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Register a new userSequences from zero entropy noncommutative toral automorphisms and Sarnak Conjecture
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In this paper we study $C^*$-algebra version of Sarnak Conjecture for noncommutative toral automorphisms. Let $A_Theta$ be a noncommutative torus and $alpha_Theta$ be the noncommutative toral automorphism arising from a matrix $Sin GL(d,mathbb{Z})$. We show that if the Voiculescu-Brown entropy of $alpha_{Theta}$ is zero, then the sequence ${rho(alpha_{Theta}^nu)}_{nin mathbb{Z}}$ is a sum of a nilsequence and a zero-density-sequence, where $uin A_Theta$ and $rho$ is any state on $A_Theta$. Then by a result of Green and Tao, this sequence is linearly disjoint from the Mobius function.
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We introduce a notion of noncommutative Choquet simplex, or briefly an nc simplex, that generalizes the classical notion of a simplex. While every simplex is an nc simplex, there are many more nc simplices. They arise naturally from C*-algebras and in noncommutative dynamics. We characterize nc simplices in terms of their geometry and in terms of structural properties of their corresponding operator systems. There is a natural definition of nc Bauer simplex that generalizes the classical definition of a Bauer simplex. We show that a compact nc convex set is an nc Bauer simplex if and only if it is affinely homeomorphic to the nc state space of a unital C*-algebra, generalizing a classical result of Bauer for unital commutative C*-algebras. We obtain several applications to noncommutative dynamics. We show that the set of nc states of a C*-algebra that are invariant with respect to the action of a discrete group is an nc simplex. From this, we obtain a noncommutative ergodic decomposition theorem with uniqueness. Finally, we establish a new characterization of discrete groups with Kazhdans property (T) that extends a result of Glasner and Weiss. Specifically, we show that a discrete group has property (T) if and only if for every action of the group on a unital C*-algebra, the set of invariant states is affinely homeomorphic to the state space of a unital C*-algebra.
A C*-dynamical system is said to have the ideal separation property if every ideal in the corresponding crossed product arises from an invariant ideal in the C*-algebra. In this paper we characterize this property for unital C*-dynamical systems over discrete groups. To every C*-dynamical system we associate a twisted partial C*-dynamical system that encodes much of the structure of the action. This system can often be untwisted, for example when the algebra is commutative, or when the algebra is prime and a certain specific subgroup has vanishing Mackey obstruction. In this case, we obtain relatively simple necessary and sufficient conditions for the ideal separation property. A key idea is a notion of noncommutative boundary for a C*-dynamical system that generalizes Furstenbergs notion of topological boundary for a group.
We show in prime dimension that for two non-commuting totally irreducible toral automorphisms the set of points that equidistribute under the first map but have non-dense orbit under the second has full Hausdorff dimension. In non-prime dimension the argument fails only if the automorphisms have strong algebraic relations.
In this paper, we reduce the logarithmic Sarnak conjecture to the ${0,1}$-symbolic systems with polynomial mean complexity. By showing that the logarithmic Sarnak conjecture holds for any topologically dynamical system with sublinear complexity, we provide a variant of the $1$-Fourier uniformity conjecture, where the frequencies are restricted to any subset of $[0,1]$ with packing dimension less than one.
We provide a systematic study of a noncommutative extension of the classical Anzai skew-product for the cartesian product of two copies of the unit circle to the noncommutative 2-tori. In particular, some relevant ergodic properties are proved for these quantum dynamical systems, extending the corresponding ones enjoyed by the classical Anzai skew-product. As an application, for a uniquely ergodic Anzai skew-product $F$ on the noncommutative $2$-torus $ba_a$, $ainbr$, we investigate the pointwise limit, $lim_{nto+infty}frac1{n}sum_{k=0}^{n-1}l^{-k}F^k(x)$, for $xinba_a$ and $l$ a point in the unit circle, and show that there exist examples for which the limit does not exist even in the weak topology.
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