In the present paper we consider a von Neumann algebra M with a faithful normal semi-finite trace $\t$, and $\{α_ t\} $ a strongly continuous extension to $L^p(M,\t)$ of a semigroup of absolute contractions on $L^1 (M,τ)$. By means of a non-commutative Banach Principle we prove for a Besicovitch function b and $x\in L^p(M,\t)$, the averages \frac{1}{T}\int_0^Tb(t)\a_t(x)dt converge bilateral almost uniform in $L^p(M,\t)$ as $T\to 0$.
We show that ergodic flows in noncommutative fully symmetric spaces (associated with a semifinite von Neumann algebra) generated by continuous semigroups of positive Dunford-Schwartz operators and modulated by bounded Besicovitch almost periodic functions converge almost uniformly. The corresponding local ergodic theorem is also discussed.
In [11], employing the technique of noncommutative interpolation, a maximal ergodic theorem in noncommutative Lp-spaces, 1 < p < infinity, was established and, among other things, corresponding maximal ergodic inequalities and individual ergodic theorems were derived. In this article, we derive maximal ergodic inequalities in noncommutative Lp-spaces directly from [25] and apply them to prove corresponding individual and Besicovitch weighted ergodic theorems. Then we extend these results to noncommutative fully symmetric Banach spaces with Fatou property and non-trivial Boyd indices, in partic…
It is known that, for a positive Dunford-Schwartz operator in a noncommutative $L^p-$space, $1\leq p<\infty$ or, more generally, in a noncommutative Orlicz space with order continuous norm, the corresponding ergodic averages converge bilaterally almost uniformly. We show that these averages converge bilaterally almost uniformly in each noncommutative symmetric space $E$ such that $μ_t(x) \to 0$ as $t \to 0$ for every $x \in E$, where $μ_t(x)$ is a non-increasing rearrangement of $x$. In particular, these averages converge bilaterally almost uniformly in all noncommutative symmetric spaces with…
In the first part of the paper the natural scheme for proving noncommutative individual ergodic theorems for multiple sequences is described and applied to obtain results on unrestricted convergence of multiaverages. In the second part convergence of ergodic averages induced by several maps satisfying specific recurrence relations, including so-called Multi Free Group Partial Sums, is proved. This is the multiindexed version of results obtained earlier jointly with V.I.Chilin and S.Litvinov.
It is known that, for a positive Dunford-Schwartz operator in a noncommutative $L^p$-space, $1\leq p<\infty$, or, more generally, in a noncommutative Orlicz space with order continuous norm, the corresponding ergodic averages converge bilaterally almost uniformly. We show that these averages converge almost uniformly in each noncommutative symmetric space $E$ such that $μ_t(x)\to 0$ as $t\to\infty$ for every $x\in E$, where $μ_t(x)$ is the non-increasing rearrangement of $x$. Noncommutative Dunford-Schwartz-type multiparameter ergodic theorems are studied. A wide range of noncommutative symmet…
It is known that, for a positive Dunford-Schwartz operator in a noncommutative $L^p-$space, $1\leq p<\infty$ or, more generally, in a noncommutative Orlicz space with order continuous norm, the corresponding ergodic averages converge bilaterally almost uniformly. We show that these averages converge almost uniformly in each noncommutative symmetric space $E$ such that $μ_t(x) \to 0$ as $t \to 0$ for every $x \in E$, where $μ_t(x)$ is a non-increasing rearrangement of $x$. Noncommutative Dunford-Schwartz-type multiparameter ergodic theorems are studied. A wide range of noncommutative symmetric …
In the paper we consider $T_{1},..., T_{d}$ absolute contractions of von Neumann algebra $\M$ with normal, semi-finite, faithful trace, and prove that for every bounded Besicovitch weight $\{a(\kb)\}_{\kb\in\bn^d}$ and every $x\in L_{p}(\M)$, ($p>1$) the averages A_{\Nb}(x)=\frac{1}{|\Nb|}\sum\limits_{\kb=1}^{\Nb}a(\kb)\Tb^{\kb}(x). converge bilaterally almost uniformly in $L_{p}(\M)$.
This paper is devoted to the study of various maximal ergodic theorems in noncommutative $L_p$-spaces. In particular, we prove the noncommutative analogue of the classical Dunford-Schwartz maximal ergodic inequality for positive contractions on $L_p$ and the analogue of Stein's maximal inequality for symmetric positive contractions. We also obtain the corresponding individual ergodic theorems. We apply these results to a family of natural examples which frequently appear in theory of von Neumann algebras and in quantum probability.
In this paper, we investigate the boundedness of a square function from ergodic theory on noncommutative $L_{p}$-spaces. The main result is a weak $(1,1)$ type estimate of this square function. We also show the $(L_{\infty},\mathrm{BMO})$ estimate, and thus strong $(L_{p},L_{p})$ estimate by interpolation. The main novel difficulty lies in the fact that the kernel of this square function does not enjoy any regularity, which is crucial in showing such endpoint estimates for standard noncommutative Calderón-Zygmund singular integral operators.
For a noncommutative Orlicz space associated with a semifnite von Neumann algebra, a faithful normal semifnite trace and an Orlicz function satisfying $(δ_2,Δ_2)-$condition, an individual ergodic theorem is proved.
We extend the noncommutative L1-maximal ergodic inequality for semifinite von Neumann algebras established by Yeadon in 1977 to the framework of noncommutative L1-spaces associated with sigma-finite von Neumann algebras. Since the semifnite case of this result is one of the two essential parts in the proof of noncommutative maximal ergodic inequality for tracial Lp-spaces (1<p<infinity) by Junge-Xu in 2007, we hope our result will be helpful to establish a complete noncommutative maximal ergodic inequality for non-tracial Lp-spaces in the future.
We prove maximal ergodic theorems for spherical averages on the Heisenberg groups acting on $L_p$ spaces over measure spaces not necessarily commutative, that is, on noncommutative $L_p$ spaces. The scale of $p$ is optimal in the reduced Heisenberg group case. We also obtain the corresponding individual ergodic theorems and differential theorems in the noncommutative setting. The results can be regarded as noncommutative analogues of Nevo-Thangavelu's ergodic theorems. The approach of proof involves recent developments in noncommutative $L_p$ spaces and in noncommutative harmonic analysis, in …
This article is devoted to studying individual ergodic theorems for subsequential weighted ergodic averages on the noncommutative Lp-spaces associated to a semifinite von Neumann algebra M. In particular, we establish the convergence of these averages along sequences with density one and certain types of block sequences with positive lower density, and we extend known results along uniform sequences in the sense of Brunel and Keane.
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 $\ba_\a$, $\a\in\br$, we investigate the pointwise limit, $\lim_{n\to+\infty}\frac1{n}\sum_{k=0}^{n-1}ł^{-k}\F^k(x)$, for $x\in\ba_\a$ and $ł$ …
In this paper, we establish the quantitative mean ergodic theorems for two subclasses of power bounded operators on a fixed noncommutative $L_p$-space with $1<p<\infty$, which mainly concerns power bounded invertible operators and Lamperti contractions. Our approach to the quantitative ergodic theorems is the noncommutative square function inequalities. The establishment of the latter involves several new ingredients such as the almost orthogonality and Calderón-Zygmund arguments for non-smooth kernels from semi-commutative harmonic analysis, the extension properties of the operators under con…
We extend in a noncommutative setting the individual ergodic theorem of Nevo and Stein concerning measure preserving actions of free groups and averages on spheres $s_{2n}$ of even radius. Here we study state preserving actions of free groups on a von Neumann algebra $A$ and the behaviour of $(s_{2n}(x))$ for $x$ in noncommutative spaces $L^p(A)$. For the Cesàro means $\frac{1}{n}\sum_{k=0}^{n-1} s_k$ and $p = +\infty$, this problem was solved by Walker. Our approach is based on ideas of Bufetov. We prove a noncommutative version of Rota ``Alternierende Verfahren'' theorem. To this end, we int…
We extend the theory of ergodic optimization and maximizing measures to the non-commutative field of C*-dynamical systems. We then employ this ergodic optimization machinery to provide an alternate characterization of unique erogdicity of C*-dynamical systems when the resident group action satisfies certain Choquet-theoretic assumptions.
We establish that particular quotients of the non-commutative Hardy algebras carry ergodic actions of convergent discrete subgroups of the group $\operatorname*{SU}(n,1)$ of automorphisms of the unit ball in $\mathbb{C}% ^{n}$. To do so, we provide a mean to compute the spectra of quotients of noncommutative Hardy algebra and characterize their automorphisms in term of biholomorphic maps of the unit ball in $\mathbb{C}^{n}$.
In this paper, we establish a multi-parameter version of Bellow and Losert's Wiener-Wintner type ergodic theorem for dynamical systems not necessarily being commutative. More precisely, we introduce a weight class $\mathcal{D}$, which is shown to strictly include the multi-parameter bounded Besicovitch weight class, thus including the set $$Λ_d=\left\{\{λ^{k_1}_1\dotsmλ^{k_d}_d\}_{(k_1,\dots,k_d)\in \mathbb{N}^d}:\quad (λ_1,\dots,λ_d) \in \mathbb{T}^d\right\};$$ then prove a multi-parameter Bellow and Losert's Wiener-Wintner type ergodic theorem for the class $\mathcal{D}$ and for noncommutati…
This paper is devoted to the study of noncommutative maximal inequalities and ergodic theorems for group actions on von Neumann algebras. Consider a locally compact group $G$ of polynomial growth with a symmetric compact subset $V$. Let $α$ be a continuous action of $G$ on a von Neumann algebra $\mathcal{M}$ by trace-preserving automorphisms. We then show that the operators defined by \begin{equation*} A_{n}x= \frac{1}{m(V^{n})} \int _{V^{n}}α_{g}x\,dm(g),\quad x\in L_{p}( \mathcal{M}),n\in \mathbb{N},1\leq p\leq \infty , \end{equation*} are of weak type $(1,1)$ and of strong type $(p,p)$ for …
In this article, we obtain a version of the noncommutative Banach Principle suitable to prove Wiener-Wintner type results for weights in W1-space. This is used to obtain noncommutative Wiener-Wintner type ergodic theorems for various types of weights for certain types of positive Dunford-Schwartz operators. We also study the b.a.u. (a.u.) convergence of some subsequential averages and moving averages of such operators
We extend the theory of ergodic optimization and maximizing measures to the non-commutative field of C*-dynamical systems. We then provide a result linking the ergodic optimizations of elements of a C*-dynamical system to the convergence of certain ergodic averages in a suitable seminorm. We also provide alternate proofs of several results in this article using the tools of nonstandard analysis.
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…
In this paper, we establish the one-sided maximal ergodic inequalities for a large subclass of positive operators on noncommutative $L_p$-spaces for a fixed $1<p<\infty$, which particularly applies to positive isometries and general positive Lamperti contractions or power bounded doubly Lamperti operators; moreover, it is known that this subclass recovers all positive contractions on the classical Lebesgue spaces $L_p([0,1])$. Our study falls into neither the category of positive contractions considered by Junge-Xu nor the class of power bounded positive invertible operators considered by Hong…
In this paper we study unique ergodicity of $C^*$-dynamical system $(\ga,T)$, consisting of a unital $C^*$-algebra $\ga$ and a Markov operator $T:\ga\mapsto\ga$, relative to its fixed point subspace, in terms of Riesz summation which is weaker than Cesaro one. Namely, it is proven that $(\ga,T)$ is uniquely ergodic relative to its fixed point subspace if and only if its Riesz means {equation*} \frac{1}{p_1+...+p_n}\sum_{k=1}^{n}p_kT^kx {equation*} converge to $E_T(x)$ in $\ga$ for any $x\in\ga$, as $n\to\infty$, here $E_T$ is an projection of $\ga$ to the fixed point subspace of $T$. It is als…
In a previous article, we extended the notion of ergodic optimization to the setting of C*-dynamical systems of countable discrete groups. Among the key results of that paper was that given an action $G \stackrelΞ{\curvearrowright} \mathfrak{M}$ of a countable discrete amenable group $G$ on a W*-probability space $(\mathfrak{M}, ρ)$ by $ρ$-preserving $*$-automorphisms of $\mathfrak{M}$, a positive element $x \in \mathfrak{M}$, and a right Følner sequence $\mathcal{F} = (F_k)_{k \in \mathbb{N} }$ for $G$, the sequence $$\left( \left\| \frac{1}{|F_k|} \sum_{g \in F_k} Ξ_g x \right\| \right)_{ k …
Recent results of M.Junge and Q.Xu on the ergodic properties of the averages of kernels in noncommutative L^p-spaces are applied to the analysis of the almost uniform convergence of operators induced by the convolutions on compact quantum groups.
We consider the reduction of problems on general noncommutative $L_p$-spaces to the corresponding ones on those associated with finite von Neumann algebras. The main tool is a unpublished result of the first named author which approximates any noncommutative $L_p$-space by tracial ones. We show that under some natural conditions a map between two von Neumann algebras extends to their crossed products by a locally compact abelian group or to their noncommutative $L_p$-spaces. We present applications of these results to the theory of noncommutative martingale inequalities by reducing most recent…
The purpose of the paper is to establish weighted maximal $L_p$-inequalities in the context of operator-valued martingales on semifinite von Neumann algebras. The main emphasis is put on the optimal dependence of the $L_p$ constants on the characteristic of the weight involved. As applications, we establish weighted estimates for the noncommutative version of Hardy-Littlewood maximal operator and weighted bounds for noncommutative maximal truncations of a wide class of singular integrals.
The noncommutative Choquet boundary and the C*-envelope of operator systems of the form Span{1,T,T*}, where T is a Hilbert space operator with normal-like features, are studied. Such operators include normal operators, k-normal operators, subnormal operators, and Toeplitz operators. Our main result is the determination of the noncommutative Choquet boundary for an operator system generated by an irreducible periodic weighted unilateral shift operator.
To any action of a locally compact group $G$ on a pair $(A,B)$ of von Neumann algebras is canonically associated a pair $(π\_A^α, π\_B^α)$ of unitary representations of $G$. The purpose of this paper is to provide results allowing to compare the norms of the operators $π\_A^α(μ)$ and $π\_B^α(μ)$ for bounded measures $μ$ on $G$. We have a twofold aim. First to point out that several known facts in ergodic and representation theory are indeed particular cases of general results about $(π\_A^α, π\_B^α)$. Second, under amenability assumptions, to obtain transference of inequalities that will be us…
Let $\mathcal{M}$ be a semifinite von Neumann algebra equipped with a semifinite normal faithful trace $τ$. Let $d$ be an injective positive measurable operator with respect to $(\mathcal{M}, τ)$ such that $d^{-1}$ is also measurable. Define $$L_p(d)={x\in L_0(\mathcal{M}) : dx+xd\in L_p(\mathcal{M})}\quad{and}\quad \|x\|_{L_p(d)}=\|dx+xd\|_p .$$ We show that for $1\le p_0<p_1\le\8$, $0<θ<1$ and $α_0\ge0, α_1\ge0$ the interpolation equality $$(L_{p_0}(d^{α_0}), L_{p_1}(d^{α_1}))_θ =L_{p}(d^α)$$ holds with equivalent norms, where $\frac1p=\frac{1-θ}{p_0}+\fracθ{p_1}$ and $α=(1-θ)α_0+θα_1$.
We initiate the study of weighted multi-Toeplitz operators associated with noncommutative regular domains in B(H)^n. These operators are acting on the full Fock space with n generators and have as symbols free pluriharmonic functions. Several classical results from complex analysis concerning harmonic functions have analogues in our noncommutative setting. In particular, we show that the bounded free pluriharmonic functions are precisely those which are noncommutative Berezin transforms of weighted multi-Toeplitz operators, and solve the Dirichlet extension problem in this setting. Using nonco…
We obtain a noncommutative multivariable analogue of Louhichi and Olofsson characterization of Toeplitz operators with harmonic symbols on the weighted Bergman space $A_m({\bf D})$, as well as Eschmeier and Langendorfer extension to the unit ball of ${\bf C}^n$. All our results are proved in the more general setting of noncommutative poly-hyperballs ${\bf D_n^m}(H)$, ${\bf n,m}\in {\bf N}^k$, and are used to characterize the bounded free $k$-pluriharmonic functions with operator coefficients on poly-hyperballs and to solve the associated Dirichlet extension problem. In particular, the results …
We present a generalization of bilateral weighted shift operators for the noncommutative multivariable setting. We discover a notion of periodicity for these shifts, which has an appealing diagramatic interpretation in terms of an infinite tree structure associated with the underlying Hilbert space. These shifts arise naturally through weighted versions of certain representations of the Cuntz C$^*$-algebras $O_n$. It is convenient, and equivalent, to consider the weak operator topology closed algebras generated by these operators when investigating their joint reducing subspace structure. We p…
We prove that unique ergodicity of tensor product of $C^*$-dynamical system implies its strictly weak mixing. By means of this result a uniform weighted ergodic theorem with respect to $S$-Besicovitch sequences for strictly weak mixing dynamical systems is proved. Moreover, we provide certain examples of strictly weak mixing dynamical systems.
We study an ergodic theorem for disjoint C*-dynamical systems, where disjointness here is a noncommutative version of the concept introduced by Furstenberg for classical dynamical systems. This is applied to W*-dynamical systems. We also consider specific examples of disjoint C*-dynamical and W*-dynamical systems, including for actions of other groups than $\mathbb{Z}$. Unique ergodicity and unique ergodicity relative to the fixed point algebra are closely related to disjointness, and are used to give examples of disjoint C*-dynamical systems.
In this paper we introduce and study the class of weighted multi-Toeplitz operators associated with noncommutative polydomains ${\bf D_f^m}$, ${\bf m}:=(m_1,\ldots, m_k)\in {\bf N}^k$, generated by $k$-tuples ${\bf f}:=(f_1,\ldots, f_k)$ of positive regular free holomorphic functions in a neighborhood of the origin. These operators are acting on the tensor product $F^2(H_{n_1})\otimes \cdots \otimes F^2(H_{n_k})$ of full Fock spaces with $n_i$ generators or, equivalently, they can be viewed as multi-Toeplitz operators acting on tensor products of weighted full Fock spaces. For a large class of…
Noncommutative multivariable versions of weighted shifts arise naturally as `weighted' left creation operators acting on Fock space. We investigate the unital weak operator topology closed algebras they generate. The unweighted case yields noncommutative analytic Toeplitz algebras. The commutant can be described in terms of weighted right creation operators when the weights satisfy a condition specific to the noncommutative setting. We prove these algebras are reflexive when the eigenvalues for the adjoint algebra include an open set in complex $n$-space, and provide a new elementary proof of …
Maximal Ergodic Theorems for Operators with Finite Peripheral Spectrum
Indian Institute of Science Education and Research Thiruvananthapuram（印度科学教育研究所特里凡得琅分校）；The Institute of Mathematical Sciences（数学科学研究院）；Homi Bhabha National Institute（霍米·巴巴国家研究所）
DSI-NRF CoE in Math. and Stat. Sci, Pure and Applied Analytics NWU（南非科学与技术部-国家研究基金会数学与统计科学卓越中心，纯与应用分析西北大学）；School of Math. & Stat. Sci., NWU（西北大学数学与统计科学学院）
