On ergodic properties of convolution operators associated with compact quantum groups

Ergodic actions of the compact quantum group $O_{-1}(2)$

Among the ergodic actions of a compact quantum group $\mathbb{G}$ on possibly non-commutative spaces, those that are {\it embeddable} are the natural analogues of actions of a compact group on its homogeneous spaces. These can be realized as {\it coideal subalgebras} of the function algebra $\mathcal{O}(\mathbb{G})$ attached to the compact quantum group. We classify the embeddable ergodic actions of the compact quantum group $O_{-1}(2)$, basing our analysis on the bijective correspondence between all ergodic actions of the classical group $O(2)$ and those of its quantum twist resulting from th…

Compact quantum metric spaces and ergodic actions of compact quantum groups

We show that for any co-amenable compact quantum group A=C(G) there exists a unique compact Hausdorff topology on the set EA of isomorphism classes of ergodic actions of G such that the following holds: for any continuous field of ergodic actions of G over a locally compact Hausdorff space T the map T->EA sending each t in T to the isomorphism class of the fibre at t is continuous if and only if the function counting the multiplicity of gamma in each fibre is continuous over T for every equivalence class gamma of irreducible unitary representations of G. Generalizations for arbitrary compact q…

Mean ergodic theorem for amenable discrete quantum groups and a Wiener type theorem for compact metrizable groups

We prove a mean ergodic theorem for amenable discrete quantum groups. As an application, we prove a Wiener type theorem for continuous measures on compact metrizable groups.

冯·诺依曼代数上群作用的算子卷积的结合性

Associativity of operator convolutions for group actions on von Neumann algebras

本文针对局部紧幺模群在半有限冯·诺依曼代数上的特定作用，证明了算子卷积的结合性，为量子调和分析一般版本奠定了基础。

Compact Quantum Ergodic Systems

We develop theory of multiplicity maps for compact quantum groups, as an application, we obtain a complete classification of right coideal $C^*$-algebras of $C(SU_q(2))$ for $q\in [-1,1]\setminus \{0\}$. They are labeled with Dynkin diagrams, but classification results for positive and negative cases of $q$ are different. Many of the coideals are quantum spheres or quotient spaces by quantum subgroups, but we do have other ones in our classification list.

Ergodic actions of compact quantum groups from solutions of the conjugate equations

We use a tensor C*-category with conjugates and two quasitensor functors into the category of Hilbert spaces to define a *-algebra depending functorially on this data. If one of them is tensorial, we can complete in the maximal C*-norm. A particular case of this construction allows us to begin with solutions of the conjugate equations and associate ergodic actions of quantum groups on the C*-algebra in question. The quantum groups involved are A_u(Q) and B_u(Q).

Berezin Quantization From Ergodic Actions of Compact Quantum Groups, and Quantum Gromov-Hausdorff Distance

Over the last 25 years, the notion of "fuzzy spaces" has become ubiquitous in the high-energy physics literature. These are finite dimensional noncommutative approximations of the algebra of functions on a classical space. The most well known examples come from the Berezin quantization of coadjoint orbits of compact semisimple Lie groups. We develop a theory of Berezin quantization for certain quantum homogeneous spaces coming from ergodic actions of compact quantum groups. This allows us to construct fuzzy versions of these quantum homogeneous spaces. We show that the finite dimensional appro…

A mean ergodic theorem for actions of amenable quantum groups

We prove a weak form of the mean ergodic theorem for actions of amenable locally compact quantum groups in the von Neumann algebra setting.

On the entangled ergodic theorem

Let $U$ be a unitary operator acting on the Hilbert space H, and $α:\{1,..., m\}\mapsto\{1,..., k\}$ a partition of the set $\{1,..., m\}$. We show that the ergodic average $$ \frac{1}{N^{k}}\sum_{n_{1},...,n_{k}=0}^{N-1} U^{n_{α(1)}}A_{1}U^{n_{α(2)}}... U^{n_{α(m-1)}}A_{m-1}U^{n_{α(m)}} $$ converges in the weak operator topology if the $A_{j}$ belong to the algebra of all the compact operators on H. We write esplicitely the formula for these ergodic averages in the case of pair--partitions. Some results without any restriction on the operators $A_{j}$ are also presented in the almost periodic…

Tannaka-Krein reconstruction and ergodic actions of easy quantum groups

We give a new alternative version of the reconstruction procedure for ergodic actions of compact quantum groups and we refine it to include characterizations of (braided commutative) Yetter-Drinfeld C*-algebras. We then use this to construct families of ergodic actions of easy quantum groups out of combinatorial data involving partitions and study them. Eventually, we use this categorical point of view to show that the quantum permutation group cannot act ergodically on a classical connected compact space, thereby answering a question of D. Goswami and H. Huang.

A duality theorem for ergodic actions of compact quantum groups on C*-algebras

The spectral functor of an ergodic action of a compact quantum group G on a unital C*-algebra is quasitensor, in the sense that the tensor product of two spectral subspaces is isometrically contained in the spectral subspace of the tensor product representation, and the inclusion maps satisfy natural properties. We show that any quasitensor *-functor from Rep(G) to the category of Hilbert spaces is the spectral functor of an ergodic action of G on a unital C*-algebra. As an application, we associate an ergodic G-action on a unital C*-algebra to an inclusion of Rep(G) into an abstract tensor C*…

Generalized Dobrushin Ergodicity Coefficient and Uniform Ergodicities of Markov Operators

In this paper the stability and the perturbation bounds of Markov operators acting on abstract state spaces are investigated. Here, an abstract state space is an ordered Banach space where the norm has an additivity property on the cone of positive elements. We basically study uniform ergodic properties of Markov operators by means of so-called a generalized Dobrushin's ergodicity coefficient. This allows us to get several convergence results with rates. Some results on quasi-compactness of Markov operators are proved in terms of the ergodicity coefficient. Furthermore, a characterization of u…

Growth rates of dimensional invariants of compact quantum groups and a theorem of Hoegh-Krohn, Landstad and Stormer

We give local upper and lower bounds for the eigenvalues of the modular operator associated to an ergodic action of a compact quantum group on a unital C*-algebra. They involve the modular theory of the quantum group and the growth rate of quantum dimensions of its representations and they become sharp if other integral invariants grow subexponentially. For compact groups, this reduces to the finiteness theorem of Hoegh-Krohn, Landstad and Stormer. Consequently, compact quantum groups of Kac type admitting an ergodic action with a non-tracial invariant state must have representations whose dim…

The Szemeredi property in ergodic W*-dynamical systems

We study weak mixing of all orders for asymptotically abelian weakly mixing state preserving C*-dynamical systems, where the dynamics is given by the action of an abelian second countable locally compact group which contains a Folner sequence satisfying the Tempelman condition. For a smaller class of groups (which include Z^q and R^q) this is then used to show that an asymptotically abelian ergodic W*-dynamical system either has the "Szemeredi property" or contains a nontrivial subsystem (a "compact factor") that does. A van der Corput lemma for Hilbert space valued functions on the group is o…

Ergodic coactions with large multiplicity and monoidal equivalence of quantum groups

We construct new examples of ergodic coactions of compact quantum groups, in which the multiplicity of an irreducible corepresentation can be strictly larger than the dimension of the latter. These examples are obtained using a bijective correspondence between certain ergodic coactions on C*-algebras and unitary fiber functors on the representation category of a compact quantum group. We classify these unitary fiber functors on the universal orthogonal and unitary quantum groups. The associated C*-algebras and von Neumann algebras can be defined by generators and relations, but are not yet wel…

关于\(\mathbb{R}^n\)上线性作用的遍历性及群冯·诺依曼代数的因子性

On ergodicity of linear actions on $\mathbb{R}^n$ and factoriality of group von Neumann algebras

研究离散可数群在李型阿贝尔局部紧群上作用，通过给出自然条件，证明其半直积的群冯·诺依曼代数的因子性，进而给出一大类局部紧群例子，其群冯·诺依曼代数是因子。

Noncommutative Ergodic Theorems for Connected Amenable Groups

This paper is devoted to the study of noncommutative ergodic theorems for connected amenable locally compact groups. For a dynamical system $(\mathcal{M},τ,G,σ)$, where $(\mathcal{M},τ)$ is a von Neumann algebra with a normal faithful finite trace and $(G,σ)$ is a connected amenable locally compact group with a well defined representation on $\mathcal{M}$, we try to find the largest noncommutative function spaces constructed from $\mathcal{M}$ on which the individual ergodic theorems hold. By using the Emerson-Greenleaf's structure theorem, we transfer the key question to proving the ergodic t…

源自遍历共作用的量子超群

Quantum hypergroups arising from ergodic coactions

本文研究了通过遍历可积作用得到的量子超群，揭示了其自然共作用结构，并通过等变对应理论刻画了这些超群的共可除性。

A Nonconventional Ergodic Theorem for quantum "diagonal measures"

We extend the Nonconventional Ergodic Theorem for generic measures by Furstenberg, to several situations of interest arising from quantum dynamical systems. We deal with the diagonal state canonically associated to the product state (i.e. quantum "diagonal measures"), or to convex combinations of diagonal measures for non ergodic cases. For the sake of completeness, we treat also the Nonconventional Ergodic Theorem for compact dynamical systems, that is when the unitary generating the dynamics in the GNS representation is almost periodic. The Nonconventional Ergodic Theorem allows in a natural…

Ergodic invariant states and irreducible representations of crossed product $C^*$-algebras

Motivated by reformulating Furstenberg's $\times p,\times q$ conjecture via representations of a crossed product $C^*$-algebra, we show that in a discrete $C^*$-dynamical system $(A,Γ)$, the space of (ergodic) $Γ$-invariant states on $A$ is homeomorphic to a subspace of (pure) state space of $A\rtimesΓ$. Various applications of this in topological dynamical systems and representation theory are obtained. In particular, we prove that the classification of ergodic $Γ$-invariant regular Borel probability measures on a compact Hausdorff space $X$ is equivalent to the classification a special type …

Ergodic theory of affine isometric actions on Hilbert spaces

The classical Gaussian functor associates to every orthogonal representation of a locally compact group $G$ a probability measure preserving action of $G$ called a Gaussian action. In this paper, we generalize this construction by associating to every affine isometric action of $G$ on a Hilbert space, a one-parameter family of nonsingular Gaussian actions whose ergodic properties are related in a very subtle way to the geometry of the original action. We show that these nonsingular Gaussian actions exhibit a phase transition phenomenon and we relate it to new quantitative invariants for affine…