A VON NEUMANN ALGEBRA CHARACTERIZATION OF PROPERTY (T) FOR GROUPOIDS

For an arbitrary discrete probability-measure-preserving groupoid G, we provide a characterization of property (T) for G in terms of the groupoid von Neumann algebra L(G). More generally, we obtain a characterization of relative property (T) for a subgroupoid H⊂G in terms of the inclusions L(H)⊂L(G)

Nonstandard analysis and the sumset phenomenon in arbitrary amenable groups

Beiglboeck, Bergelson and Fish proved that if subsets A,B of a countable discrete amenable group G have positive Banach densities a and b respectively, then the product set AB is piecewise syndetic, i.e. there exists k such that the union of k-many left translates of AB is thick. Using nonstandard analysis we give a shorter alternative proof of this result that does not require G to be countable, and moreover yields the explicit bound that k is not greater than 1/ab. We also prove with similar methods that if {A_i} are finitely many subsets of G having positive Banach densities a_i and G is countable, then there exists a subset B whose Banach density is at least the product of the densities a_i and such that the product B(B^−1) is a subset of the intersection of the product sets A_i(A_i^−1). In particular, the latter set is piecewise Bohr

Nonlocal games and quantum permutation groups

We present a strong connection between quantum information and the theory of quantum permutation groups. Specifically, we define a notion of quantum isomorphisms of graphs based on quantum automorphisms from the theory of quantum groups, and then show that this is equivalent to the previously defined notion of quantum isomorphism corresponding to perfect quantum strategies to the isomorphism game. Moreover, we show that two connected graphs X and Y are quantum isomorphic if and only if there exists x is an element of V(X) and y is an element of V(Y) that are in the same orbit of the quantum automorphism group of the disjoint union of X and Y. This connection links quantum groups to the more concrete notion of nonlocal games and physically observable quantum behaviours. In this work, we exploit this by using ideas and results from quantum information in order to prove new results about quantum automorphism groups of graphs, and about quantum permutation groups more generally. In particular, we show that asymptotically almost surely all graphs have trivial quantum automorphism group. Furthermore, we use examples of quantum isomorphic graphs from previous work to construct an infinite family of graphs which are quantum vertex transitive but fail to be vertex transitive, answering a question from the quantum permutation group literature. Our main tool for proving these results is the introduction of orbits and orbitals (orbits on ordered pairs) of quantum permutation groups. We show that the orbitals of a quantum permutation group form a coherent configuration/algebra, a notion from the field of algebraic graph theory. We then prove that the elements of this quantum orbital algebra are exactly the matrices that commute with the magic unitary defining the quantum group. We furthermore show that quantum isomorphic graphs admit an isomorphism of their quantum orbital algebras which maps the adjacency matrix of one graph to that of the other. We hope that this work will encourage new collaborations among the communities of quantum information, quantum groups, and algebraic graph theory. (C) 2020 Elsevier Inc. All rights reserved

Canine substitution of a missing maxillary lateral incisor in an orthodontic re-treatment case: Long term follow up

Abstract Introduction: This case report describes the orthodontic re-treatment of a case with a severely compromised maxillary lateral incisor requiring removal and canine substitution. The treatment included creative asymmetric treatment mechanics and a careful management of anchorage. Case Presentation: Pre-treatment, post treatment and 5 years follow-up records are shown. The treatment outcomes proved to be stable at the follow-up with acceptable aesthetic and functional results. Conclusion: Through careful management of anchorage it was possible to successfully use asymmetric treatment mechanics to achieve a good functional occlusion