than.
Amenable actions, (bi-)exactness, and local reflexivity
A countable discrete group Γ has Guoliang Yu's property A [104] if and only if Γ acts amenable on its Stone–Cech limitβ0Γ =βΓ\Γ. A C∗-algebra A is said to be locally reflexive if for every finite-dimensional operator systemE ⊂A∗∗ there exists a network (ϕi)i∈I of contractive completely positive (c.c.p.) mappings ϕi:E →A that converge to the pointwise identity -ultraweak topologies. We can see that if Γ is biexact in the sense of Definition 15.1.2 of [7], if and only if Γ is biexact in the sense of the above definition.
With the same proof that "property A ⇒ rough embeddability in Hilbert space" (cf. we have the following.
A proof of Proposition IV.2.2
Gaboriau: Examples of groups that are measure equivalent to the free group, Ergodic Theory Dynam. Haagerup: Injectivity and Decomposition of Completely Bounded Maps, Operator Algebras and Their Connections with Topology and Ergodic Theory (Bu¸steni, 1983), Lecture Notes in Math Springer, Berlin. Vaes: Cocycle and orbit superrigidity for lattices in SL(n,R) acting on homogeneous spaces, preprint (2008), to appear at the conference in honor of Bob Zimmer's 60th birthday, "Geometry, Rigidity and Group Actions", Sept .
Schmidt: Amenability, Kazhdan's property (T), strong ergodicity and invariant means for ergodic groups of actions, Ergodic Theory Dynam.
