A computational approach to the Thompson group FF

Let $F$ denote the Thompson group with standard generators $A=x_0$, $B=x_1$. It is a long standing open problem whether $F$ is an amenable group. By a result of Kesten from 1959, amenability of $F$ is equivalent to $$(i)\qquad ||I+A+B||=3$$ and to $$(ii)\qquad ||A+A^{-1}+B+B^{-1}||=4,$$ where in both cases the norm of an element in the group ring $\mathbb{C} F$ is computed in $B(\ell^2(F))$ via the regular representation of $F$. By extensive numerical computations, we obtain precise lower bounds for the norms in $(i)$ and $(ii)$, as well as good estimates of the spectral distributions of $(I+A+B)^*(I+A+B)$ and of $A+A^{-1}+B+B^{-1}$ with respect to the tracial state $\tau$ on the group von Neumann Algebra $L(F)$. Our computational results suggest, that $$||I+A+B||\approx 2.95 \qquad ||A+A^{-1}+B+B^{-1}||\approx 3.87.$$ It is however hard to obtain precise upper bounds for the norms, and our methods cannot be used to prove non-amenability of $F$.Comment: appears in International Journal of Algebra and Computation (2015

Explicit lower and upper bounds on the entangled value of multiplayer XOR games

XOR games are the simplest model in which the nonlocal properties of entanglement manifest themselves. When there are two players, it is well known that the bias --- the maximum advantage over random play --- of entangled players can be at most a constant times greater than that of classical players. Recently, P\'{e}rez-Garc\'{i}a et al. [Comm. Math. Phys. 279 (2), 2008] showed that no such bound holds when there are three or more players: the advantage of entangled players over classical players can become unbounded, and scale with the number of questions in the game. Their proof relies on non-trivial results from operator space theory, and gives a non-explicit existence proof, leading to a game with a very large number of questions and only a loose control over the local dimension of the players' shared entanglement. We give a new, simple and explicit (though still probabilistic) construction of a family of three-player XOR games which achieve a large quantum-classical gap (QC-gap). This QC-gap is exponentially larger than the one given by P\'{e}rez-Garc\'{i}a et. al. in terms of the size of the game, achieving a QC-gap of order $\sqrt{N}$ with $N^2$ questions per player. In terms of the dimension of the entangled state required, we achieve the same (optimal) QC-gap of $\sqrt{N}$ for a state of local dimension $N$ per player. Moreover, the optimal entangled strategy is very simple, involving observables defined by tensor products of the Pauli matrices. Additionally, we give the first upper bound on the maximal QC-gap in terms of the number of questions per player, showing that our construction is only quadratically off in that respect. Our results rely on probabilistic estimates on the norm of random matrices and higher-order tensors which may be of independent interest.Comment: Major improvements in presentation; results identica

A new look at C*-simplicity and the unique trace property of a group

We characterize when the reduced C*-algebra of a group has unique tracial state, respectively, is simple, in terms of Dixmier-type properties of the group C*-algebra. We also give a simple proof of the recent result by Breuillard, Kalantar, Kennedy and Ozawa that the reduced C*-algebra of a group has unique tracial state if and only if the amenable radical of the group is trivial.Comment: 8 page

Mutually unbiased bases in dimension six: The four most distant bases

We consider the average distance between four bases in dimension six. The distance between two orthonormal bases vanishes when the bases are the same, and the distance reaches its maximal value of unity when the bases are unbiased. We perform a numerical search for the maximum average distance and find it to be strictly smaller than unity. This is strong evidence that no four mutually unbiased bases exist in dimension six. We also provide a two-parameter family of three bases which, together with the canonical basis, reach the numerically-found maximum of the average distance, and we conduct a detailed study of the structure of the extremal set of bases.Comment: 10 pages, 2 figures, 1 tabl

On the uniqueness of the injective III1 factor

We give a new proof of a theorem due to Alain Connes, that an injective factor N of type III1 with separable predual and with trivial bicentralizer is isomorphic to the Araki-Woods type III1 factor R∞. This, combined with the author's solution to the bicentralizer problem for injective III1 factors provides a new proof of the theorem that up to *-isomorphism, there exists a unique injective factor of type III1 on a separable Hilbert space

On the uniqueness of the injective III1 factor

We give a new proof of a theorem due to Alain Connes, that an injective factor N of type III1 with separable predual and with trivial bicentralizer is isomorphic to the Araki-Woods type III1 factor R∞. This, combined with the author's solution to the bicentralizer problem for injective III1 factors provides a new proof of the theorem that up to *-isomorphism, there exists a unique injective factor of type III1 on a separable Hilbert space