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We show that a partition of the unity (or POVM) on a Hilbert space that is almost orthogonal is close to an orthogonal POVM in the same von Neumann algebra. This generalizes to infinite dimension previous results in matrix algebras by Kempe–Vidick and Ji–Natarajan–Vidick–Wright–Yuen. Quantitatively, our result are also finer, as we obtain a linear dependance, which is optimal.We also generalize to infinite dimension a duality result between POVMs and minimal majorants of finite subsets in the predual of a von Neumann algebra.

Answering a question by Chatterji–Druţu–Haglund, we prove that, for every locally compact group $G$ , there exists a critical constant $p_G \in [0,\infty ]$ such that $G$ admits a continuous affine isometric action on an $L_p$ space ($0) with unbounded orbits if and only if $p \geq p_G$. A similar result holds for the existence of proper continuous affine isometric actions on $L_p$ spaces. Using a representation of cohomology by harmonic cocycles, we also show that such unbounded orbits cannot occur when the linear part comes from a measure-preserving action, or more generally a state-preserving action on a von Neumann algebra and $p>2$. We also prove the stability of this critical constant $p_G$ under $L_p$ measure equivalence, answering a question of Fisher. [ABSTRACT FROM AUTHOR]

It is well known that a complete Riemannian manifold M which is locally isometric to a symmetric space is covered by a symmetric space. Here, we prove that a discrete version of this property (called local to global rigidity) holds for a large class of vertex-transitive graphs, including Cayley graphs of torsion-free lattices in simple Lie groups, and Cayley graphs of torsion-free virtually nilpotent groups. By contrast, we exhibit various examples of Cayley graphs of finitely presented groups (for example, SL4(Z)) which fail to have this property, answering a question of Benjamini and Georgakopoulos. Answering a question of Cornulier, we also construct a continuum of pairwise nonisometric large-scale simply connected locally finite vertex-transitive graphs. This question was motivated by the fact that large-scale simply connected Cayley graphs are precisely Cayley graphs of finitely presented groups and therefore have countably many isometric classes. [ABSTRACT FROM AUTHOR]

This article contains two rigidity type results for SL(n,Z) for large n that share the same proof. Firstly, we prove that for every p ∈ [1,∞] different from 2, the noncommutative Lp-space associated with SL(n,Z) does not have the completely bounded approximation property for sufficiently large n depending on p. The second result concerns the coarse embeddability of expander families constructed from SL(n,Z). Let X be a Banach space and suppose that there exist β < 1/2 and C > 0 such that the Banach-Mazur distance to a Hilbert space of all k-dimensional subspaces of X is bounded above by Ckβ. Then the expander family constructed from SL(n,Z) does not coarsely embed into X for sufficiently large n depending on X. More generally, we prove that both results hold for lattices in connected simple real Lie groups with sufficiently high real rank. [ABSTRACT FROM AUTHOR]

Extensive amenability is a property of group actions which has recently been used as a tool to prove amenability of groups. We study this property and prove that it is preserved under a very general construction of semidirect products. As an application, we establish the amenability of all subgroups of the group IET of interval exchange transformations that have angular components of rational rank less than or equal to two. In addition, we obtain a reformulation of extensive amenability in terms of inverted orbits and use it to present a purely probabilistic proof that recurrent actions are extensively amenable. Finally, we study the triviality of the Poisson boundary for random walks on IET and show that there are subgroups G < IET admitting no finitely supported measure with trivial boundary. [ABSTRACT FROM PUBLISHER]

We prove that λg → e-t|g|r λg defines a multiplier on the von Neuman algebra of hyperbolic groups with a complete bound = r, for any 0 < t < ∞, 1 < r < ∞. In the proof we observe that a construction of Ozawa allows us to characterize the radial multipliers that are bounded on every hyperbolic graph, partially generalizing results of Haagerup-Steenstrup-Szwarc and Wysocza'nski. Our argument is also based on the work of Peller. [ABSTRACT FROM AUTHOR]

We give an algorithm to compute the stable lengths of pseudo- Anosovs on the curve graph, answering a question of Bowditch. We also give a procedure to compute all invariant tight geodesic axes of pseudo-Anosovs. Along the way we show that there are constants 1 < a1 < a2 such that the minimal upper bound on 'slices' of tight geodesics is bounded below and above by aξ(S)1 and aξ(S)2, where ξ(S) is the complexity of the surface. As a consequence, we give the first computable bounds on the asymptotic dimension of curve graphs and mapping class groups. Our techniques involve a generalization of Masur-Minsky's tight geodesics and a new class of paths on which their tightening procedure works. [ABSTRACT FROM AUTHOR]

We prove that connected higher-rank simple Lie groups have Lafforgue's strong property (T) with respect to a certain class of Banach spaces 10 containing many classical superreflexive spaces and some non-reflexive spaces as well. This generalizes the result of Lafforgue asserting that SL(3,R) has strong property (T) with respect to Hilbert spaces and the more recent result of the second-named author asserting that SL(3,R) has strong property (T) with respect to a certain larger class of Banach spaces. For the generalization to higher-rank groups, it is sufficient to prove strong property (T) for Sp(2,R) and its universal covering group. As consequences of our main result, it follows that for X 10, connected higher-rank simple Lie groups and their lattices have property (FX) of Bader, Furman, Gelander and Monod, and that the expanders constructed from a lattice in a connected higher-rank simple Lie group do not admit a coarse embedding into X. [ABSTRACT FROM AUTHOR]

Given a metric space (X, d), the wobbling group of X is the group of bijections g : X → X satisfying sup x→X d(g(x), x) < ∞. We study algebraic and analytic properties of W(X) in relation with the metric space structure of X, such as amenability of the action of the lamplighter group ⊕X Z/2Z...W(X) on ⊕X Z/2Z and property (T). [ABSTRACT FROM AUTHOR]