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Register a new userSkew-product dynamical systems for crossed product $C^*$-algebras and their ergodic properties
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Starting from a discrete $C^*$-dynamical system $(mathfrak{A}, theta, omega_o)$, we define and study most of the main ergodic properties of the crossed product $C^*$-dynamical system $(mathfrak{A}rtimes_alphamathbb{Z}, Phi_{theta, u},om_ocirc E)$, $E:mathfrak{A}rtimes_alphamathbb{Z}rightarrowga$ being the canonical conditional expectation of $mathfrak{A}rtimes_alphamathbb{Z}$ onto $mathfrak{A}$, provided $ainaut(ga)$ commute with the $*$-automorphism $th$ up tu a unitary $uinga$. Here, $Phi_{theta, u}inaut(mathfrak{A}rtimes_alphamathbb{Z})$ can be considered as the fully noncommutative generalisation of the celebrated skew-product defined by H. Anzai for the product of two tori in the classical case.
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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.
We consider a family of dynamical systems (A,alpha,L) in which alpha is an endomorphism of a C*-algebra A and L is a transfer operator for alpha. We extend Exels construction of a crossed product to cover non-unital algebras A, and show that the C*-algebra of a locally finite graph can be realised as one of these crossed products. When A is commutative, we find criteria for the simplicity of the crossed product, and analyse the ideal structure of the crossed product.
Let $(G, P)$ be an abelian, lattice ordered group and let $X$ be a compactly aligned product system over $P$. We show that the C*-envelope of the Nica tensor algebra $mathcal{N}mathcal{T}^+_X$ coincides with both Sehnems covariance algebra $mathcal{A} times_X P$ and the co-universal C*-algebra $mathcal{N}mathcal{O}^r_X$ for injective, gauge compatible, Nica-covariant representations of Carlsen, Larsen, Sims and Vittadello. We give several applications of this result on both the selfadjoint and non-selfadjoint operator algebra theory. First we guarantee the existence of $mathcal{N}mathcal{O}^r_X$, thus settling a problem of Carlsen, Larsen, Sims and Vittadello which was open even for abelian, lattice ordered groups. As a second application, we resolve a problem posed by Skalski and Zacharias on dilating isometric representations of product systems to unitary representations. As a third application we characterize the C*-envelope of the tensor algebra of a finitely aligned higher-rank graph which also holds for topological higher-rank graphs. As a final application we prove reduced Hao-Ng isomorphisms for generalized gauge actions of discrete groups on C*-algebras of product systems. This generalizes recent results that were obtained by various authors in the case where $(G, P) =(mathbb{Z},mathbb{N})$.
A cosystem consists of a possibly nonselfadoint operator algebra equipped with a coaction by a discrete group. We introduce the concept of C*-envelope for a cosystem; roughly speaking, this is the smallest C*-algebraic cosystem that contains an equivariant completely isometric copy of the original one. We show that the C*-envelope for a cosystem always exists and we explain how it relates to the usual C*-envelope. We then show that for compactly aligned product systems over group-embeddable right LCM semigroups, the C*-envelope is co-universal, in the sense of Carlsen, Larsen, Sims and Vittadello, for the Fock tensor algebra equipped with its natural coaction. This yields the existence of a co-universal C*-algebra, generalizing previous results of Carlsen, Larsen, Sims and Vittadello, and of Dor-On and Katsoulis. We also realize the C*-envelope of the tensor algebra as the reduced cross sectional algebra of a Fell bundle introduced by Sehnem, which, under a mild assumption of normality, we then identify to the quotient of the Fock algebra by the image of Sehnems strong covariance ideal. In another application, we obtain a reduced Hao-Ng isomorphism theorem for the co-universal algebras.
We investigate some ergodic and spectral properties of general (discrete) $C^*$-dynamical systems $({mathfrak A},Phi)$ made of a unital $C^*$-algebra and a multiplicative, identity-preserving $*$-map $Phi:{mathfrak A}to{mathfrak A}$, particularising the situation when $({mathfrak A},Phi)$ enjoys the property of unique ergodicity with respect to the fixed-point subalgebra. For $C^*$-dynamical systems enjoying or not the strong ergodic property mentioned above, we provide conditions on $lambda$ in the unit circle ${zin{mathbb C}mid |z|=1}$ and the corresponding eigenspace ${mathfrak A}_lambdasubset{mathfrak A}$ for which the sequence of Cesaro averages $left(frac1{n}sum_{k=0}^{n-1}lambda^{-k}Phi^kright)_{n>0}$, converges point-wise in norm. We also describe some pivotal examples coming from quantum probability, to which the obtained results can be applied.
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