Noncommutative maximal ergodic inequalities associated with doubling conditions

TL;DR

This paper establishes noncommutative maximal ergodic inequalities for group actions on von Neumann algebras, extending previous results to more general doubling conditions and employing quantum probabilistic and random walk techniques.

Contribution

It introduces new maximal inequalities for noncommutative spaces under doubling conditions, surpassing prior work limited to Dunford-Schwartz operators.

Findings

Proved weak and strong type inequalities for averaging operators on von Neumann algebras.

Established almost uniform convergence of ergodic averages in noncommutative Lp spaces.

Extended maximal ergodic theorems to broader classes of actions beyond previous frameworks.

Abstract

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 $\alpha $ 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}}\alpha _{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 $1 < p<\infty $. Consequently, the sequence $(A_{n}x)_{n\geq 1}$ converges almost uniformly for $x\in L_{p}(\mathcal{M})$ for $1\leq p<\infty $. Also, we establish the noncommutative maximal and individual ergodic theorems associated with more general doubling conditions, and we prove the corresponding results for general actions on one fixed noncommutative $L_{p}$-space which are beyond the class of Dunford-Schwartz operators considered previously by Junge and Xu. As key ingredients, we also obtain the Hardy-Littlewood maximal inequality on metric spaces with doubling measures in the operator-valued setting. After the groundbreaking work of Junge and Xu on the noncommutative Dunford-Schwartz maximal ergodic inequalities, this is the first time that more general maximal inequalities are proved beyond Junge and Xu's setting. Our approach is based on quantum probabilistic methods as well as random walk theory.

Equations

$$A_{n}x=\frac{1}{m(V^n)}{\int }_{V^n}{\alpha }_{g}xdm(g),x\in L_p(\mathcal{M}),n\in \mathbb{N},1\le p\le \infty ,$$

$$m(B_{2r})\le Cm(B_r),r>0,$$

$$\underset{r\to \infty }{\lim }\frac{m((B_{r}s)△B_r)}{m(B_r)}=0,$$

$$A_{r}x=\frac{1}{m(B_r)}{\int }_{B_r}{\alpha }_{s}xdm(s),x\in \mathcal{M},r>0.$$

$$A_{n}x=\frac{1}{m(V^n)}{\int }_{V^n}{\alpha }_{s}xdm(s),x\in L_p(\mathcal{M}),n\in \mathbb{N}$$

$$A_{n}x=\frac{1}{m(V^n)}{\int }_{V^n}{\alpha }_{s}xdm(s),x\in \mathcal{M},n\in \mathbb{N}$$

$$A_{r}f(x)=\frac{1}{\mu(B(x,r))}\int_{B(x,r)}f\,d\mu,\quad f\in L_{p}\bigl{(}X;L_{p}(\mathcal{M})\bigr{)},x\in X,r>0$$

$$\frac{1}{m(V^n)}{\int }_{V^n}{\alpha }_{s}xdm(s)\le \frac{c}{n^2}\sum _{k=1}^{2n^2}{T}^{k}x,x\ge 0,$$

$$\|x\|_{p}=\bigl{[}\tau\bigl{(}|x|^{p}\bigr{)}\bigr{]}^{1/p},\quad x\in{\mathcal{S}},$$

$$x_n=a{y}_{n}b,\forall n\ge 0.$$

$$\|x\|_{L_{p}(\mathcal{M};\ell_{\infty})}=\inf\bigl{\{}\|a\|_{2p}\sup_{n\geq 0}\|y_{n}\|_{\infty}\|b\|_{2p}\bigr{\}},$$

$$\bigl{\|}{\sup_{n}}^{+}x_{n}\bigr{\|}_{p}=\inf\bigl{\{}\|a\|_{p}:a\in L_{p}(\mathcal{M}),a\geq 0\text{ and }x_{n}\leq a\text{ for any }n\geq 0\bigr{\}}.$$

$$x_n=y_{n}a,\forall n\ge 0.$$

$$\|x\|_{L_{p}(\mathcal{M};\ell_{\infty}^{c})}=\inf\bigl{\{}\sup_{n\geq 0}\|y_{n}\|_{\infty}\|a\|_{p}\bigr{\}},$$

$$\sup_{J\mathrm{finite}}\bigl{\|}\mathop{{\sup_{i\in J}}^{+}}x_{i}\bigr{\|}_{p}<\infty,$$

$$\bigl{\|}\mathop{{\sup_{i\in I}}^{+}}x_{i}\bigr{\|}_{p}=\sup_{J\mathrm{finite}}\bigl{\|}\mathop{{\sup_{i\in J}}^{+}}x_{i}\bigr{\|}_{p}.$$

$$L_{p}(\mathcal{M};\ell_{\infty})=\bigl{(}L_{p_{0}}(\mathcal{M};\ell_{\infty}),L_{p_{1}}(\mathcal{M};\ell_{\infty})\bigr{)}_{\theta},$$

$$L_{p}(\mathcal{M};\ell_{\infty}^{c})=\bigl{(}L_{p_{0}}(\mathcal{M};\ell_{\infty}^{c}),L_{p_{1}}(\mathcal{M};\ell_{\infty}^{c})\bigr{)}_{\theta},$$

$$\tau (1-e)\le \frac{C^p}{{\lambda }^p}\Vert x{\Vert }_p^p,e{S}_i(x)e\le \lambda e,i\in I.$$

$$\bigl{\|}{\sup_{i\in I}}^{+}S_{i}x\bigr{\|}_{p}\leq C\|x\|_{p},\quad x\in L_{p}(\mathcal{M}).$$

$$\tau (1-e_J)\le c^p{\lambda }^{-p},\Vert e_J{x}_i{e}_J{\Vert }_{\infty }\le \lambda ,i\in J,$$

$$\tau (1-e)\le 2c^p{\lambda }^{-p},\Vert e{x}_{i}e{\Vert }_{\infty }\le \lambda ,i\in I.$$

$$\Vert e{x}_{i}e{\Vert }_{\infty }$$

$$\le \Vert u{x}_{i}u{\Vert }_{\infty }=\Vert x_i^{1/2}u{\Vert }_{\infty }^2=\underset{y\in L_1(\mathcal{M})\cap \mathcal{M}}{\sup }\frac{|\tau (x_i^{1/2}uy){|}^2}{\Vert y{\Vert }_1^2}$$

$$=\underset{y\in L_1(\mathcal{M})}{\sup }\underset{J}{\sup }\frac{|\tau (x_i^{1/2}{e}_{J}y){|}^2}{\Vert y{\Vert }_1^2}$$

$$\le \underset{J}{\sup }\Vert x_i^{1/2}{e}_J{\Vert }_{\infty }^2\le \lambda$$

$$e={\chi }_{[\frac{1}{2},1]}(u)\le 2u,1-e\le 2(1-u).$$

$$\alpha :G\to B(E),s\mapsto {\alpha }_s$$

$$\underset{i}{\lim }\frac{m((F_{i}s)△F_i)}{m(F_i)}=0.$$

$$\underset{i}{\lim }\frac{m(F_{i}K)}{m(F_i)}=1.$$

$$m(V^n)\le k{n}^r,n\ge 1,$$

$$A_{i}x=\frac{1}{m(F_i)}{\int }_{F_i}{\alpha }_{s}xdm(s),x\in L_p(\mathcal{M}).$$

Full text

[TABLE]

are of weak type $(1,1)$ and of strong type $(p,p)$ for $1<p<\infty$. Consequently, the sequence $(A_{n}x)_{n\geq 1}$ converges almost uniformly for $x\in L_{p}(\mathcal{M})$ for $1\leq p<\infty$. Also, we establish the noncommutative maximal and individual ergodic theorems associated with more general doubling conditions, and we prove the corresponding results for general actions on one fixed noncommutative $L_{p}$-space which are beyond the class of Dunford–Schwartz operators considered previously by Junge and Xu. As key ingredients, we also obtain the Hardy–Littlewood maximal inequality on metric spaces with doubling measures in the operator-valued setting. After the groundbreaking work of Junge and Xu on the noncommutative Dunford–Schwartz maximal ergodic inequalities, this is the first time that more general maximal inequalities are proved beyond Junge and Xu’s setting. Our approach is based on quantum probabilistic methods as well as random walk theory.

1. Introduction

This article studies maximal inequalities and ergodic theorems for group actions on noncommutative $L_{p}$-spaces. The connection between ergodic theory and von Neumann algebras goes back to the very beginning of the theory of operator algebras. However, the study of individual ergodic theorems in the noncommutative setting only took off with Lance’s pioneering work [Lan76] in 1976. The topic was then extensively investigated in a series of works of Conze, Dang-Ngoc, Kümmerer, Yeadon, and others (see [CDN78, Küm78, Yea77, Jaj85] and references therein). Among them, Yeadon [Yea77] obtained a maximal ergodic theorem in the preduals of semifinite von Neumann algebras. But the corresponding maximal inequalities in $L_{p}$-spaces remained open until the celebrated work of Junge and Xu [JX07], which established the noncommutative analogue of the Dunford–Schwartz maximal ergodic theorem. This breakthrough has motivated further research on noncommutative ergodic theorems, such as [AD06, Bek08, HS16, Hu08], and [Lit14]. Note that all these works essentially remain in the class of Dunford–Schwartz operators, that is, do not go beyond Junge and Xu’s setting.

On the other hand, in classical ergodic theory, a number of significant developments related to individual ergodic theorems for group actions have been established in recent years. In particular, Breuillard [Bre14] and Tessera [Tes07] studied the balls in groups of polynomial growth; they proved that for any invariant metric quasi-isometric to a word metric (such as invariant Riemannian metrics on connected nilpotent Lie groups), the balls are asymptotically invariant and satisfy the doubling condition, and hence satisfy the individual ergodic theorem. This settled a long-standing problem in ergodic theory since Calderón’s classical paper [Cal53] in 1953. Also, Lindenstrauss [Lin01] proved the individual ergodic theorem for tempered Følner sequences, which resolves the problem of constructing pointwise ergodic sequences on an arbitrary amenable group. (We refer to [Nev06] for more details.)

Thus, it is natural to extend Junge and Xu’s work to actions of more general amenable groups rather than the integer group. As in the classical case, the first natural step would be to establish the maximal ergodic theorems for doubling conditions. However, since we do not have an appropriate analogue of covering lemmas in the noncommutative setting, no significant progress has been made in this direction. In this paper we provide a new approach to this problem. It is based on both classical and quantum probabilistic methods, and allows us to go beyond the class of Dunford–Schwartz operators considered by Junge and Xu.

Our main results establish the noncommutative maximal and individual ergodic theorems for ball averages under the doubling condition. Let $G$ be a locally compact group equipped with a right Haar measure $m$. Recall that for an invariant metric $d$ on $G$,111In this paper, we always assume that $d$ is a measurable function on $G\times G$ and $m$ is a Radon Borel measure with respect to $(G,d)$. we say that $(G,d)$ satisfies the doubling condition if the balls $B_{r}\coloneqq\{s\in G:d(s,e)\leq r\}$ satisfy

[TABLE]

where $C$ is a constant independent of $r$. We say that the balls are asymptotically invariant under right translation if for every $s\in G$,

[TABLE]

where $\bigtriangleup$ denotes the usual symmetric difference of subsets. To state the noncommutative ergodic theorems, we consider a von Neumann algebra $\mathcal{M}$ equipped with a normal semifinite trace $\tau$. We also consider an action $\alpha$ of $G$ on the associated noncommutative $L_{p}$-spaces $L_{p}(\mathcal{M})$, under some mild assumptions clarified in later sections. In particular, if $\alpha$ is a continuous action of $G$ on $\mathcal{M}$ by $\tau$-preserving automorphisms of $\mathcal{M}$, then $\alpha$ extends to isometric actions on the spaces $L_{p}(\mathcal{M})$. The following is one of our main results.

Theorem 1.1**.**

Assume that $(G,d)$ satisfies (1.1) and (1.2). Let $\alpha$ be a continuous action of $G$ on $\mathcal{M}$ by $\tau$-preserving automorphisms. Let $A_{r}$ be the averaging operators

[TABLE]

Then $(A_{r})_{r>0}$ is of weak type $(1,1)$ and of strong type $(p,p)$ for $1<p<\infty$. Moreover, for all $1\leq p<\infty$, the sequence $(A_{r}x)_{r>0}$ converges almost uniformly for $x\in L_{p}(\mathcal{M})$.

Here we refer to Section 2.1 for the notion of weak and strong type $(p,p)$ inequalities in the noncommutative setting. Also, the notion of almost uniform convergence is a noncommutative analogue of the notion of almost everywhere convergence. We refer to Definition 6.1 for the relevant definitions.

There exist a number of examples satisfying assumptions (1.1) and (1.2) of the above theorem, for which we refer to the previously mentioned works [Bre14, Nev06], and [Tes07]. In particular, if we take $G$ to be the integer group $\mathbb{Z}$ and $d$ to be the usual word metric, then we recover the usual ergodic average $A_{n}=\frac{1}{2n+1}\sum_{k=-n}^{n}T^{k}$ for an invertible operator $T$, as is treated in [JX07]. More generally, we may consider groups of polynomial growth.

Theorem 1.2**.**

Assume that $G$ is generated by a symmetric compact subset $V$ and is of polynomial growth.

1. (1)

Fix $1<p<\infty$. Let $\alpha$ be a strongly continuous and uniformly bounded action of $G$ on $L_{p}(\mathcal{M})$ such that $\alpha_{s}$ is a positive map for each $s\in G$. Then the operators defined by

[TABLE]

are of strong type $(p,p)$. The sequence $(A_{n}x)_{n\geq 1}$ converges bilaterally almost uniformly for $x\in L_{p}(\mathcal{M})$. 2. (2)

Let $\alpha$ be a strongly continuous action of $G$ on $\mathcal{M}$ by $\tau$-preserving automorphisms. Then the operators defined by

[TABLE]

are of weak type $(1,1)$ and of strong type $(p,p)$ for all $1<p<\infty$. The sequence $(A_{n}x)_{n\geq 1}$ converges almost uniformly for $x\in L_{p}(\mathcal{M})$ for all $1\leq p<\infty$.

The theorems rely on several key results obtained in this work. We address the following subjects.

(i) Noncommutative transference principles. Our first key ingredient is a noncommutative variant of Calderón’s transference principle in [Cal68] (see also [CW76, Fen98]), given in Theorems 3.1 and 3.3. More precisely, we prove that for actions by an amenable group, in order to establish the noncommutative maximal ergodic inequalities, it suffices to show the inequalities for translation actions on operator-valued functions. We remark that the particular case of certain actions by $\mathbb{R}$ is also discussed in [Hon17] by the first author.

(ii) Noncommutative Hardy–Littlewood maximal inequalities on metric measure spaces. As for the second key ingredient, we prove in Theorem 4.1 a noncommutative extension of Hardy–Littlewood maximal inequalities on metric measure spaces. For a doubling metric measure space $(X,d,\mu)$, denote by $B(x,r)$ the ball with center $x$ and radius $r$ with respect to the metric $d$. Our result asserts that the Hardy–Littlewood averaging operators on the $L_{p}(\mathcal{M})$-valued functions

[TABLE]

satisfy the weak type $(1,1)$ and strong type $(p,p)$ inequalities. We remark that the classical argument via covering lemmas does not seem to fit into this operator-valued setting. Instead, our approach is based on the study of random dyadic systems by Naor and Tao in [NT10] and Hytönen and Kairema in [HK12]. The key idea is to relate the desired inequality to noncommutative martingales, and to use the available results in quantum probability developed in [Cuc71] and [Jun02]. The approach is inspired by Mei’s famous work in [Mei03] and [Mei07], which asserts that the usual continuous BMO space is the intersection of several dyadic BMO spaces.

(iii) Domination by Markov operators. In the study of ergodic theorems for actions by free groups or free abelian groups, it is a key fact that the associated ergodic averages can be dominated by the standard averaging operators of the form $\frac{1}{n}\sum_{k=1}^{n}T^{k}$ for some map $T$ (see [Bru73, NS94]). Also, in [SS83], Stein and Strömberg apply the Markov semigroup with heat kernels to estimate the maximal inequalities on Euclidean spaces with large dimensions. In this paper, we build a similar result for groups of polynomial growth. Our approach is new and the construction follows easily from some typical Markov chains on these groups. More precisely, we show in Proposition 4.8 that for a group $G$ of polynomial growth with a symmetric compact generating subset $V\subset G$, and for an action $\alpha$ of $G$, there exists a constant $c$ such that

[TABLE]

where $T=\frac{1}{m(V)}\int_{V}\alpha_{s}\,dm(s)$. The result will help us to improve the weak type inequalities in Theorem 1.2.

(iv) Individual ergodic theorems for $L_{p}$ representations. In the classical setting, the individual ergodic theorem holds for positive contractions on $L_{p}$-spaces with one fixed $p\in(1,\infty)$ (see [IT64, Akc75]). The results can be also generalized for positive power-bounded operators and more general Lamperti operators (see, e.g., [Kan78, MRDlT88, Tem15]). However, in the noncommutative setting, the individual ergodic theorems on $L_{p}$-spaces were only known for operators which can be extended to $L_{1}+L_{\infty}$. In Section 6 we will develop some new methods to prove the individual ergodic theorems for operators on one fixed $L_{p}$-space.

Apart from the above approach, we also provide in Section 5 an alternative proof of Theorem 1.1 for discrete groups of polynomial growth. Compared to the previous approach, this proof is much more group-theoretical and has its own interests. It relies essentially on the concrete structure of groups of polynomial growth discovered by Bass, Gromov, and Wolf.

We remark that although our results are stated in the setting of tracial $L_{p}$-spaces, a large number of the results can be extended to the general nontracial case without difficulty. Since the standard methods for these generalizations are already well developed in [HJX10] and [JX07], we leave the details to the reader and restrict our attention to the semifinite case for simplicity of exposition.

We end this Introduction with a brief description of the organization of the paper. In the next section, we recall some basics on noncommutative maximal operators as well as actions by amenable groups. Section 3 is devoted to the proof of the noncommutative variant of Calderón’s transference principle. In Section 4, we prove the Hardy–Littlewood maximal inequalities mentioned above and deduce the maximal inequalities in Theorem 1.1. We also use similar ideas to establish the ergodic theorems for increasing sequences of compact subgroups (Theorem 4.7). In the last part of the section, we will provide an approach based on the random walk theory, which relates the ball averages to the classical ergodic averages of Markov operators. In Section 5, we provide an alternative group-theoretical approach to Theorem 1.2. In Section 6, we discuss the individual ergodic theorems, which prove the bilateral almost uniform convergences in Theorem 1.1. Also, we give new results on almost uniform convergences associated with actions on one fixed $L_{p}$-space.

2. Preliminaries

2.1. Noncommutative $L_{p}$-spaces and noncommutative maximal norms

Throughout the paper, unless explicitly stated otherwise, $\mathcal{M}$ will always denote a semifinite von Neumann algebra equipped with a normal semifinite trace $\tau$. Let $\mathcal{S}_{+}$ denote the set of all $x\in{\mathcal{M}}_{+}$ such that $\tau(\operatorname{supp}x)<\infty$, where $\operatorname{supp}x$ denotes the support of $x$. Let ${\mathcal{S}}$ be the linear span of ${\mathcal{S}}_{+}$. Given $1\leq p<\infty$, we define

[TABLE]

where $|x|=(x^{*}x)^{1/2}$ is the modulus of $x$. Then $({\mathcal{S}},\|\cdot\|_{p})$ is a normed space, whose completion is the noncommutative $L_{p}$-space associated with $({\mathcal{M}},\tau)$, denoted by $L_{p}(\mathcal{M})$. As usual, we set $L_{\infty}(\mathcal{M})={\mathcal{M}}$ equipped with the operator norm. Let $L_{0}(\mathcal{M})$ denote the space of all closed densely defined operators on $H$ measurable with respect to $(\mathcal{M},\tau)$ ($H$ being the Hilbert space on which $\mathcal{M}$ acts). Then $L_{p}(\mathcal{M})$ can be viewed as closed densely defined operators on $H$. We denote by $L_{0}^{+}(\mathcal{M})$ the positive part of $L_{0}(\mathcal{M})$, and set $L_{p}^{+}(\mathcal{M})=L_{0}^{+}(\mathcal{M})\cap L_{p}(\mathcal{M})$. We refer to [PX03] for more information on noncommutative $L_{p}$-spaces.

For a $\sigma$-finite measure space $(X,\Sigma,\mu)$, we consider the von Neumann algebraic tensor product $L_{\infty}(X)\bar{\otimes}\mathcal{M}$ equipped with the trace $\int\otimes\tau$, where $\int$ denotes the integral against $\mu$. For $1\leq p<\infty$, the space $L_{p}(L_{\infty}(X)\bar{\otimes}\mathcal{M})$ isometrically coincides with $L_{p}(X;L_{p}(\mathcal{M}))$, the usual $L_{p}$-space of $p$-integrable functions from $X$ to $L_{p}(\mathcal{M})$. In this paper we will not distinguish these two notions unless specified otherwise.

Maximal norms in the noncommutative setting require a specific definition. The subtlety is that $\sup_{n}|x_{n}|$ does not make any sense for a sequence $(x_{n})_{n}$ of arbitrary operators. This difficulty is overcome by considering the spaces $L_{p}(\mathcal{M};\ell_{\infty})$, which are the noncommutative analogues of the usual Bochner spaces $L_{p}(X;\ell_{\infty})$. These vector-valued $L_{p}$-spaces were first introduced by Pisier [Pis98] for injective von Neumann algebras and then extended to general von Neumann algebras by Junge [Jun02]. The descriptions and properties below can be found in [JX07, Section 2]. Given $1\leq p\leq\infty$, $L_{p}(\mathcal{M};\ell_{\infty})$ is defined as the space of all sequences $x=(x_{n})_{n\geq 0}$ in $L_{p}(\mathcal{M})$ which admit a factorization of the following form: there are $a,b\in L_{2p}(\mathcal{M})$ and a bounded sequence $y=(y_{n})\subset L_{\infty}(\mathcal{M})$ such that

[TABLE]

We then define

[TABLE]

where the infimum runs over all factorizations as above. We will adopt the convention that the norm $\|x\|_{L_{p}(\mathcal{M};\ell_{\infty})}$ is denoted by $\|\sup_{n}^{+}x_{n}\|_{p}$. As an intuitive description, we remark that a positive sequence $(x_{n})_{n\geq 0}$ of $L_{p}(\mathcal{M})$ belongs to $L_{p}(\mathcal{M};\ell_{\infty})$ if and only if there exists a positive $a\in L_{p}(\mathcal{M})$ such that $x_{n}\leq a$ for any $n\geq 0$ and in this case,

[TABLE]

Also, we denote by $L_{p}(\mathcal{M};c_{0})$ the closure of finite sequences in $L_{p}(\mathcal{M};\ell_{\infty})$ for $1\leq p<\infty$. On the other hand, we may also define the space $L_{p}(\mathcal{M};\ell_{\infty}^{c})$, which is the space of all sequences $x=(x_{n})_{n\geq 0}$ in $L_{p}(\mathcal{M})$ which admit a factorization of the following form: there are $a\in L_{p}(\mathcal{M})$ and $y=(y_{n})\subset L_{\infty}(\mathcal{M})$ such that

[TABLE]

And we define

[TABLE]

where the infimum runs over all factorizations as above. Similarly, we denote by $L_{p}(\mathcal{M};c_{0}^{c})$ the closure of finite sequences in $L_{p}(\mathcal{M};\ell_{\infty}^{c})$. (We refer to [DJ04] and [Mus03] for more information.)

Indeed, for any index $I$, we can define the spaces $L_{p}(\mathcal{M};\ell_{\infty}(I))$ of families $(x_{i})_{i\in I}$ in $L_{p}(\mathcal{M})$ with similar factorizations as above. We omit the details and we will simply denote the spaces by the same notation $L_{p}(\mathcal{M};\ell_{\infty})$ and $L_{p}(\mathcal{M};c_{0})$ if no confusion can occur.

The following properties will be useful here.

Proposition 2.1**.**

1. (1)

A family $(x_{i})_{i\in I}\subset L_{p}(\mathcal{M})$ belongs to $L_{p}(\mathcal{M};\ell_{\infty})$ if and only if

[TABLE]

and in this case

[TABLE] 2. (2)

Let $1\leq p_{0}<p_{1}\leq\infty$, and let $0<\theta<1$. Then we have isometrically

[TABLE]

where $\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}$. If additionally $p_{0}\geq 2$, then we have isometrically

[TABLE]

where $\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}$.

Based on these notions we can discuss the noncommutative maximal inequalities.

Definition 2.2**.**

Let $1\leq p\leq\infty$, and let $S=(S_{i})_{i\in I}$ be a family of maps from $L_{p}^{+}(\mathcal{M})$ to $L_{0}^{+}(\mathcal{M})$.

1. (1)

For $p<\infty$, we say that $S$ is of weak type $(p,p)$ with constant $C$ if there exists a constant $C>0$ such that, for all $x\in L_{p}^{+}(\mathcal{M})$ and $\lambda>0$, there is a projection $e\in\mathcal{M}$ satisfying

[TABLE] 2. (2)

For $1\leq p\leq\infty$, we say that $S$ is of strong type $(p,p)$ with constant $C$ if there exists a constant $C>0$ such that

[TABLE]

We will also need a reduction below for weak type inequalities.

Lemma 2.3**.**

If for all finite subsets $J\subset I$, $(S_{i})_{i\in J}$ is of weak type $(p,p)$ with constant $C$, then $(S_{i})_{i\in I}$ is of weak type $(p,p)$ with constant $2^{1/p}C$.

Proof.

Indeed, the proof of [Hon17, Lemma 3.2] shows the following general property: if a family $(x_{i})_{i\in I}\subset L_{0}^{+}(\mathcal{M})$ and a constant $c>0$ satisfy that for all $\lambda>0$ and all finite subsets $J\subset I$, there exists a projection $e_{J}\in\mathcal{M}$ with

[TABLE]

then there exists a projection $e\in\mathcal{M}$ with

[TABLE]

To see this, we note that $\mathcal{S}_{+}$ is dense in $L_{0}^{+}(\mathcal{M})$ with respect to the topology of convergence in measure. So we may assume without loss of generality that $(x_{i})_{i\in I}\subset\mathcal{S}_{+}$. Then it suffices to take a w*-accumulation point $u$ of $(e_{J})_{J}$ and set the spectral projection $e=\chi_{[\frac{1}{2},1]}(u)$. In this case we have

[TABLE]

and

[TABLE]

Then we obtain the claimed property. The lemma follows by taking $x_{i}=S_{i}(x)$ and $c=C\|x\|_{p}$. ∎

The following noncommutative Doob inequalities will play a crucial role in our proof.

Lemma 2.4** ([Cuc71, Proposition 6],[Jun02, Theorem 0.2]).**

Let $(\mathcal{M}_{n})_{n\in\mathbb{Z}}$ be an increasing sequence of von Neumann subalgebras of $\mathcal{M}$ such that $\bigcup_{n\in\mathbb{Z}}\mathcal{M}_{n}$ is w-dense in $\mathcal{M}$. Denote by $\mathbb{E}_{n}$ the $\tau$-preserving conditional expectation from $L_{p}(\mathcal{M})$ onto $L_{p}(\mathcal{M}_{n})$. Then $(\mathbb{E}_{n})_{n\in\mathbb{Z}}$ is of weak type $(1,1)$ with a universal constant and is of strong type $(p,p)$ with constants depending only on $p$ for all $1<p<\infty$.*

Note that only monotone sequences of the form $(\mathbb{E}_{n})_{n\geq 0}$ are concerned in the statement of [Cuc71] and [Jun02]. However, we can easily deduce the general result in Lemma 2.4 from the previous case. Indeed, for a sequence of the form $(\mathbb{E}_{n})_{n\in\mathbb{Z}}$ in the above lemma, the statement in [Cuc71] and [Jun02] yields that for all $k\in\mathbb{Z}$, the sequence $(\mathbb{E}_{n})_{n\geq k}$ is of weak type $(1,1)$ and strong type $(p,p)$ with constants independent of $k$ by reindexing the sequence. In particular, $(\mathbb{E}_{n})_{-k\leq n\leq k}$ satisfies the same maximal inequalities. Then by Proposition 2.1(1) and Lemma 2.3, we see that the same property holds for the sequence $(\mathbb{E}_{n})_{n\in\mathbb{Z}}$ as well.

2.2. Actions by amenable groups

Unless explicitly stated otherwise, throughout $G$ will denote a locally compact group with neutral element $e$, equipped with a fixed right invariant Haar measure $m$. For a Banach space $E$, we say that

[TABLE]

is an action if $\alpha_{s}\circ\alpha_{h}=\alpha_{st}$ for all $s,t\in G$. Let $(\mathcal{M},\tau)$ be as before. For a fixed $1\leq p\leq\infty$, we will be interested in actions $\alpha=(\alpha_{s})_{s\in G}$ on $L_{p}(\mathcal{M})$ with the following conditions:

1. $(\mathbf{A}^{p}_{1})$

Continuity: for all $x\in L_{p}(\mathcal{M})$, the map $s\mapsto\alpha_{s}x$ from $G$ to $L_{p}(\mathcal{M})$ is continuous. Here we take the norm topology on $L_{p}(\mathcal{M})$ if $1\leq p<\infty$ and the w*-topology if $p=\infty$. 2. $(\mathbf{A}^{p}_{2})$

Uniform boundedness: $\sup_{s\in G}\|\alpha_{s}:L_{p}(\mathcal{M})\to L_{p}(\mathcal{M})\|<\infty$. 3. $(\mathbf{A}^{p}_{3})$

Positivity: for all $s\in G$, $\alpha_{s}x\geq 0$ if $x\geq 0$ in $L_{p}(\mathcal{M})$.

As a natural example, if $\alpha$ is an action on $\mathcal{M}$ satisfying the condition:

1. $(\mathbf{A}^{\prime})$

for all $x\in\mathcal{M}$, the map $s\mapsto\alpha_{s}x$ from $G$ to $\mathcal{M}$ is continuous with respect to the w*-topology on $\mathcal{M}$; and for all $s\in G$, $\alpha_{s}$ is an automorphism of $\mathcal{M}$ (in the sense of $*$-algebraic structures) such that $\tau=\tau\circ\alpha_{s}$,

then $\alpha$ extends naturally to actions on $L_{p}(\mathcal{M})$ with conditions $(\mathbf{A}^{p}_{1})$–$(\mathbf{A}^{p}_{3})$ for all $1\leq p\leq\infty$, still denoted by $\alpha$ (see, e.g., [JX07, Lemma 1.1]). In this case for each $s\in G$, $\alpha_{s}$ is an isometry on $L_{p}(\mathcal{M})$. We refer to [Bek15, Oli12], and [Oli13] for other natural examples of group actions on noncommutative $L_{p}$-spaces.

Recall that $G$ is said to be amenable if $G$ admits a Følner net, that is, a net $(F_{i})_{i\in I}$ of measurable subsets of $G$ with $m(F_{i})<\infty$ such that for all $s\in G$,

[TABLE]

Note that the above condition is a reformulation of the asymptotic invariance (1.2) for the general setting. It is known that $(F_{i})_{i\in I}$ is a Følner net if for all compact measurable subsets $K\subset G$,

[TABLE]

Recall that $G$ is a compactly generated group of polynomial growth if the compact generating subset $V\subset G$ satisfies

[TABLE]

where $k>0$ and $r\in\mathbb{N}$ are constants independent of $n$. It is well known that any group of polynomial growth is amenable and the sequence $(V^{n})_{n\geq 1}$ satisfies the above Følner condition (see, e.g., [Bre14, Tes07]). We refer to [Pat88] for more information on amenable groups.

Now let $G$ be amenable, and let $(F_{i})_{i\in I}$ be a Følner net in $G$. Let $1<p<\infty$. Let $\alpha=(\alpha_{s})_{s\in G}$ be an action of $G$ on $L_{p}(\mathcal{M})$ satisfying $(\mathbf{A}^{p}_{1})$–$(\mathbf{A}^{p}_{3})$. Denote by $A_{i}$ the corresponding averaging operators

[TABLE]

According to the mean ergodic theorem for amenable groups (see, e.g., [ADAB*+*10, Théorème 2.2.7]), we have a canonical splitting on $L_{p}(\mathcal{M})$:

[TABLE]

with

[TABLE]

Let $P$ be the bounded positive projection from $L_{p}(\mathcal{M})$ onto $\mathcal{F}_{p}$. Then $(A_{i}x)_{i}$ converges to $Px$ in $L_{p}(\mathcal{M})$ for all $x\in L_{p}(\mathcal{M})$.

Assume additionally that $\alpha$ extends to an action on $\bigcup_{1\leq p\leq\infty}L_{p}(\mathcal{M})$ satisfying $(\mathbf{A}^{p}_{1})$–$(\mathbf{A}^{p}_{3})$ for every $1\leq p\leq\infty$. Note that the convergence in $L_{p}(\mathcal{M})$ yields the convergence in measure in $L_{0}(\mathcal{M})$, and in particular for $x\in L_{1}^{+}(\mathcal{M})\cap\mathcal{M}_{+}$ and for $p_{0}=1$ or $p_{0}=\infty$,

[TABLE]

so by [JX07, Lemma 1.1] and [Yea77, Proposition 1], $P$ admits a continuous extension on $L_{1}(\mathcal{M})$ and $\mathcal{M}$, still denoted by $P$. The splitting (2.4) is also true in this case. Note then, however, that $\mathcal{F}_{\infty}{}^{\perp}$ is the w*-closure of the space spanned by $\{x-\alpha_{s}x:s\in G,x\in\mathcal{M}\}$.

3. Noncommutative Calderón’s transference principle

In this section, we discuss a noncommutative variant of Calderón’s transference principle. Fix $1\leq p<\infty$. Let $G$ be a locally compact group, and let $\alpha$ be an action satisfying $(\mathbf{A}^{p}_{1})$–$(\mathbf{A}^{p}_{3})$ in the previous section. Let $(\mu_{n})_{n\geq 1}$ be a sequence of Radon probability measures on $G$. We consider the following averages:

[TABLE]

Let us also consider the natural translation action of $G$ on itself. We are interested in the following averages: for all $f\in L_{p}(G;L_{p}(\mathcal{M}))$,

[TABLE]

where the integration denotes the usual integration of Banach space-valued functions.

3.1. Strong type inequalities

We begin with the transference principle for strong type $(p,p)$ inequalities.

Theorem 3.1**.**

Assume that $G$ is amenable. Fix $1\leq p<\infty$. If there exists a constant $C>0$ such that

[TABLE]

then there exists a constant $C^{\prime}>0$ depending on $\alpha$ such that

[TABLE]

Proof.

Note that we may take an increasing net of compact subsets $K_{i}\subset G$ such that $\lim_{i}\mu_{n}(K_{i})=\mu_{n}(G)$. Then for

[TABLE]

we have

[TABLE]

So for $x,y\in L_{p}^{+}(\mathcal{M})$,

[TABLE]

Hence $\|\sup_{n,i}^{+}A_{n,i}x_{n,i}\|_{p}=\|\sup_{n}^{+}A_{n}x_{n}\|_{p}$. So without loss of generality we may assume that the $\mu_{n}$’s are of compact support.

We fix $x\in L_{p}(\mathcal{M})$ and $N\geq 1$. Choose a compact subset $K\subset G$ such that $\mu_{n}$ is supported in $K$ for all $1\leq n\leq N$. Since $\alpha_{s}:L_{p}(\mathcal{M})\to L_{p}(\mathcal{M})$ is positive for all $s\in G$, we see that $(\alpha_{s}\otimes\mathrm{Id})_{s\in G}$ extends to a uniformly bounded family of maps on $L_{p}(\mathcal{M};\ell_{\infty})$ (see, e.g., [HJX10, Proposition 7.3]). So we may choose a constant $C^{\prime}>0$ such that

[TABLE]

Let $F$ be a compact subset. Then we have

[TABLE]

We define a function $f\in L_{p}(G;L_{p}(\mathcal{M}))$ as

[TABLE]

Then for all $s\in F$,

[TABLE]

We consider $(A_{n}^{\prime}f)_{1\leq n\leq N}\in L_{p}(L_{\infty}(G)\bar{\otimes}\mathcal{M};\ell_{\infty})$, and for any $\varepsilon>0$ we take a factorization $A_{n}^{\prime}f=aF_{n}b$ such that $a,b\in L_{2p}(L_{\infty}(G)\bar{\otimes}\mathcal{M})$, $F_{n}\in L_{\infty}(G)\bar{\otimes}\mathcal{M}$, and

[TABLE]

Then we have

[TABLE]

Since $\varepsilon$ is arbitrarily chosen, we obtain

[TABLE]

Thus, together with (3.3), (3.4), and the assumption, we see that

[TABLE]

Since $G$ is amenable, for any $\varepsilon>0$ we may choose the above subset $F$ such that $m(FK)/m(F)\leq 1+\varepsilon$. Therefore, we obtain

[TABLE]

Note that $N$, $\varepsilon$, $x$ are all arbitrarily chosen, so we establish the theorem.∎

Remark 3.2*.*

Applying the same argument, we may obtain several variants of the above theorem.

1. (1)

The sequence of measures $(\mu_{n})_{n\geq 1}$ can be replaced by any family $(\mu_{i})_{i\in I}$ of Radon probability measures for an arbitrary index set $I$. 2. (2)

The positivity of the action $\alpha$ can be replaced by more general assumptions. It suffices to assume that

[TABLE]

If $\mathcal{M}$ is commutative, then this is equivalent to saying that the operators $(\alpha_{s})_{s\in G}$ are regular with uniformly bounded regular norm (see [MN91]). In the noncommutative setting, one may assume that $(\alpha_{s})_{s\in G}$ are uniformly bounded decomposable maps, and we refer to [JR04] and [Pis95] for more details. 3. (3)

One may also state similar properties for transference of linear operators; in this case the assumption on positivity of $\alpha$ can be ignored, and the semigroup actions can be included. We have the following noncommutative analogue of the transference result in [CW76, Theorem 2.4]. Assume that $G$ and $\alpha$ satisfy one of the following conditions:

1. (a)

$G$ is an amenable locally compact group, and $\alpha$ satisfies $(\mathbf{A}^{p}_{1})$ and $(\mathbf{A}^{p}_{2})$; 2. (b)

$G$ is a discrete amenable semigroup or $G=\mathbb{R}_{+}$, $\alpha$ satisfies $(\mathbf{A}^{p}_{1})$, and each $\alpha_{s}$ is an isometry on $L_{p}(\mathcal{M})$ (or more generally, there exist $K_{1},K_{2}>0$ such that for all $s\in G$, we have $K_{1}\|x\|_{p}\leq\|\alpha_{s}x\|_{p}\leq K_{2}\|x\|_{p}$).

Let $\mu$ be a bounded Radon measure on $G$. Define

[TABLE]

and

[TABLE]

Then we have

[TABLE]

3.2. Weak type inequalities

Now we discuss the transference principle for weak type $(p,p)$ inequalities. In this case we will only consider the special case of group actions on von Neumann algebras. We assume that $\alpha$ is given by an action on $\mathcal{M}$ satisfying the condition $(\mathbf{A}^{\prime})$ in Section 2.2.

Theorem 3.3**.**

Assume that $G$ is amenable. Let $(A_{n})_{n\geq 1}$ and $(A_{n}^{\prime})_{n\geq 1}$ be the associated sequences of maps given in (3.1) and (3.2). Fix $1\leq p<\infty$. If the sequence $(A_{n}^{\prime})_{n\geq 1}$ is of weak type $(p,p)$, then $(A_{n})_{n\geq 1}$ is of weak type $(p,p)$ too.

Proof.

As in the last subsection, we may assume without loss of generality that $\mu_{n}$ is of compact support. Assume that the sequence $(A_{n}^{\prime})_{n\geq 1}$ is of weak type $(p,p)$ with constant $C$. By Lemma 2.3, it suffices to show that there exists a constant $C^{\prime}>0$ such that for all $\lambda>0$, $x\in L_{p}^{+}(\mathcal{M})$, $N\geq 1$, there exists a projection $e\in\mathcal{M}$ such that

[TABLE]

We fix $\lambda>0$, $x\in L_{p}(\mathcal{M})$, and $N\geq 1$. Choose a compact subset $K\subset G$ such that $\mu_{n}$ is supported in $K$ for all $1\leq n\leq N$. Let $F$ be a compact subset. We define a function $f\in L_{p}(G;L_{p}(\mathcal{M}))$ as

[TABLE]

Then for all $s\in F$,

[TABLE]

Since the sequence $(A_{n}^{\prime})_{n\geq 1}$ is of weak type $(p,p)$ with constant $C$, we may choose a projection $e\in L_{\infty}(G)\bar{\otimes}\mathcal{M}$ such that

[TABLE]

where we regard $e$ as a measurable operator-valued function on $G$. Therefore, by (3.5) for $s\in G$, we have

[TABLE]

Recall that each $\alpha_{s^{-1}}$ is a unital $\tau$-preserving automorphism of $\mathcal{M}$. In particular, for an arbitrary $\varepsilon>0$, we may choose $s_{0}\in G$ and a projection $\tilde{e}\coloneqq\alpha_{s_{0}^{-1}}e(s_{0})\in\mathcal{M}$ such that

[TABLE]

Then we have

[TABLE]

Since $G$ is amenable, for any $\varepsilon>0$ we may choose the above subset $F$ such that $m(FK)/m(F)\leq 1+\varepsilon$. Therefore, we obtain

[TABLE]

Note that $N$, $\varepsilon$, $x$ are all arbitrarily chosen, so we establish the theorem.∎

Remark 3.4*.*

We remark that the above result also holds for other index sets. For example, $(\mu_{n})_{n\geq 1}$ can be replaced by a one-parameter family $(\mu_{r})_{r\in\mathbb{R}_{+}}$ such that $r\mapsto A_{r}x$ is continuous for all $x\in L_{p}(\mathcal{M})$. The only ingredient needed in the proof is that the condition (3.6) hold true almost everywhere on $G$.

4. Maximal inequalities: Probabilistic approach

This section is devoted to the proof of the maximal inequalities in Theorem 1.1. To this end, we will first establish noncommutative Hardy–Littlewood maximal inequalities on doubling metric spaces.

4.1. Hardy–Littlewood maximal inequalities on metric measure spaces

Throughout, a metric measure space $(X,d,\mu)$ refers to a metric space $(X,d)$ equipped with a Radon measure $\mu$. We denote $B(x,r)=\{y\in X:d(y,x)\leq r\}$, and we say that $\mu$ satisfies the doubling condition if there exists a constant $K>0$ such that

[TABLE]

In the sequel, we always assume the nondegeneracy property $0<\mu(B(x,r))<\infty$ for all $r>0$. The following theorem can be regarded as an operator-valued analogue of the Hardy–Littlewood maximal inequalities.

Theorem 4.1**.**

Let $(X,d,\mu)$ be a metric measure space. Suppose that $\mu$ satisfies the doubling condition. Let $1\leq p<\infty$, and let $A_{r}$ be the averaging operators

[TABLE]

Then $(A_{r})_{r\in\mathbb{R}_{+}}$ is of weak type $(1,1)$ and of strong type $(p,p)$ for $1<p<\infty$.

The key ingredient of the proof is the following construction of dyadic systems of metric measure spaces, which is established in [HK12, Corollary 7.4].

Lemma 4.2**.**

Let $(X,d)$ be a metric space, and let $\mu$ be a Radon measure on $X$ satisfying the doubling condition. Then there exists a finite collection of families $\mathscr{P}^{1},\mathscr{P}^{2},\ldots,\mathscr{P}^{N}$, where each $\mathscr{P}^{i}\coloneqq(\mathscr{P}_{k}^{i})_{k\in\mathbb{Z}}$ is a sequence of partitions of $X$, such that the following conditions hold true:

1. (1)

for each $1\leq i\leq N$ and for each $k\in\mathbb{Z}$, the partition $\mathscr{P}_{k+1}^{i}$ is a refinement of the partition $\mathscr{P}_{k}^{i}$; 2. (2)

there exists a constant $C>0$ such that for all $x\in X$ and $r>0$, there exist $1\leq i\leq N$, $k\in\mathbb{Z}$ and an element $Q\in\mathscr{P}_{k}^{i}$ such that

[TABLE]

Remark 4.3*.*

The lemma dates back to the construction of dyadic systems in the case of $X=\mathbb{R}^{d}$, which is due to Mei [Mei03, Mei07] (see also [Pis16]). We remark that Mei’s construction also works for the discrete space $\mathbb{Z}^{d}$ as follows. For $0\leq i\leq d$ and $k\geq 0$, we set $\tilde{\mathscr{P}}_{-k}^{i}$ to be the following family of intervals in $\mathbb{Z}$:

[TABLE]

where $\alpha_{k}^{(i)}=\sum_{j=0}^{k-1}2^{j}\xi_{j}^{(i)}$ modulo $2^{k}$, with

[TABLE]

And we set $\tilde{\mathscr{P}}_{k}^{i}=\tilde{\mathscr{P}}_{0}^{i}$ for all $k\geq 0$. Consider the usual word metric $d$ and the counting measure $\mu$ on $X=\mathbb{Z}^{d}$. Then the partitions

[TABLE]

satisfy the conditions in Lemma 4.2, with constant $C\leq 2^{3d(d+2)}$.

Proof of Theorem 4.1.

Let $\Sigma$ be the $\sigma$-algebra of Borel sets on $X$. For $1\leq i\leq N$ and $k\in\mathbb{Z}$, we define $\Sigma_{k}^{i}\subset\Sigma$ to be the $\sigma$-subalgebra generated by the elements of $\mathscr{P}_{k}^{i}$. Denote by $\mathbb{E}_{k}^{i}$ the conditional expectation from $L_{\infty}(X,\Sigma,\mu)\bar{\otimes}\mathcal{M}$ to $L_{\infty}(X,\Sigma_{k}^{i},\mu|_{\Sigma_{k}^{i}})\bar{\otimes}\mathcal{M}$. For each $x\in X$, let $\mathscr{P}_{k}^{i}(x)$ be the unique element of $\mathscr{P}_{k}^{i}$ which contains $x$. Then we have

[TABLE]

By Lemma 2.4, there exists a constant $C>0$ such that for $\lambda>0$ and $f\in L_{1}^{+}(X;L_{1}(\mathcal{M}))$, there exists a projection $e_{i}\in L_{\infty}(X,\Sigma,\mu)\bar{\otimes}\mathcal{M}$ satisfying

[TABLE]

Take $e=\bigwedge_{i}e_{i}$ to be the infimum of $(e_{i})_{i=1}^{N}$, that is, the projection onto $\bigcap_{i}e_{i}H$ ($H$ being the Hilbert space on which $\mathcal{M}$ acts). Note that $(\bigwedge_{i}e_{i})^{\bot}\leq\sum_{i}e_{i}^{\bot}$. Then we have

[TABLE]

By Lemma 4.2, there exists a constant $C^{\prime}>0$ such that for each $x\in X$ and $r>0$, there exist $1\leq i\leq N$ and $k\in\mathbb{Z}$ such that $B(x,r)\subset\mathscr{P}_{k}^{i}(x)$, $\mu(\mathscr{P}_{k}^{i}(x))\leq C^{\prime}\mu(B(x,r))$; in particular,

[TABLE]

Then together with (4.2) we see that

[TABLE]

Therefore, $(A_{r})_{r>0}$ is of weak type $(1,1)$.

On the other hand, for $1<p<\infty$, according to the proof of (4.3), we have for $f\in L_{p}^{+}(X;L_{p}(\mathcal{M}))$,

[TABLE]

Since each $(\mathbb{E}_{k}^{i})_{k\in\mathbb{Z}}$ on the right-hand side is of strong type $(p,p)$ by Lemma 2.4, we see that $(A_{r})_{r\in\mathbb{R}_{+}}$ is of strong type $(p,p)$, as desired. ∎

Remark 4.4*.*

There is another approach to random dyadic systems of metric measure spaces, which is proved by Naor and Tao [NT10, Lemma 3.1]. The construction is motivated by the study of Hardy–Littlewood maximal inequalities on large-dimensional doubling spaces. Their result replaces the families $\mathscr{P}^{1},\mathscr{P}^{2},\ldots,\mathscr{P}^{N}$ in Lemma 4.2 by an infinite random collection $(\mathscr{P}^{\omega})_{\omega\in\Omega}$ for a probability space $(\Omega,P)$, and assumes a positive probability for the coverings of balls. In this case we may find a random family of martingales $\{(\mathbb{E}_{k}^{\omega})_{k\in\mathbb{Z}},\omega\in\Omega\}$ such that for some $k:\mathbb{R}_{+}\to\mathbb{Z}$ and for some fixed constant $c$, we have

[TABLE]

This yields as well the strong type $(p,p)$ inequalities of $(A_{r})_{r\in\mathbb{R}_{+}}$ for $1<p<\infty$. We omit the details.

4.2. Maximal ergodic inequalities

Based on the previous result, we are now ready to deduce the following maximal ergodic theorems. We say that a metric $d$ on $G$ is invariant if $d(e,g)=d(h,hg)$ for all $g,h\in G$. We denote $B_{r}=B(e,r)$ for $r>0$. As before we consider an action $\alpha$ on $L_{p}(\mathcal{M})$ satisfying the conditions $(\mathbf{A}^{p}_{1})$–$(\mathbf{A}^{p}_{3})$ for a fixed $p$ in Section 2.2. The following result establishes the maximal inequalities in Theorem 1.1.

Theorem 4.5**.**

Let $G$ be an amenable locally compact group, and let $d$ be an invariant metric on $G$. Assume that $(G,d)$ satisfies the doubling condition (1.1). Fix $1\leq p<\infty$. Let $A_{r}$ be the averaging operators

[TABLE]

Then $(A_{r})_{r>0}$ is of strong type $(p,p)$ if $1<p<\infty$. If, in addition, $\alpha$ satisfies the condition $(\mathbf{A}^{\prime})$, then $(A_{r})_{r>0}$ is of weak type $(1,1)$.

Proof.

By Theorems 3.1 and 3.3 and the remarks following them, it suffices to prove the maximal inequalities for the averaging operators

[TABLE]

Since the condition (1.1) holds, $G$ must be unimodular (see, e.g., [Cal53]). In other words, the measure $m$ is also invariant under left translation. Note that $sB_{r}=B(s,r)$ by the invariance of $d$. Thus $m(B_{r})=m(sB_{r})=m(B(s,r))$ and

[TABLE]

And by Theorem 4.1, the right-hand side is of weak type $(1,1)$ and of strong type $(p,p)$ for $1<p<\infty$. Thus $(A_{r}^{\prime})_{r>0}$ is of weak type $(1,1)$ and of strong type $(p,p)$ for $1<p<\infty$ as well. The theorem is proved.∎

Example 4.6**.**

The theorem comprises noncommutative variants of classical results due to Wiener [Wie39], Calderón [Cal53], and Nevo [Nev06]. If $G$ is a compactly generated group of polynomial growth, then the theorem applies to a large class of invariant metrics on $G$, such as distance functions derived from an invariant Riemann metric or word metrics. We refer to [Nev06, Sections 4 and 5] for more examples. Here, we list several typical examples satisfying the doubling condition.

1. (1)

Let $G$ be a compactly generated group of polynomial growth, and let $V$ be a symmetric compact generating subset. The word metric defined by

[TABLE]

satisfies (1.1) and (1.2). Note that the integer groups and finitely generated nilpotent groups are of polynomial growth.

1. (i)

The averaging operators

[TABLE]

are of strong type $(p,p)$ if $1<p<\infty$. If, in addition, $\alpha$ satisfies the condition $(\mathbf{A}^{\prime})$, then $(A_{n})_{n\geq 1}$ is of weak type $(1,1)$. This in particular establishes the maximal inequalities in Theorem 1.2. 2. (ii)

Let $T:L_{p}(\mathcal{M})\to L_{p}(\mathcal{M})$ be a positive invertible operator with positive inverse such that $\sup_{n\in\mathbb{Z}}\|T^{n}\|<\infty$. Then

[TABLE]

is of strong type $(p,p)$ if $1<p<\infty$. If $T$ is an automorphism of $\mathcal{M}$ which leaves $\tau$ invariant, then $(A_{n})_{n\geq 1}$ is of weak type $(1,1)$. 3. (2)

Let $G$ be a compactly generated group of polynomial growth, and let $d$ be a metric on $G$. If $d$ is invariant under a cocompact subgroup of $G$ and if $d$ satisfies a weak kind of the existence of geodesics axiom (see [Bre14, Definition 4.1]), then $(G,d)$ satisfies (1.1) and (1.2).

We remark that a natural generalization of the doubling condition (1.1) is given by Tempelman [Tem67] as follows. A sequence $(F_{k})_{k\geq 1}$ of sets of finite measure in $G$ satisfies Tempelman’s regular condition if

[TABLE]

for some $C>0$ independent of $k$. We refer to [Tem92, Chapter 5] for more details. It is unclear for us how to establish the noncommutative maximal inequalities in this setting. In the following, we provide a typical example for which the inequalities hold true.

Theorem 4.7**.**

Let $G$ be an increasing union of compact subgroups $(G_{n})_{n\geq 1}$. Fix $1\leq p<\infty$. Let $(A_{n})_{n\geq 1}$ be the averaging operators

[TABLE]

Then $(A_{n})_{n\geq 1}$ is of strong type $(p,p)$ if $1<p<\infty$. If, in addition, $\alpha$ satisfies the condition $(\mathbf{A}^{\prime})$, then $(A_{n})_{n\geq 1}$ is of weak type $(1,1)$.

Proof.

By Theorems 3.1 and 3.3, it suffices to prove the maximal inequalities for the averaging operators

[TABLE]

Set $\Sigma$ to be the $\sigma$-algebra of Borel sets on $G$. For each $n\geq 1$, we define $\Sigma_{n}\subset\Sigma$ to be the $\sigma$-subalgebra generated by the cosets of $G_{n}$

[TABLE]

We see that $\Sigma_{n+1}\subset\Sigma_{n}$ for all $n\geq 1$. Let $\mathbb{E}_{n}$ be the conditional expectation from $L_{\infty}(G,\Sigma,m)\bar{\otimes}\mathcal{M}$ to $L_{\infty}(G,\Sigma_{n},m|_{\Sigma_{n}})\bar{\otimes}\mathcal{M}$. Then it is easy to see that

[TABLE]

According to Lemma 2.4, we see that $(\mathbb{E}_{n})_{n\geq 1}$ is of weak type $(1,1)$ and of strong type $(p,p)$ for $1<p<\infty$. This yields the desired inequalities. ∎

4.3. A random walk approach

In this subsection, we provide an alternative approach to maximal inequalities for groups of polynomial growth. This approach is based on a Gaussian lower bound of random walks on groups (see [HSC93]). Independent of the previous approaches, in this method we do not need the results on dyadic decompositions of the group, nor do we use the transference principle. The key observation is that we may relate the ball averages on groups with the ergodic averages of a Markov operator.

Proposition 4.8**.**

Let $G$ be a locally compact group of polynomial growth, and let $V$ be a compact generating set. Let $\alpha$ be a strongly continuous action of $G$ on an ordered Banach space $E$ such that $\alpha_{s}x\geq 0$ for all $s\in G$ and $x\in E_{+}$. Define an operator $T$ on $E$ by

[TABLE]

Then there exists a constant $c$ depending only on $G$ such that

[TABLE]

We remark that for actions by abelian semigroups, it is known by Brunel [Bru73] that the ergodic averages of multioperators can be related to averages of some Markov operators; Nevo and Stein [NS94] showed that similar observations hold for spherical averages of free group actions. These results play an essential role in the proof of ergodic theorems therein. In the case of groups of polynomial growth, our construction of Markov operators is different from theirs, which is inspired by [SS83]. The argument is relatively easy and based on the Markov chains on groups of polynomial growth.

To prove the proposition, we consider a locally compact group $G$ and a measure $\mu$ on $G$. For an integer $k$, we denote by $\mu^{\star k}$ the $k$th convolution of $\mu$, that is, the unique measure $\nu$ on $G$ satisfying

[TABLE]

If $f$ is the density function of $\mu$, then we still denote by $f^{\star k}$ the density function of $\mu^{\star k}$.

In the following $d$ will denote the word metric with respect to $V$ introduced in Example 4.6(1), and $B_{r}=\{x\in G:d(e,x)\leq r\}$ for $r>0$.

Lemma 4.9** ([HSC93, Theorem 5.1]).**

Let $G$ be a locally compact group of polynomial growth, and let $V$ be a compact generating set. Let $f$ be the density function of a symmetric continuous probability measure on $G$ such that $\operatorname{supp}(f)$ is bounded and $V\subset\operatorname{supp}(f)$. Then there exists a constant $c>0$ such that for any integer $k$,

[TABLE]

Lemma 4.10**.**

Let $G$, $V$, and $f$ be as in the previous lemma. Then there exists a constant $c>0$ such that for any integer $n$,

[TABLE]

Proof.

It suffices to prove the inequality in the lemma for sufficiently large $n$. By the previous lemma, there exists $c>0$ such that

[TABLE]

Therefore, for $s\in B_{n}(e)$,

[TABLE]

where $c^{\prime}>0$ is a constant depending only on the doubling condition of $G$. ∎

Proof of Proposition 4.8.

We apply Lemma 4.10 with $f=\chi_{V}/m(V)$. Then we obtain for $x\in E_{+}$,

[TABLE]

By the definition of $f^{\star k}$, we have

[TABLE]

Recall that $\alpha$ is a group action and $f=\chi_{V}/m(V)$, so we obtain

[TABLE]

We have therefore established the desired inequality. ∎

Proposition 4.8 allows us to deduce maximal ergodic theorems for group actions from well-known ergodic theorems for Markov operators. As a corollary, we may obtain the maximal inequalities in Example 4.6(1). Moreover, for maximal ergodic inequalities on $L_{1}$, it is not necessary to assume that $\alpha$ is an action by automorphisms as in $(\mathbf{A}^{\prime})$.

Corollary 4.11**.**

Let $G$ and $V$ be as above. Let $\alpha$ be a continuous $\tau$-preserving action of $G$ on $\mathcal{M}$ such that $\alpha_{s}$ is a positive isometry on $\mathcal{M}$ for each $s\in G$. Then $\alpha$ extends to an action on $L_{1}(\mathcal{M})$. The operators defined by

[TABLE]

are of weak type $(1,1)$.

Proof.

Note that the operator

[TABLE]

is a positive contraction on $\mathcal{M}$, which preserves $\tau$. Then it is well known that the averages $\frac{1}{n}\sum_{k=1}^{n}T^{k}$ are of weak type $(1,1)$ (see [Yea77]). Thus, by Proposition 4.8, $(A_{n})_{n\geq 0}$ is of weak type $(1,1)$ as well. ∎

5. Maximal inequalities: Group-theoretic approach

In this section we provide an alternative approach to Theorem 4.5 in the case where $G$ is a finitely generated discrete group of polynomial growth, $d$ is the word metric, and $p\neq 1$. The argument follows from a structural study of nilpotent groups.

We first recall some well-known facts on the structure of nilpotent groups. Let $G$ be a discrete finitely generated nilpotent group with lower central series

[TABLE]

Each quotient group $G_{i}/G_{i+1}$ is an abelian group of rank $r_{i}$, that is, there is a group isomorphism

[TABLE]

with a finite abelian group $F_{i}$. It was shown in [Bas72] that $G$ is of polynomial growth. We summarize below some facts in the argument of [Bas72]. We may choose a finite generating set $T$ of $G$ such that

[TABLE]

and take

[TABLE]

Then

[TABLE]

For each $1\leq j\leq K$, we order the elements in $T_{j}\setminus T_{j+1}$ as

[TABLE]

so that $\pi_{j}([t_{1}^{(j)}]),\ldots,\pi_{j}([t_{r_{j}}^{(j)}])$ are the generators of $\mathbb{Z}^{r_{j}}$. Let $N_{j}$ be the index of the subgroup $\langle t_{1}^{(j)},t_{2}^{(j)},\ldots,t_{r_{j}}^{(j)}\rangle G_{j+1}$ in $G_{j}$.

By a word in a subset $T^{\prime}\subset T$ we mean a sequence of elements $w=(s_{1},s_{2},\ldots,\allowbreak s_{n})$ with $s_{1},s_{2},\ldots,s_{n}\in T^{\prime}$, and we denote by $|w|=s_{1}s_{2}\cdots s_{n}\in G$ the resulting group element in $G$. If $w=(s_{1},s_{2},\ldots,s_{n})$ is a word in $T^{\prime}$ and if $T^{\prime\prime}\subset T^{\prime}$, then we let $\deg_{T^{\prime\prime}}(w)$ be the cardinality of $\{k:1\leq k\leq n,s_{k}\in T^{\prime\prime}\}$. We say that

[TABLE]

if we have $\deg_{T_{j}\setminus T_{i+1}}(w)\leq d_{i}$ for all $j\leq i\leq K$. Denote by $G_{j}(d)$ the set of all words $w$ in $T_{j}$ such that $\deg^{(j)}(w)\leq d$, and denote by $G_{j}^{\prime}(d)$ the subset of the words $w\in G_{j}(d)$ of the form

[TABLE]

where $v$ is a word in $T_{j+1}$ and the element $t_{k}^{(j)}$ does not appear more than $N_{j}$ times in $w$ for $r_{j}<k\leq l_{j}$.

The key observation in [Bas72] for proving the polynomial growth of $G$ is as follows (see the assertions (6) and (7) in [Bas72, p. 613]).

Lemma 5.1**.**

Let $c>0$ be a constant, and let $m\geq 1$. For each $1\leq j\leq K$, we have

[TABLE]

where $c^{\prime}>0$ is a constant depending only on $c$ and $G_{j}$.

In particular, we take $g\in T^{n}$. Hence $g$ corresponds to a word $w$ in $T=T_{1}$ such that

[TABLE]

Using the lemma inductively, we may find another word $w^{\prime}=(w_{1},\ldots,w_{K})$ in $T$ (where each $w_{j}$ in the bracket stands for a subword in $T_{j}\setminus T_{j+1}$) and a constant $c>0$ such that

[TABLE]

In other words, we have the following observation.

Lemma 5.2**.**

Let $n\in\mathbb{N}$. Each element $g\in T^{n}$ can be written in the form

[TABLE]

where there exists a constant $c>0$ such that for $1\leq j\leq K$,

[TABLE]

and

[TABLE]

In [Bas72] and [Wol68] it is proved that $G$ satisfies the following strict polynomial growth condition.

Lemma 5.3**.**

We have two constants $c_{1},c_{2}>0$ such that

[TABLE]

where $|\,|$ denotes the cardinality of a subset and

[TABLE]

Note that the upper bound in the above lemma follows directly from Lemma 5.2.

Now we will prove the following maximal inequalities, which are particular cases studied in Theorem 4.5 and Example 4.6(1).

Proposition 5.4**.**

Let $G$ be a finitely generated discrete group of polynomial growth, and let $S\subset G$ be a finite generating set. Fix $1<p<\infty$, and let $\alpha$ be an action $\alpha=(\alpha_{s})_{s\in G}$ of $G$ on $L_{p}(\mathcal{M})$ which satisfies $(\mathbf{A}^{p}_{1})$–$(\mathbf{A}^{p}_{3})$. We consider the averaging operators

[TABLE]

where $|\,|$ denotes the cardinality of a subset. Then $(A_{n})_{n\geq 1}$ is of strong type $(p,p)$.

The proposition relies on the following characterization of groups of polynomial growth by Gromov [Gro81].

Lemma 5.5**.**

Any finitely generated discrete group of polynomial growth contains a finitely generated nilpotent subgroup of finite index.

We also need the following fact.

Lemma 5.6**.**

Let $G$ be a finitely generated group of polynomial growth. Let $H$ be a normal subgroup of $G$ of finite index. Then $H$ is finitely generated. Let $U\subset G$ be a finite system of representatives of the cosets $G/H$ with $e\in U$. Let $T\subset H$ be a finite generating set of $H$. Write $V=U\cup T$. Then there exists an integer $N$ such that

[TABLE]

Proof.

This is given in the proof of [Wol68, Theorem 3.11]. Let $N$ be an integer large enough such that for all $u_{1}^{\epsilon},u_{2}^{\eta}\in U$ with $\epsilon,\eta\in\{\pm 1\}$, there exist $u\in U$ and $t\in T^{N}$ satisfying $u_{1}^{\epsilon}u_{2}^{\eta}=ut$. Then $N$ satisfies the desired condition. ∎

Now we deduce the desired result.

Proof of Proposition 5.4.

By Theorem 3.1, it suffices to consider the case where $\alpha$ is an action on $L_{p}(G;\break L_{p}(\mathcal{M}))$ by translation. By Lemma 5.5, we may find a nilpotent subgroup $H\subset G$ of finite index. As is explained in [Wol68, Theorem 3.11], $H$ can be taken normal by replacing $H$ with $\bigcap_{s\in G}sHs^{-1}$. Now let $T=\{t_{k}^{(j)}:1\leq k\leq l_{j},1\leq j\leq K\}$ be a finite generating set of the nilpotent group $H$, where $T$ and the indices $k$, $j$ are chosen in the same manner as in (5.1). Also, let $U$ and $V$ be given as in the previous lemma. Consider $x\in L_{p}^{+}(G;L_{p}(\mathcal{M}))$, and write

[TABLE]

Since the operators $\alpha_{s}$ extend to positive operators on $L_{p}(G;L_{p}(\mathcal{M}))$, by Lemma 5.2 and Lemma 5.6, there exists a constant $c>0$ for all $n\in\mathbb{N}$ such that

[TABLE]

Recall that by Lemma 5.3 we may find a constant $c^{\prime}>0$ such that

[TABLE]

So we may find a constant $c^{\prime\prime}>0$ satisfying

[TABLE]

Note that by [JX07, Theorem 4.1], for each $1\leq j\leq K$ and $1\leq k\leq r_{j}$ there exists a constant $C_{p}$ depending only on $p$ such that

[TABLE]

Applying the inequality iteratively, we obtain a constant $C_{p}^{\prime}>0$ such that

[TABLE]

Since $S$ and $V$ are both finite, we may find two integers $k$ and $k^{\prime}$ with

[TABLE]

So the strong type $(p,p)$ inequality for $A_{n}$ follows as well. ∎

6. Individual ergodic theorems

In this section, we apply the maximal inequalities to study the pointwise ergodic convergence in Theorems 1.1 and 1.2.

We will use the following analogue for the noncommutative setting of the usual almost everywhere convergence. The definition is introduced by Lance [Lan76] (see also [Jaj85]).

Definition 6.1**.**

Let $\mathcal{M}$ be a von Neumann algebra equipped with a normal semifinite faithful trace $\tau$. Let $x_{n},x\in L_{0}(\mathcal{M})$. We say that $(x_{n})_{n\geq 1}$ converges bilaterally almost uniformly (b.a.u. for short) to $x$ if for every $\varepsilon>0$ there is a projection $e\in\mathcal{M}$ such that

[TABLE]

and that it converges almost uniformly (a.u. for short) to $x$ if for every $\varepsilon>0$ there is a projection $e\in\mathcal{M}$ such that

[TABLE]

In the case of classical probability spaces, the definition above is equivalent to the usual almost everywhere convergence in terms of Egorov’s theorem.

Now let $G$ be an amenable locally compact group, and let $(F_{n})_{n\geq 1}$ be a Følner sequence in $G$. Let $1\leq p\leq\infty$. Assume that $\alpha=(\alpha_{s})_{s\in G}$ is an action on $L_{p}(\mathcal{M})$ which satisfies $(\mathbf{A}^{p}_{1})$–$(\mathbf{A}^{p}_{3})$. Denote by $A_{n}$ the corresponding averaging operators

[TABLE]

We keep the notation $\mathcal{F}_{p}\subset L_{p}(\mathcal{M})$ and $P$ introduced in Section 2.2.

We first consider the case where $\alpha$ extends to an action on $L_{1}(\mathcal{M})+\mathcal{M}$. In this case we use a standard argument for b.a.u. convergences adapted from [Hon17, JX07], and [Yea77]. The following lemma from [DJ04] will be useful.

Lemma 6.2**.**

Let $1\leq p<\infty$. If $(x_{n})\in L_{p}(\mathcal{M};c_{0})$, then $x_{n}$ converges b.a.u. to [math]. If $(x_{n})\in L_{p}(\mathcal{M};c_{0}^{c})$ with $2\leq p<\infty$, then $x_{n}$ converges a.u. to [math].

We will also use the following noncommutative analogue of the Banach principle given by [Lit17] and [CL16, Theorem 3.1].

Lemma 6.3**.**

Let $1\leq p<\infty$, and let $S=(S_{n})_{n\geq 1}$ be a sequence of additive maps from $L_{p}^{+}(\mathcal{M})$ to $L_{0}^{+}(\mathcal{M})$. Assume that $S$ is of weak type $(p,p)$. Then the set

[TABLE]

is closed in $L_{p}(\mathcal{M})$.

Proposition 6.4**.**

Assume that $\alpha=(\alpha_{s})_{s\in G}$ is an action well defined on $\bigcup_{1\leq p\leq\infty}L_{p}(\mathcal{M})$ which satisfies $(\mathbf{A}^{p}_{1})$–$(\mathbf{A}^{p}_{3})$ for every $1\leq p\leq\infty$. Let $(A_{n})_{n\geq 1}$ be as above, and let $1\leq p_{0}<p_{1}\leq\infty$. Assume that $(A_{n})_{n\geq 1}$ is of strong type $(p,p)$ for all $p_{0}<p<p_{1}$.

1. (1)

For all $x\in L_{p}(\mathcal{M})$ with $p_{0}<p\leq 2p_{0}$, $(A_{n}x-Px)_{n\geq 1}\in L_{p}(\mathcal{M};c_{0})$, and hence $(A_{n}x)_{n\geq 1}$ converges b.a.u. to $Px$. 2. (2)

For all $x\in L_{p}(\mathcal{M})$ with $2p_{0}<p<p_{1}$, $(A_{n}x-Px)_{n\geq 1}\in L_{p}(\mathcal{M};c_{0}^{c})$, and hence $(A_{n}x)_{n\geq 1}$ converges a.u. to $Px$.

Proof.

According to the splitting (2.4) and the discussion after it, we know that

[TABLE]

is dense in $\overline{(\mathrm{Id}-P)(L_{p}(\mathcal{M}))}$ for all $1\leq p<\infty$. Also, observe that for all $x\in S$,

[TABLE]

To see this, take an arbitrary $x\in S$ of the form $x=y-\alpha_{s_{0}}y$ for some $s_{0}\in G$ and $y\in L_{1}(\mathcal{M})\cap\mathcal{M}$. Then

[TABLE]

Therefore, according to $(\mathbf{A}_{2}^{\infty})$,

[TABLE]

which converges to [math] as $n\to\infty$ according to the Følner condition. This therefore yields the a.u. convergence of $(A_{n}x)_{n}$ in (6.1), as desired.

Now we prove assertion (1). Take $x\in L_{p}(\mathcal{M})$. Since $S$ is dense in$\overline{(\mathrm{Id}-P)(L_{p}(\mathcal{M}))}$, there are $x_{k}\in S$ such that

[TABLE]

Since $(A_{n})_{n\geq 1}$ is of strong type $(p,p)$, there exists a constant $C>0$ independent of $x$ such that

[TABLE]

Thus,

[TABLE]

Since $L_{p}(\mathcal{M};c_{0})$ is closed in $L_{p}(\mathcal{M};\ell_{\infty})$, it suffices to show that $(A_{n}x_{k})_{n}\in L_{p}(\mathcal{M};c_{0})$ for all $k$. To this end, we take an arbitrary $z\in S$ of the form $z=y-\alpha_{s_{0}}y$ for some $s_{0}\in G$ and $y\in L_{1}(\mathcal{M})\cap\mathcal{M}$. Take some $p_{0}<q<p$. Note that $z\in L_{q}(\mathcal{M})$ and that $(A_{n})_{n}$ is of strong type $(q,q)$ by assumption, so $(A_{n}(z))_{n}$ belongs to $L_{q}(\mathcal{M};\ell_{\infty})$. Then by (2.1) and (6.2), for any $m<n$,

[TABLE]

Thus, $\|\mathop{{\sup}^{+}}_{j\geq m}A_{j}z\|_{p}$ tends to [math] as $m\to\infty$. Therefore, the finite sequence $(A_{1}z,\ldots,A_{m}z,0,\ldots)$ converges to $(A_{n}z)_{n}$ in $L_{p}(\mathcal{M};\ell_{\infty})$ as $m\to\infty$. As a result $(A_{n}(z))_{n}\in L_{p}(\mathcal{M};c_{0})$, as desired.

Assertion (2) is similar. It suffices to note that by the classical Kadison inequality in [Kad52],

[TABLE]

and hence by the strong type $(p,p)$ inequality and the definition of $L_{p}(\mathcal{M};\ell_{\infty}^{c})$, there exists a constant $C$ such that

[TABLE]

Then a similar argument yields that $(A_{n}x-Px)_{n\geq 1}\in L_{p}(\mathcal{M};c_{0}^{c})$. ∎

Remark 6.5*.*

The above argument certainly works as well for a Følner sequence $(F_{r})_{r>0}$ indexed by $r\in\mathbb{R}_{+}$, provided that $r\mapsto A_{r}x$ is continuous for $x\in L_{p}(\mathcal{M})$.

As a corollary, we obtain the individual ergodic theorems for actions on $L_{1}(\mathcal{M})+\mathcal{M}$. We complete the proof of Theorem 1.1.

Theorem 6.6**.**

Let $d$ be an invariant metric on $G$. Assume that $(G,d)$ satisfies (1.1) and (1.2). Let $\alpha$ be an action of $G$ well defined on $\bigcup_{1\leq p\leq\infty}L_{p}(\mathcal{M})$ which satisfies $(\mathbf{A}^{p}_{1})$–$(\mathbf{A}^{p}_{3})$ for every $1\leq p\leq\infty$. Denote

[TABLE]

Then $(A_{r}x)_{r>0}$ converges a.u. to $Px$ as $r\to\infty$ for all $1<p<\infty$.

Moreover, if $\alpha$ is a continuous action of $G$ on $\mathcal{M}$ by $\tau$-preserving automorphisms (i.e., $\alpha$ satisfies $(\mathbf{A}^{\prime})$), then $(A_{r}x)_{r>0}$ converges a.u. to $Px$ as $r\to\infty$ for all $x\in L_{1}(\mathcal{M})$.

Proof.

Note that for $1\leq p\leq 2$ and $2<p^{\prime}<\infty$, $L_{p}(\mathcal{M})\cap L_{p^{\prime}}(\mathcal{M})$ is dense in $L_{p}(\mathcal{M})$, and $(A_{r}x)_{r>0}$ converges a.u. to $Px$ for all $x\in L_{p^{\prime}}(\mathcal{M})$ according to Proposition 6.4. Then the theorem is an immediate consequence of Lemma 6.3 and Theorem 4.5. ∎

For word metrics on groups of polynomial growth, it is well known that the associated balls satisfy the Følner condition (see [Bre14, Tes07]). Together with Corollary 4.11 we obtain the following result. This also proves the a.u. convergence on $L_{1}(\mathcal{M})$ stated in Theorem 1.2.

Theorem 6.7**.**

Assume that $G$ is of polynomial growth, and is generated by a symmetric compact subset $V$. Let $\alpha$ be an action of $G$ well defined on $\bigcup_{1\leq p\leq\infty}L_{p}(\mathcal{M})$ which satisfies $(\mathbf{A}^{p}_{1})$–$(\mathbf{A}^{p}_{3})$ for every $1\leq p\leq\infty$. Denote

[TABLE]

Then $(A_{n}x)$ converges a.u. to $Px$ for all $1<p<\infty$.

Moreover, if $\alpha$ is a continuous $\tau$-preserving action of $G$ on $\mathcal{M}$ such that $\alpha_{s}$ is a positive isometry on $\mathcal{M}$ for each $s\in G$, then $(A_{n}x)$ converges a.u. to $Px$ for all $x\in L_{1}(\mathcal{M})$.

Also, it is obvious that an increasing sequence of compact subgroups always satisfies the Følner condition. Together with Theorem 4.7 we obtain the following.

Theorem 6.8**.**

Let $G$ be an increasing union of compact subgroups $(G_{n})_{n\geq 1}$. Let $\alpha$ be an action of $G$ well defined on $\bigcup_{1\leq p\leq\infty}L_{p}(\mathcal{M})$ which satisfies $(\mathbf{A}^{p}_{1})$–$(\mathbf{A}^{p}_{3})$ for every $1\leq p\leq\infty$. Denote

[TABLE]

Then $(A_{n}x)$ converges a.u. to $Px$ for all $1<p<\infty$.

Moreover, if $\alpha$ is a continuous action of $G$ on $\mathcal{M}$ by $\tau$-preserving automorphisms (i.e., $\alpha$ satisfies $(\mathbf{A}^{\prime})$), then $(A_{n}x)$ converges a.u. to $Px$ for all $x\in L_{1}(\mathcal{M})$.

Note that all the above arguments rely on the assumption that the action $\alpha$ extends to a uniformly bounded action on $L_{\infty}(\mathcal{M})$ with conditions $(\mathbf{A}_{1}^{\infty})$–$(\mathbf{A}_{3}^{\infty})$, though our strong type $(p,p)$ inequalities in previous sections do not require this assumption. Also, in general this assumption does not hold for bounded representations on one fixed $L_{p}$-space. In Theorem 6.11 we will give a stronger result for Følner sequences associated with doubling conditions. This also completes the proof of Theorem 1.2.

Lemma 6.9**.**

Let $(X,d,\mu)$ be a metric measure space satisfying the doubling condition (4.1). Take $i\in\mathbb{N}$ and $k\leq 2^{i}$. Then there exists $2^{i}\leq r_{i}<2^{i+1}$ such that

[TABLE]

where $C$ depends only on the doubling constant.

Proof.

The result and the argument are adapted from [Tes07, Proposition 17]. For each $r\in\mathbb{N}$, we denote

[TABLE]

Then

[TABLE]

Therefore,

[TABLE]

Thus, the lemma follows thanks to the doubling condition (4.1). ∎

Lemma 6.10** ([Bre14, Theorem 1.1], [Tes07, Corollary 10]).**

Let $G$ be a locally compact group of polynomial growth, generated by a symmetric compact subset $V$. Then

[TABLE]

where $d(G)$ is the rank of $G$ and $c$ is a constant depending on $V$. And there exist $\delta>0$ and a constant $C$ such that

[TABLE]

Theorem 6.11**.**

Fix $1<p<\infty$. Let $\alpha=(\alpha_{s})_{s\in G}$ be an action on $L_{p}(\mathcal{M})$ which satisfies $(\mathbf{A}^{p}_{1})$–$(\mathbf{A}^{p}_{3})$.

1. (1)

Assume that there exists an invariant metric $d$ on $G$ and that $(G,d)$ satisfies (1.1) and (1.2). Denote

[TABLE]

Then there exists a lacunary sequence $(r_{k})_{k\geq 1}$ with $2^{k}\leq r_{k}<2^{k+1}$ such that $(A_{r_{k}}x)_{k\geq 1}$ converges b.a.u. to $Px$ for all $x\in L_{p}(\mathcal{M})$. If additionally $p\geq 2$, then $(A_{r_{k}}x)_{k\geq 1}$ converges a.u. to $Px$ for all $x\in L_{p}(\mathcal{M})$. 2. (2)

Assume that $G$ is a locally compact group of polynomial growth, generated by a symmetric compact subset $V$. Then the sequence

[TABLE]

converges b.a.u. to $Px$ for all $x\in L_{p}(\mathcal{M})$. 3. (3)

Assume that $G$ is an increasing union of compact subgroups $(G_{n})_{n\geq 1}$. Then the sequence

[TABLE]

converges a.u. to $Px$ for all $x\in L_{p}(\mathcal{M})$.

Proof.

(1) By Lemma 6.9, there exists $(r_{i})_{i\geq 1}$ such that $2^{i}\leq r_{i}\leq 2^{i+1}$ and such that

[TABLE]

That is to say,

[TABLE]

We show that $(A_{r_{i}}x)_{i\geq 1}$ converges b.a.u. to $Px$. By (2.4) and Lemma 6.2, it suffices to show that $(A_{r_{i}}x)_{i\geq 1}\in L_{p}(\mathcal{M};c_{0})$ for $x\in\mathcal{F}_{p}^{\bot}$. By Theorem 4.5, it is enough to consider the case where $x=y-\alpha_{s_{0}}y$ with $y\in L_{p}^{+}(\mathcal{M})$ and $s_{0}\in G$. Indeed, if $(A_{r_{i}}x)_{i\geq 1}\in L_{p}(\mathcal{M};c_{0})$ for $x$’s of the aforementioned form $y-\alpha_{s_{0}}y$, then the same holds for all $x\in\operatorname{span}\{x-\alpha_{s}x:s\in G,x\in L_{p}(\mathcal{M})\}$. Now by the definition of $\mathcal{F}_{p}^{\bot}$, for any $\varepsilon>0$ and $x\in\mathcal{F}_{p}^{\bot}$, we may find an element $y\in\operatorname{span}\{x-\alpha_{s}x:s\in G,x\in L_{p}(\mathcal{M})\}$ with $\|x-y\|_{p}<\varepsilon$. So, by Theorem 4.5, we see that

[TABLE]

with a constant $C$ independent of $x$ and $y$. Thus, $(A_{r_{i}}x)$ belongs to $L_{p}(\mathcal{M};c_{0})$ since $L_{p}(\mathcal{M};c_{0})$ is a Banach space. In the following we prove the claim $(A_{r_{i}}x)_{i\geq 1}\in L_{p}(\mathcal{M};c_{0})$ with $x=y-\alpha_{s_{0}}y$. Denote $|s_{0}|=d(e,s_{0})$. Note that

[TABLE]

where

[TABLE]

By (6.3) we have for $i$ so that $(3/2)^{i}\geq|s_{0}|$,

[TABLE]

On the other hand, for any $m\leq j\leq n$,

[TABLE]

and by the previous argument

[TABLE]

Hence, $\|(A_{r_{j}}^{1}y)_{m\leq j\leq n}\|_{L_{p}(\mathcal{M};c_{0})}$ tends to [math] as $m,n\to\infty$. Similarly, $(A_{r_{i}}^{2}y)_{i\geq 1}$ converges in the same manner. Therefore, $(A_{r_{i}}x)_{i\geq 1}\in L_{p}(\mathcal{M};c_{0})$, as desired.

Moreover, if $p\geq 2$, then

[TABLE]

and hence we can find contractions $u_{j}\in L_{\infty}(\mathcal{M})$ such that for $m$ large enough,

[TABLE]

Therefore, $\|(A_{r_{j}}^{1}y)_{m\leq j\leq n}\|_{L_{p}(\mathcal{M};c_{0}^{c})}$ tends to [math] as $m,n\to\infty$, and $(A_{r_{i}}^{1}y)_{i\geq 1}$ converges a.u. to [math] according to Lemma 6.2. Similarly, $(A_{r_{i}}^{2}y)_{i\geq 1}$ converges in the same manner. Thus, we obtain that $(A_{r_{i}}x)_{i\geq 1}$ converges a.u. to [math]. Then by (2.4) and Lemma 6.3, $(A_{r_{i}}x)_{i\geq 1}$ converges a.u. for all $x\in L_{p}(\mathcal{M})$.

(2) We keep the notation $x$, $y$, $s_{0}$ in (1), and denote as before

[TABLE]

Write $\delta^{\prime}=[\delta^{-1}]+1$. Note that by Lemma 6.10, there exists a constant $C>0$ such that for $y\in L_{p}(\mathcal{M})$ and $k\geq 1$,

[TABLE]

where $|s_{0}|=d(e,s_{0})$ and $d$ refers to the word metric defined in Example 4.6(1). Hence,

[TABLE]

Then, by an argument similar to that in (1), we see that $(A_{n^{\delta^{\prime}}}x)_{n\geq 1}$ converges b.a.u. to $Px$ for all $x\in L_{p}(\mathcal{M})$, and if $p\geq 2$, then $(A_{n^{\delta^{\prime}}}x)_{n\geq 1}$ converges a.u. to $Px$.

For the general case, we consider $x\in L_{p}^{+}(\mathcal{M})$. For each $k$, let $n(k)$ be the number such that $n(k)^{\delta^{\prime}}\leq k<(n(k)+1)^{\delta^{\prime}}$. Then

[TABLE]

Also note that according to Lemma 6.10, $m(V^{n(k)^{\delta^{\prime}}})/m(V^{k})$ tends to $1$. Therefore, it is easy to see from the definition of b.a.u. convergence that $A_{k}x$ converges b.a.u. to $Px$.

(3) Note that for $x=y-\alpha_{s_{0}}y$ with $y\in L_{p}(\mathcal{M})$ and $s_{0}\in G$, we have for $n$ large enough so that $G_{n}\ni s_{0}$,

[TABLE]

That is to say, $A_{n}x$ converges a.u. to [math] as $n\to\infty$. Then by (2.4) and Lemma 6.3, we see that $A_{n}x$ converges a.u. for all $x\in L_{p}(\mathcal{M})$. ∎

In particular, the above arguments give the individual ergodic theorem for positive invertible operators on $L_{p}$-spaces.

Corollary 6.12**.**

Let $1<p<\infty$. Let $T:L_{p}(\mathcal{M})\to L_{p}(\mathcal{M})$ be a positive invertible operator with positive inverse such that $\sup_{n\in\mathbb{Z}}\|T^{n}\|<\infty$. Denote

[TABLE]

Then $(A_{n}x)_{n\geq 1}$ converges b.a.u. to $Px$ for all $x\in L_{p}(\mathcal{M})$. If additionally $p\geq 2$, then $(A_{n}x)_{n\geq 1}$ converges a.u. to $Px$.

Proof.

The assertion follows from the proof of Theorem 6.11(2) for the case where $G$ equals the integer group $\mathbb{Z}$. It suffices to notice that in this case we may choose $\delta=1$ in Lemma 6.10 and take $\delta^{\prime}=1$ in (6.4). ∎

Remark 6.13*.*

Note that the above result is not true for $p=1$, even for positive invertible isometries on classical $L_{1}$-spaces (see, e.g., [IT64]). So it is natural to assume that $p\neq 1$ in the above discussions.

The following conjecture for mean bounded maps is still open. The result for classical $L_{p}$-spaces is given by [MRDlT88].

Conjecture 6.14**.**

Let $1<p<\infty$. Let $T:L_{p}(\mathcal{M})\to L_{p}(\mathcal{M})$ be a positive invertible operator with positive inverse such that $\sup_{n\in\mathbb{Z}}\|\frac{1}{2n+1}\sum_{k=-n}^{n}T^{k}\|<\infty$. Denote

[TABLE]

Then $(A_{n}x)_{n\geq 1}$ converges b.a.u. to $Px$ for all $x\in L_{p}(\mathcal{M})$. If additionally $p\geq 2$, then $(A_{n}x)_{n\geq 1}$ converges a.u. to $Px$.

Acknowledgment

The subject of this paper came from a suggestion made several years ago to the authors by Professor Quanhua Xu, whom we thank for many fruitful discussions. We would also like to thank the anonymous referees for their helpful comments and suggestions. Part of the work was done during the Summer Working Seminar on Noncommutative Analysis at Wuhan University (2015) and the Institute for Advanced Study in Mathematics at Harbin Institute of Technology (2016 and 2017).

This work was supported by National Science Foundation (NSF) of China grant 11431011. Hong’s work was partially supported by NSF of China grants 11601396 and 12071355. Liao’s work was partially supported by the National Science Foundation and by Coalition for National Science Funding grant 11420101001. Wang’s work was partially supported by the European Research Council Advanced Grant “Non-Commutative Distributions in Free Probability” No. 339760, Agence Nationale de la Recherche grant ANR-19-CE40-0002, and a public grant as part of the Fondation Mathématique Jacques Hadamard.