Paley’s Inequality for Discrete Groups

Abstract

This article studies Paley’s theory of lacunary Fourier series for von Neumann algebra of discrete groups. The results unify and generalize the work of Rudin (Fourier Analysis on Groups, Reprint of the 1962 original. Wiley Classics Library, A Wiley-Interscience Publication, Wiley, New York, 1990, Section 8) for abelian discrete ordered groups and the work of Lust-Piquard and Pisier (Ark Mat 29(2):241–260, 1991) for operator valued functions, and provide new examples of Paley sets and $\Lambda (p)$ sets on free groups.

1 Introduction

Denote by ${\mathbb {T}}$ the unit circle. Consider a sequence $(j_k)_{ k\in {\mathbb {N}}}$ of elements of ${\mathbb {Z}}$ which is lacunary à la Hadamard, i.e. there exists $\delta > 0$ such that for all $k \in {\mathbb {N}}$,

$$\begin{aligned} \frac{|j_{k+1}|}{|j_k|}>1+\delta . \end{aligned}$$

A classical Khintchine-Pisier type inequality states that there exists $C_{\delta } <\infty$ such that

$$\begin{aligned} \left\| \sum _{k=1}^{\infty } c_kz^{j_k} \right\| _{L^1({\mathbb {T}})} \le \left( \sum _{k=1}^{\infty } |c_k|^2 \right) ^\frac{1}{2} \le C_{\delta } \left\| \sum _{k=1}^{\infty } c_kz^{j_k} \right\| _{L^1({\mathbb {T}})} . \end{aligned}$$

This shows that $\ell _2({\mathbb {N}})$ embeds into $L^1({\mathbb {T}})$. However, the map

$$\begin{aligned} P: f \mapsto \big (\hat{f}(j_k)\big )_{k\in {\mathbb {N}}} \end{aligned}$$

does not extend to a bounded map from the whole space $L^1({\mathbb {T}})$ to $\ell _2$. Here $\hat{f}$ denotes the Fourier transform of f. This can be easily seen by looking at the so-called Riesz products,

$$\begin{aligned} f(z)= \prod _{k=1}^N (1+\frac{z^{2^k}+z^{-2^k}}{2}), \end{aligned}$$

which have norm $\Vert f\Vert _{L^1({\mathbb {T}})}=\hat{f}(0)=1$ while $(\hat{f}(2^k))_{1\le k\le N}$ has norm $\frac{\sqrt{N}}{2}$ since $\hat{f}(2^k)=\frac{1}{2}$ for $k=1,\cdots ,N$. Paley’s theory [33] is a variant of Khintchine’s inequality. Let $H^1({\mathbb {T}})$ be the real Hardy space on the unit circle, that consists of integrable functions such that both their analytic and the anti-analytic parts are integrable. Equivalently,

$$\begin{aligned} H^1({\mathbb {T}})= \left\{ f\in L^1({\mathbb {T}}) : \Vert f\Vert _{H^1}=\Vert f\Vert _{L^1}+\Vert H (f)\Vert _{L^1}<\infty \right\} , \end{aligned}$$

with H the Hilbert transform of f. Paley’s theory says that

$$\begin{aligned} \left( \sum _k |c_k|^2 \right) ^\frac{1}{2}\simeq _{\delta }\inf \left\{ \Vert f\Vert _{H^1}: f\in H^1({\mathbb {T}}), \hat{f}(j_k)=c_k \right\} \end{aligned}$$

(0.1)

This shows that the map P is bounded from $H^1({\mathbb {T}})$ (or the analytic $L^1$) to $\ell _2$. These Khintichine/Paley type inequalities for lacunary series have important applications to Grothendieck’s theory (see [40, Section 5], [26, Appendix]).

A subset $E\subset {\mathbb {N}}$ is called a Paley set (see [37, Section 3]) if the above equivalence (0.1) holds for all choices of $(c_k)_k\in \ell _2, j_k\in E$ with constants depending only on E. Rudin [48, Section 8] proved that E is a Paley set only if

$$\begin{aligned} \sup _{n\in {\mathbb {N}}} \# E\cap [2^n,2^{n+1}]<C \end{aligned}$$

which is equivalent to say that E is a finite union of lacunary sequences.

By Fefferman-Stein’s $H^1$-BMO duality theory, (0.1) has an equivalent formulation that, for any $(c_k) \in \ell _2$,

$$\begin{aligned} \left( \sum _k |c_k|^2 \right) ^\frac{1}{2}\simeq _{\delta } \left\| \sum _k c_kz^{j_k} \right\| _{BMO({\mathbb {T}})}. \end{aligned}$$

(0.2)

Here, BMO$({\mathbb {T}})$ denotes the bounded mean oscillation (semi)norm

$$\begin{aligned} \Vert f\Vert _{BMO({\mathbb {T}})}=\sup _I \frac{1}{|I|}\int _I |f-f_I| \ d s \end{aligned}$$

(0.3)

with the supremum taking over all arcs $I \subseteq {\mathbb {T}}.$

In the first part of this article, we give an interpretation of Paley’s theory in the semigroup language which allows an extension to non-abelian discrete groups. For each $t > 0$, let $P_t$ be the Poisson integral operator that sends $e^{ik\theta }$ to $e^{- |k| t} e^{ik\theta }$. Here is an equivalent characterization of the classical BMO and $H^1$-norms by $P_t$’s. For $f\in L^1({\mathbb {T}}$),

$$\begin{aligned}&\Vert f\Vert _{BMO({\mathbb {T}})}\simeq \sup _{t > 0} \left\| P_{t} \left[ |f - P_{t}(f)|^2 \right] \right\| _{L^\infty ({\mathbb {T}})}^\frac{1}{2} \end{aligned}$$

(0.4)

$$\begin{aligned}&\Vert f\Vert _{H^{1} ({\mathbb {T}})}\simeq \left\| \left( \int _0^\infty \left| \frac{\partial }{\partial t} P_tf \right| ^2tdt \right) ^\frac{1}{2} \right\| _{L^1({\mathbb {T}})}. \end{aligned}$$

(0.5)

Consider a discrete group G and a conditionally negative definite length $\psi$ on G. By that, we mean $\psi$ is a ${\mathbb {R}}_+$-valued function on G satisfying $\psi (g)=0$ if and only if $g=e$, $\psi (g)=\psi (g^{-1})$, and

$$\begin{aligned} \sum _{g,h}\overline{a_g}a_h\psi (g^{-1}h)\le 0 \end{aligned}$$

(0.6)

for any finite collection of coefficients $a_g\in {\mathbb {C}}$ with $\sum _g a_g=0$. We say a sequence $(h_{k})_{k \in {\mathbb {N}}}$ of elements of G is $\psi$-lacunary if there exists a constant $\delta >0$ such that

$$\begin{aligned} {\psi (h_{k})}\ge & {} (1 + \delta ){\psi (h_j )} \end{aligned}$$

(0.7)

$$\begin{aligned} \psi (h_j^{-1}h_{k})\ge & {} \delta \psi (h_k). \end{aligned}$$

(0.8)

for any $k>j$.

Note that the condition (0.8) follows from (0.7) (with a smaller constant) if we further require $\psi$ to be sub-additive, i.e. $\psi (hg)\le \psi (h)+\psi (g)$. In fact, assuming ${\psi (h_{k})}=(1+\varepsilon )\psi (h_j)\ge (1+\delta )\psi (h_{j})$, then the sub-additivity of $\psi$ implies that $\psi (h_k)=\psi (h_{j} h_j^{-1} h_k)\le \psi (h_j)+ \psi (h_j^{-1} h_{k})$, thus $\psi (h_{j}^{-1} h_k)\ge \psi (h_k)-\psi (h_j)\ge \varepsilon \psi (h_j)=\frac{\varepsilon }{1+\varepsilon }\psi (h_{k}) \ge \frac{\delta }{1+\delta }\psi (h_{k})$.

Let $\lambda$ be the left regular representation of G. Given a sequence $c_k\in \ell ^2({\mathbb {C}})$, we view $f=\sum _k c_k\lambda _{h_k}$ as a lacunary “Fourier series” and will study the related Paley’s theory. To state our first main result, let us recall the semigroup type $H^1$ and BMO-norms introduced in [20, 28], which have been frequently used in recent study of noncommutative analysis (see [8,9,10, 12, 21, 23, 34] etc.). Let

$$\begin{aligned} T_t:\lambda _g\mapsto e^{-t\psi (g)}\lambda _g \end{aligned}$$

(0.9)

be the semigroup of operators on the group von Neumann algebra $\hat{G}$ associated with $\psi$. For $f=\sum _g c_g\lambda _g\in L^1(\hat{G})$, let

$$\begin{aligned}&\Vert f\Vert _{H^1_c(\psi )}= \tau \left[ \left( \int _0^\infty \left| \frac{\partial }{\partial s} T_{s} (f) \right| ^2s \ ds \right) ^\frac{1}{2} \right] \\&\Vert f\Vert _{\mathrm{BMO}_c(\psi )}=\sup _{s > 0} \left\| T_{s} \left[ |f-T_{s}(f) |^2 \right] \right\| ^\frac{1}{2}, \end{aligned}$$

with $\tau$ being the canonical trace on the group von Neumann algebra of G. Please see Sect. 1.1 for the introduction of the group von Neumann algebra and $L^1(\hat{G})$.

Theorem 0.1

Assume $(h_k)$ is a $\psi$-lacunary sequence. Then, for any sequence $(c_k)_{k = 1}^{\infty }\in \ell ^2({\mathbb {C}})$, the series $\sum _{k=1}^\infty c_{k}\lambda _{h_k}$ converges in BMO$_c(\psi )$ and

$$\begin{aligned} \left\| \sum _{k=1}^\infty c_{k}\lambda _{h_k} \right\| _{BMO_c(\psi )}^2 \simeq _{\delta } \sum _{k=1}^\infty |c_k|^2 . \end{aligned}$$

(0.10)

with a constant of equivalence depending only on the lacunary constant $\delta$. Moreover,

$$\begin{aligned} \inf \left\{ \Vert f\Vert _{H^1_c(\psi )} : \hat{f}(h_k)=c_k, \text { for all } k\in {\mathbb {N}} \right\} \simeq _{\delta } \left( \sum _k |c_k|^2 \right) ^\frac{1}{2}. \end{aligned}$$

(0.11)

By the interpolation result proved in [20] (see Lemma 1.2 below), one obtains that every subset of a $\psi$-lacunary sequence is a $\Lambda (p)$ set for all $2<p<\infty$. More precisely, we have that, for any $2<p<\infty$ and any f of the form $f=\sum _kc_k\lambda _ {h_k}$,

$$\begin{aligned} \Vert f\Vert _{L^p(\hat{G})}\simeq _{\delta } p \left( \sum _k |c_k|^2 \right) ^\frac{1}{2}. \end{aligned}$$

(0.12)

Passing to the dual, we have for any $1<p<2$ and any f of the form $f=\sum _kc_k\lambda _ {h_k}$,

$$\begin{aligned} \Vert f\Vert _{L^p(\hat{G})}\simeq _{\delta } \frac{1}{p-1}\left( \sum _k |c_k|^2 \right) ^\frac{1}{2}. \end{aligned}$$

(0.13)

We will prove Theorem 0.1 for operator valued $c_k$. See Theorem 2.3.

In the second part of the article, we assume that the group G is equipped with a bi-invariant order “$\le$”, that is a total order which is invariant under the left and right multiplications. Let $G_+=\{g\in G : e\le g\}$. Following Rudin’s terminology [48, Section 8.6], we say a subset $E \subseteq G_+$ is lacunary if there exists a constant K such that

$$\begin{aligned} N(E) := \sup _{g\in G_+}\#\{h\in E: g\le h\le g^2\}\le K. \end{aligned}$$

Theorem 0.2

For any sequence $(c_{k})_{k = 1}^{\infty }\in {\mathbb {C}}$, and any sequence $(h_{k})_{k=1}^\infty$ in a lacunary subset $E\subseteq G_+$, we have

$$\begin{aligned} \inf \left\{ \tau (|f|) : \hat{f}(h_k)=c_k, {{\,\mathrm{supp}\,}}(\hat{f}) \subset G_+ \right\} \simeq \left( \sum _k|c_k|^2 \right) ^\frac{1}{2}, \end{aligned}$$

(0.14)

Theorem 0.2 follows from a factorization theorem of noncommutative analytic Hardy spaces and an adaptation of Lust-Piquard and Pisier’s argument [26] to Rudin’s terminology [48, Section 8.6] of lacunary sets. We also provide interesting examples of Paley sequences and $\Lambda (p)$ sets (see e.g. Corollary 4.3) on free groups.

We prove all these results for operator-valued functions f, that is, for f of the form $f=\sum _kc_k\otimes \lambda _ {h_k}$ with $c_k$ being operators. See Theorems 2.4 and 3.2. Sect. 1 introduces some preliminaries. The main results are proved in Sects. 2 and 3. In Sect. 4, we apply our theories to the case of free groups and construct new examples of Paley sets and $\Lambda (p)$ sets.

2 Preliminaries

2.1 Noncommutative $L^p$-Space

Let ${\mathcal {M}}$ be a semifinite von Neumann algebra acting on a Hilbert space ${\mathcal {H}}$ with a normal semifinite faithful trace $\tau$. For $0<p<\infty$, denote by $L^p({\mathcal {M}})$ the noncommutative $L^p$ space associated with the (quasi)norm $\Vert f\Vert _p =\left[ \tau (|f|^p) \right] ^{\frac{1}{p}}$. As usual, we set $L^\infty ( {\mathcal {M}})={\mathcal {M}}$ equipped with the operator norm. For a (possibly nonabelian) discrete group G, the von Neumann algebra is the closure of the linear span of the elements $\lambda _{g}$ given by the left regular representation of G with respect to the weak operator topology. The trace $\tau$ is defined by

$$\begin{aligned} \tau (f) =c_e, \end{aligned}$$

for $f=\sum _g c_g\lambda _g$. The associated $L^{p}$ norm is defined as

$$\begin{aligned} \Vert f\Vert _p = \left[ \tau (|f|^p) \right] ^\frac{1}{p} \end{aligned}$$

for $1\le p<\infty$.

When G is abelian, e.g. $G={\mathbb {Z}}^d$, the non-commutative $L^{p}$ space obtained is isometrically isomorphic to the classical $L^{p}$ space on the dual group. We denote by $\hat{G}$ the group von Neumann algebra of G, and by $L^p(\hat{G})$ the associated non-commutative $L^p$ spaces. So, $\hat{G}$ will also be denoted by $L^\infty (\hat{G})$ sometimes. Let ${\mathcal {M}}={\mathcal {B}}({\mathcal {H}})$, the algebra of all bounded operators on ${\mathcal {H}}$ equipped with the usual trace tr, then the associated $L^p$-space $L^p({\mathcal {M}})$ is the Schatten-p class $S^p({\mathcal {H}})$. We denote $\tau$ defined above as the canonical trace on the group von Neumann algebra $\hat{G}$ and $\mathrm{tr}$ as the usual trace on ${\mathcal {B}}({\mathcal {H}})$. We will often consider the tensor product of von Neumann algebra $B(H)\bar{\otimes }\hat{G}$ with the tensor trace $tr\otimes \tau$ and the associated noncommutative $L^p$ space $L^p( B(H)\bar{\otimes }\hat{G} )$. For $1\le p<\infty$, this is the closure of the collection of all finite sums $f=\sum _g c_g\otimes \lambda _g$ with $c_g\in S^p$ with respect to the $L^p$-norm

$$\begin{aligned} \Vert f\Vert _{L^p}=\left( (tr\otimes \tau ) (|f|^p)\right) ^\frac{1}{p}. \end{aligned}$$

The classical Cauchy–Schwartz inequality and Hölder’s inequality extend to the noncommutative setting. In particular, we have the Kadison-Cauchy-Schwartz inequality

$$\begin{aligned} |(id\otimes \tau ) (f) |^2 \le (id\otimes \tau ) (|f|^2), \end{aligned}$$

(1.1)

for all $f\in B(H)\bar{\otimes }\hat{G}$, and Hölder’s inequality,

$$\begin{aligned} \Vert f \varphi \Vert _{L^r}\le & {} \Vert f\Vert _{L^p}\Vert \varphi \Vert _{L^q}, \end{aligned}$$

(1.2)

for all $f\in L^p( B(H)\bar{\otimes }\hat{G} ), \varphi \in L^q( B(H)\bar{\otimes }\hat{G} )$, where $\frac{1}{p} +\frac{1}{q} = \frac{1}{r}$. Here id denotes the identity operator on B(H). For the brevity of notation, we will omit the “id” in “$id\otimes \tau$”, and omit the “$\otimes$” in “$c_k\otimes \lambda _{h_k}$” in the statement of theorems or proofs and simply write $\tau$ and $c_k\lambda _{h_k}$ when no confusion arises.

We refer readers to the survey paper [44] for information on non-commutative $L^{p}$ spaces.

2.2 Column and Row Spaces

Let $0<p\le \infty$ and let $(c_{n})_{n\ge 0}$ be a finite sequence in $L^p({\mathcal {M}})$. Define

$$\begin{aligned} \Vert (c_{n})\Vert _{L^p({\mathcal {M}}, \ell _c^2)}= \left\| \left( \sum \limits _{n\ge 0} |c_{n}|^{2} \right) ^{1/2} \right\| _p,~~ \Vert (c_{n})\Vert _{L^p({\mathcal {M}}, \ell _r^2)}= \left\| \left( \sum \limits _{n\ge 0} |c_{n}^{*}|^{2} \right) ^{1/2} \right\| _p. \end{aligned}$$

For $0<p<\infty ,$ we define $L^p({\mathcal {M}}, \ell ^2_c)$(respectively $L^p({\mathcal {M}}, \ell ^2_r)$) as the completion of the family of all finite sequences in $L^p({\mathcal {M}})$ with respect to $\Vert \cdot \Vert _{L^p({\mathcal {M}}, \ell _c^2)}$ (respectively $\Vert \cdot \Vert _{L^p({\mathcal {M}}, \ell _r^2)}$). For $p=\infty$, we define $L^\infty ({\mathcal {M}}, \ell ^2_c)$ (respectively $L^\infty ({\mathcal {M}}, \ell ^2_r)$) as the Banach space of (possible infinite) sequences in ${\mathcal {M}}$ such that $\sum _{n}c_{n}^*c_n$ (respectively $\sum _{n}c_{n}c_n^*$) converges in the $\text {weak}^*$-topology.

Let $0< p\le \infty .$ We define the space $L^p({\mathcal {M}}, \ell _{cr}^2)$ as follows:

• (1) If $0<p<2$,

  $$\begin{aligned} L^p({\mathcal {M}}, \ell _{cr}^2)=L^p({\mathcal {M}}, \ell _c^2) +L^p({\mathcal {M}}, \ell _r^2) \end{aligned}$$

  equipped with the norm:

  $$\begin{aligned} \Vert (c_k)_{n\ge 0}\Vert _{L^p({\mathcal {M}}, \ell _{cr}^2)} =\inf _{c_k=c_k^\prime +c_k^{\prime \prime }} \left\{ \Vert (c_k^{\prime })\Vert _{L^p({\mathcal {M}}, \ell _c^2)} +\Vert (c_k^{\prime \prime })\Vert _{L^p({\mathcal {M}}, \ell _r^2)} \right\} \end{aligned}$$

  where the infimum is taken over all decompositions for which

  $$\begin{aligned} \Vert (c_k^{\prime })\Vert _{L^p({\mathcal {M}}, \ell _c^2)}<\infty ~\text{ and } ~\Vert (c_k^{\prime \prime })\Vert _{L^p({\mathcal {M}}, \ell _r^2)}<\infty . \end{aligned}$$

• (2) If $p\ge 2$,

  $$\begin{aligned} L^p({\mathcal {M}}, \ell _{cr}^2)=L^p({\mathcal {M}}, \ell _c^2) \cap L^p({\mathcal {M}}, \ell _r^2) \end{aligned}$$

  equipped with the norm:

  $$\begin{aligned} \Vert (c_k)\Vert _{L^p({\mathcal {M}}, \ell _{cr}^2)} =\max \left\{ \Vert (c_k)\Vert _{L^p({\mathcal {M}}, \ell _r^2)}, \Vert (c_k)\Vert _{L^p({\mathcal {M}}, \ell _c^2)} \right\} . \end{aligned}$$

The spaces $L^p({\mathcal {M}}, \ell _{cr}^2)$ (resp. $L^p({\mathcal {M}}, \ell _{c}^2)$, $L^p({\mathcal {M}}, \ell _{r}^2)$) form an interpolation scale for the complex interpolation method.

Lemma 1.1

$$\begin{aligned} L^p({\mathcal {M}}, \ell _{c}^2)= & {} [L^\infty ({\mathcal {M}}, \ell _{c}^2),L^1({\mathcal {M}}, \ell _{c}^2)]_{\frac{1}{p}} \end{aligned}$$

(1.3)

$$\begin{aligned} L^p({\mathcal {M}}, \ell _{r}^2)= & {} [L^\infty ({\mathcal {M}}, \ell _{r}^2),L^1({\mathcal {M}}, \ell _{r}^2)]_{\frac{1}{p}} \end{aligned}$$

(1.4)

$$\begin{aligned} L^p({\mathcal {M}}, \ell _{cr}^2)\simeq & {} [L^\infty ({\mathcal {M}}, \ell _{cr}^2),L^1({\mathcal {M}}, \ell _{cr}^2)]_{\frac{1}{p}} \end{aligned}$$

(1.5)

for $1<p<\infty$.

Proof

$L^p({\mathcal {M}}, \ell _{c}^2)$ are complemented subspaces of $L^p({\mathcal {M}}\bar{\otimes } B(H))$ via the embedding

$$\begin{aligned} c_k:\mapsto c_k\otimes e_{1,k}. \end{aligned}$$

Therefore, they inherit the duality relation and the interpolation relation (1.3) from $L^p({\mathcal {M}}\bar{\otimes } B(H))$. The interpolation (1.4) holds because of a similar reason. The interpolation equivalence (1.5) is proved by Pisier in [38, Theorem 8.4.8] for ${\mathcal {M}}=B(H)$. The argument for the general case is in the same spirit, which we sketch here. Let ${\mathbb {F}}_\infty$ be the free nonabelian group generated by countable many free generators $\{ g_k, k\in {\mathbb {N}} \}$, and denote by $L^\infty (\hat{{\mathbb {F}}}_\infty )$ the associated group von Neumann algebra. Let $E_p, 1\le p< \infty$ ($p=\infty$) be the norm (weak $*$) closure of $span\{\lambda _{g_k}, k\in {\mathbb {N}}\}$. Then $E_p$ are complemented subspaces of $L^p({\mathcal {M}}\bar{\otimes } L^\infty (\hat{{\mathbb {F}}}_\infty ))$ and form an interpolation scale for complex interpolation. This is [38, Corollary 8.3.3]. By [38, Theorem 8.4.10] and the duality relation, $L^p({\mathcal {M}}, \ell _{cr}^2)$ is completely isomorphic to $E_p$ for all $1\le p\le \infty$ via the map

$$\begin{aligned} c_k:\mapsto c_k\otimes \lambda _{g_k}. \end{aligned}$$

Therefore, the interpolation relation (1.5) holds for $L^p({\mathcal {M}}, \ell _{cr}^2)$. $\square$

We denote by $S^p(\ell _c^2)$, $S^p(\ell _r^2)$ and $S^p(\ell _{cr}^2)$ the spaces $L^p({\mathcal {M}}, \ell _c^2)$, $L^p({\mathcal {M}}, \ell _r^2)$ and $L^p({\mathcal {M}}, \ell _{cr}^2)$ when ${\mathcal {M}}={\mathcal {B}}({\mathcal {H}})$, respectively. Please refer to [19, 38, 44] for details on these spaces.

2.3 Semigroup BMO Spaces

Given a conditionally negative definite length $\psi$ on discrete group G, Schoenberg’s theorem [2, p. 74, Theorem2.2] says that the functions $\phi _t(g)=e^{-t\psi (g)}$ are positive definite on G for all $t>0$. Therefore, the family of operators,

$$\begin{aligned} T_t: \lambda _g=e^{-t\psi (g)}\lambda _g \end{aligned}$$

extends to a symmetric Markov semigroup of completely positive operators on the group von Neumann algebra $\hat{G}$. That is to say, for every $t>0$, $T_t$ is

• (1) Unital, i.e. $T_t(\lambda _e)=\lambda _e$.

• (2) Normal, i.e. $T_t$ is weak $*$ continuous.

• (3) Symmetric, i.e. $\tau (T_t(f)\varphi )=\tau (fT_t(\varphi ))$, $\forall f,\varphi \in \hat{G}$.

• (4) Completely positive, i.e. $id\otimes T_t$ is positive preserving on ${\mathcal {K}}(\ell ^2)\otimes \hat{G}$ with ${\mathcal {K}}(\ell _2)$ the algebra of compact operators on $\ell ^2({\mathbb {N}})$.

and the family $(T_t)_{t>0}$ is weak-$*$ continuous at $0+$, i.e. $\tau (T_t(f)\varphi )\rightarrow \tau (f\varphi )$ as $t\rightarrow 0$ for all $f\in {\mathcal {L}}(G), \varphi \in L^1(\hat{G})$. It is well known that a unital map on $C^*$-algebras (in particular on von Neumann algebras) is (completely) positive preserving iff it is (completely) contractive. Because of this fact and the symmetric assuption (2), each $T_t$ extends to a contraction on $L^1(\hat{G})$ by duality. By interpolation, each $T_t$ extends to a contraction on $L^p (\hat{G})$ for every $1\le p\le \infty$.

Given a semifinite von Neumann algebra ${\mathcal {M}}$, denote by ${\mathcal {N}}={\mathcal {M}}\bar{\otimes } \hat{G}$. Following [20] and [28], for finite sums $f=\sum _{k} c_g\otimes \lambda _g$ with $c_g\in {\mathcal {M}}$ set

$$\begin{aligned} \Vert f\Vert _{\mathrm{BMO}_c(\psi )} =\sup _{0<t<\infty } \left\| (id\otimes T_{t}) \left[ |f-(id\otimes T_{t })(f) |^2 \right] \right\| _{\mathcal {N}}^\frac{1}{2}. \end{aligned}$$

(1.6)

We will often omit id and simply write $T_t$ instead of $T_{t}\otimes id$ when no confusion arises in the context. Set

$$\begin{aligned} \Vert f\Vert _{BMO(\psi )}=\max \{\Vert f\Vert _{ BMO_c(\psi )},\Vert f^*\Vert _{BMO_c(\psi )}\}. \end{aligned}$$

(1.7)

When G is the integer group ${\mathbb {Z}}$, $L^p( \hat{\mathbb {Z}})$ is the $L^p$-space of p-integrable functions with respect to the Haar measure on the torus ${\mathbb {T}}$. The semigroup $T_t$ is the heat semigroup (respectively Poisson semigroup) if we set $\psi (g)=|g|^2$ (respectively |g|) for $g\in {\mathbb {Z}}$. It is an elementary calculation that the semigroup BMO norm defined above coincides with those defined in (0.3), (0.4). When f is operator-valued, the semigroup BMO norm defined above coincides with the ones studied in [27, Section 1.3]. The semigroup BMO norms may differ from each other for different semigroups, see [12, Section 4] for examples.

As in [20] and [28], we define the space BMO$(\psi )$ as an abstract closure of all the finite sums $f=\sum _{k} c_g\otimes \lambda _g$. We omit the definition of this closure as it will not be needed in this article. We refer the readers to [28, p. 3377] or [20, Sect. 5.2] for the definition of this closure. The following interpolation result was proved in [20, Theorem 0.2]. We refer the readers to Theorem 5.2 and Remark 5.5 of [20] for related results. E. Ricard proved in [46] that the type of semigroup of operators considered in this article satisfies the Markov dilation assumption in [20, Theorem 0.2].

Lemma 1.2

[20, Theorem 0.2] Let $L^p_0({\mathcal {N}})=\{f\in L^p({\mathcal {N}}), \tau (f)=0\}$. The following interpolation result holds for $1<p<\infty$,

$$\begin{aligned} {[}BMO(\psi ),L^1_0({\mathcal {N}})]_{\frac{1}{p}} \simeq L^p_0({\mathcal {N}}) \end{aligned}$$

2.4 Analytic $H^p$-Space on Ordered Groups

The theory of analytic noncommutative Hardy spaces were developed by Marsalli and West etc. in [3, 5, 29,30,31] in the general context of Arveson’s subdiagonal operator algebras [1]. We recall here the definition of these spaces and the corresponding factorization, duality and interpolation results in the case of group von Neuman algebra associated with ordered groups. Please see [44, Sect. 8] for a survey of these results.

Let $(G, \le )$ be a countable (possibly non-abelian) discrete group with a total order. Denote by e the unit element. Let $L^p( \hat{G})$ be the noncommutative $L^p$ spaces associated with the canonical trace $\tau$.

For $1\le p\le \infty$, let ${\mathcal {A}}_p\subset L^p( \hat{G})$ be the collection of all the finite sum $\sum _{g\ge e} c_g\lambda _g$. Let $H^p(\hat{G})$ be the norm (respectively weak operator) closure of ${\mathcal {A}}_p$ in $L^p(\hat{G})$ for $1\le p<\infty$ (respectively $p=\infty$).

Like the classical case, one can define an analytic BMO space as the dual of $H^1(\hat{G})$ using an analogue of the Hilbert transform, that we will introduce at below. Let H be the linear map on $L^2(\hat{G})$ such that

$$\begin{aligned} H\left( \sum _gc_g\lambda _{g}\right) = -i\left( \sum _{g\ge e}c_g\lambda _g-\sum _{g\le e}c_g\lambda _g\right) . \end{aligned}$$

(1.8)

It is clear that H is bounded on $L^2(\hat{G})$. It was proved in [30] that H extends to a bounded map on $L^p (\hat{G})$ for all $1<p<\infty .$¹ So $H^p(\hat{G})$ is complemented in $L^p(\hat{G})$, and the dual of $H^p(\hat{G})$ is isomorphic to $H^{p'}(\hat{G})$ for $\frac{1}{p}+\frac{1}{p'}=1, 1<p<\infty$. For $f=\sum _gc_g\lambda _{g}\in L^2( \hat{G})$, set

$$\begin{aligned} \Vert f \Vert _{BMO(\hat{G})}= \inf \{\Vert u\Vert _{L^\infty (\hat{G})}+\Vert v\Vert _{L^\infty ({\hat{G}})}: f=u+H{v}\} \end{aligned}$$

where the infimum is taken over all $u, v\in L^\infty (\hat{G})$. Note $H{v}\in L^2(\hat{G})$ as $v\in L^\infty (\hat{G})\subset L^2(\hat{G})$. Let $BMOA(\hat{G})$ be the space of all $f \in H^2({\hat{G}})$ with finite $\Vert \cdot \Vert _{BMO(\hat{G})}$-norms. It is easy to show that the dual of $H^1(\hat{G})$ is isomorphic to BMOA$(\hat{G})$ following the classical arguments. This was explained in [31, Theorem 5] and [44, p. 1499]. Please note that BMOA was denoted by A, and the Hilbert transform was called the conjugate map and was denoted by “$\sim$” in Theorem 5 of [31].

We will also need the operator-valued version of $H^p(\hat{G})$ and BMOA$(\hat{G})$. Let $({\mathcal {M}},tr)$ be a semifinite von Neumann algebra. Let ${\mathcal {N}}={\mathcal {M}}\overline{\otimes } \hat{G}$ with the trace $tr\otimes \tau$. For $1\le p\le \infty$, let $H^p({\mathcal {N}})$ be the norm (respectively weak operator) closure in $L^p({\mathcal {N}})$ of the collection of all finite sums $\sum _{g\ge e} c_g\otimes \lambda _g$ with $c_g\in L^p({\mathcal {M}}).$ In this case, $H^1({\mathcal {N}})$ coincides with the projective tensor product $L^1({\mathcal {M}})\hat{\otimes } H^1(\hat{G})$, and its dual is isomorphic to BMOA$({\mathcal {N}})={\mathcal {M}}\bar{\otimes } BMOA(\hat{G})$ the injective tensor product. The Hilbert transform $id \otimes H$ extends to a bounded map on $L^p({\mathcal {N}})$ for all $1<p<\infty$. So, for $1<p<\infty$, $H^p({\mathcal {N}})$ is a complemented subspace of $L^p({\mathcal {N}})$, and we have the following equivalence for $f=\sum _g c_g\otimes \lambda _g\in L^p({\mathcal {N}})$,

$$\begin{aligned} \begin{aligned} \Vert f\Vert _p\simeq \left\| \sum _{g\ge e} c_g\otimes \lambda _g\right\| _p +\left\| \sum _{g< e} c_g\otimes \lambda _g\right\| _p. \end{aligned} \end{aligned}$$

(1.9)

$H^p({\mathcal {N}})$ and BMOA$({\mathcal {N}})$ were studied in [3, 29,30,31, 44] etc. in the general context of Arveson’s subdiagonal operator algebras. The following factorization theorem was proved in [30, Theorem 4.3] and [3, Theorem 3.2].

Lemma 1.3

Given any $f\in H^1({\mathcal {N}})$ and $\varepsilon >0$, there exist $y,z\in H^2({\mathcal {N}})$ such that $f=yz$ and $\Vert y\Vert _2\Vert z\Vert _2\le \Vert f\Vert _1+\varepsilon$.

It was proved in [31] and [3] that the dual of $H^1({\mathcal {N}})$ is isomorphic to BMOA$({\mathcal {N}})$, and $H^p({\mathcal {N}})$ is an interpolation space between them.

Lemma 1.4

[3, Proposition 4.1] Let $1<p< \infty$ then

$$\begin{aligned} H^p({\mathcal {N}})\simeq [ BMOA({\mathcal {N}}),H^{1}({\mathcal {N}})]_{\frac{1}{p}} \end{aligned}$$

with equivalent norms.

Lemma 1.4 has been known to experts after the work of Pisier [36], Marsallie/West [30], and Pisier/Xu [44, p. 1499]. It was formally proved in [3] for the semifinite von Neumann algebras.

3 Proof of Theorem 0.1

Lemma 2.1

For $a_s\in {\mathbb {R}}_+$, $c_s,b_s\in B(H)$, we have, for any $0<p,q,r<\infty , \frac{1}{p}+\frac{1}{q}=\frac{1}{r}$,

$$\begin{aligned} \left\| \sum _s a_sc^*_sb_s \right\|\le & {} \left\| \sum _s a_{s} |c_s|^2 \right\| ^\frac{1}{2} \left\| \sum _s a_{s} |b_s|^2 \right\| ^\frac{1}{2} \end{aligned}$$

(2.1)

$$\begin{aligned} \left\| \sum _k a_sc^*_sb_s \right\| _{S^r }\le & {} \left\| \sum _s a_{s} |c_s|^2 \right\| _{S^p}^\frac{1}{2} \left\| \sum _s a_{s} |b_s|^2 \right\| _{S^q}^\frac{1}{2}. \end{aligned}$$

(2.2)

Proof

Let $e_{s,t}$ be the canonical basis of B(H). Let $f=\sum _s a_s^\frac{1}{2}c^*_s\otimes e_{1,s}$ and $\varphi =\sum _s a_s^\frac{1}{2}b_s\otimes e_{s,1}$. Then (2.1) follows from the fact that $\Vert f \varphi \Vert \le \Vert f\Vert \Vert \varphi \Vert =\Vert ff^*\Vert ^\frac{1}{2} \Vert \varphi ^*\varphi \Vert ^\frac{1}{2}$. The inequality (2.2) follows from Hölder’s inequality (1.2), $\Vert f\varphi \Vert _{S^r} \le \Vert f\Vert _{S^p}\Vert \varphi \Vert _{S^q}$. $\square$

Lemma 2.2

Let $f=\sum _k c_k\otimes \lambda _{h_k}\in L^2(B(H)\bar{\otimes } \hat{G})$, we have

$$\begin{aligned} \frac{1}{2} \left( \sum _k |c_k|^2 \right) ^\frac{1}{2}\ge \tau \left[ \left( \int _0^\infty \left| \frac{\partial }{\partial s} T_sf \right| ^2sds \right) ^\frac{1}{2} \right] \end{aligned}$$

(2.3)

This means that subtracting the right hand side from the left hand gives a nonnegative self-adjoint element of B(H). Moreover, if we assume $(h_k)$ is a $\psi$-lacunary sequence, then

$$\begin{aligned} \left\| \int _0^\infty \left| \frac{\partial }{\partial s} T_sf \right| ^2sds \right\| \le \left( 1+\frac{2}{\delta }\right) \left\| \sum _k |c_k|^2 \right\| . \end{aligned}$$

(2.4)

Proof

An elementary calculation shows that

$$\begin{aligned} \int _0^\infty \left| \frac{\partial }{\partial s} T_sf \right| ^2sds= & {} \int _0^\infty \sum _{k} (c_k\lambda _{h_k})^*\psi (h_k)e^{-s\psi (h_k)}\sum _{j} (c_j\lambda _{h_j})\psi (h_j)e^{-s\psi (h_j)}sds \nonumber \\= & {} \sum _{k,j} (c_k\lambda _{h_k})^*c_j\lambda _{h_j}\psi (h_{k})\psi (h_j)\int _0^\infty e^{-s(\psi (h_k)+\psi (h_j))}sds \nonumber \\= & {} \sum _{k,j} a_{k,j}(c_k\lambda _{h_k})^*c_j\lambda _{h_j}, \end{aligned}$$

(2.5)

with

$$\begin{aligned} a_{k,j}= \frac{\psi (h_k)\psi (h_j)}{(\psi (h_k)+\psi (h_j))^2}\ge 0 \end{aligned}$$

since $\int _0^\infty e^{- \alpha s} s \ d s = \frac{1}{\alpha ^{2}}$. So, by Cauchy-Schwartz inequality (1.1), we get

$$\begin{aligned} \tau \left[ \left( \int _0^\infty \left| \frac{\partial }{\partial s} T_sf \right| ^2sds \right) ^\frac{1}{2} \right]\le & {} \left[ \tau \left( \int _0^\infty \left| \frac{\partial }{\partial s} T_sf \right| ^2sds \right) \right] ^\frac{1}{2}\\= & {} \left( \sum _{k}|c_k|^ 2a_{k,k} \right) ^\frac{1}{2}=\frac{1}{2} \left( \sum _k |c_k|^2 \right) ^\frac{1}{2}. \end{aligned}$$

On the other hand, for any given j, we have that

$$\begin{aligned} \sum _k a_{k,j}= & {} \sum _{k\le j}\frac{\psi (h_k)\psi (h_j)}{(\psi (h_k) +\psi (h_j))^2}+\sum _{k>j}\frac{\psi (h_k)\psi (h_j)}{(\psi (h_k)+\psi (h_j))^2}\\\le & {} \sum _{k\le j}\frac{\psi (h_k)}{ \psi (h_j)}+\sum _{k>j} \frac{ \psi (h_j)}{\psi (h_k) }\\\le & {} \frac{1+\delta }{\delta }+\frac{1}{\delta }=1+\frac{2}{\delta }. \end{aligned}$$

In the last step of the estimate above, we have used the $\psi$-lacunary property (0.7). Taking the supremum over j, we get

$$\begin{aligned} \sup _j \sum _k a_{k,j} \le 1+\frac{2}{\delta }. \end{aligned}$$

A similar estimate shows,

$$\begin{aligned} \sup _k \sum _j a_{k,j} \le 1+\frac{2}{\delta }. \end{aligned}$$

Applying Lemma 2.1 to (2.5), we have

$$\begin{aligned} \left\| \int _0^\infty \left| \frac{\partial }{\partial s} T_sf \right| ^2sds \right\|\le & {} \left\| \left( \sum _{k,j} |c_k|^2a_{k,j} \right) ^\frac{1}{2} \right\| \left\| \left( \sum _{k,j}|c_j|^ 2a_{k,j} \right) ^\frac{1}{2} \right\| \\= & {} \left\| \left( \sum _{k} |c_k|^2\sum _ja_{k,j} \right) ^\frac{1}{2} \right\| \left\| \left( \sum _{j}|c_j|^ 2\sum _ka_{k,j} \right) ^\frac{1}{2} \right\| \\\le & {} \left( 1+\frac{2}{\delta }\right) \left\| \sum _k |c_k|^2 \right\| . \end{aligned}$$

$\square$

Theorem 2.3

Suppose that $(h_k)_k \subseteq G$ is a $\psi$-lacunary sequence. Then, for any $N\in {\mathbb {N}}$ and $f~=~\sum _{k=1}^N c_k \lambda _ {h_k}$ with $c_k\in B(H)$, we have

$$\begin{aligned} \Vert f\Vert _{BMO_c(\psi )}^2\simeq _{\delta } \left\| \sum _{k,h_k\ne e} |c_k|^2 \right\| . \end{aligned}$$

(2.6)

At the other end, we have, for any $(c_k)\in S^1(\ell ^2_c)$,

$$\begin{aligned} \mathrm{tr} \left[ \left( \sum _k |c_k|^2 \right) ^\frac{1}{2} \right] \simeq _\delta \inf \left\{ ( tr \otimes \tau ) \left[ \left( \int _0^\infty \left| \frac{\partial }{\partial s} T_sf \right| ^2s \ d s \right) ^\frac{1}{2} \right] : \tau (f \lambda ^*_{ h_k})=c_k \right\} . \end{aligned}$$

(2.7)

where the infimum runs over all $f\in L^1(B(H)\bar{\otimes } \hat{G})$.

Proof

We prove the BMO estimate (2.6) first. An easy calculation shows that

$$\begin{aligned} T_t\left[ |f -T_{t}(f)|^2 \right]= & {} T_t \left[ \left( \sum _{k}c_k(1-e^{-t\psi (h_k)} \lambda _{h_k})\right) ^* \left( \sum _{k}c_k(1-e^{-t\psi (h_k)} \lambda _{h_k})\right) \right] \nonumber \\= & {} T_t\left[ \left( \sum _{k,j} (1-e^{-t\psi (h_k)})(1-e^{-t\psi (h_j)}) (c_k\lambda _{h_k})^*c_j\lambda _{h_j}\right) \right] \nonumber \\= & {} \sum _{k,j} a_{k,j}(c_k\lambda _{h_k})^*c_j\lambda _{h_j}, \end{aligned}$$

(2.8)

with

$$\begin{aligned} a_{k,j}= e^{-t\psi (h^{-1}_kh_j )} (1-e^{-t\psi (h_k^{-1})})(1-e^{-t\psi (h_j)})\ge 0. \end{aligned}$$

Note

$$\begin{aligned} a_{k,j}\le \min \{e^{-t\psi (h^{-1}_kh_j )}, 1-e^{-t\psi (h_k^{-1})}\} =\min \{e^{-t\psi (h^{-1}_kh_j )}, 1-e^{-t\psi (h_k)}\}. \end{aligned}$$

We have, for a fixed j,

$$\begin{aligned} \sum _{k} a_{k,j} \le \sum _{t\psi (h_k)\le 1} 1-e^{-t\psi (h_k)} +\sum _{t\psi (h_k)> 1} e^{-t\psi (h^{-1}_kh_j )}. \end{aligned}$$

By the lacunary property (0.8) $\psi (h_k^{-1}h_j)\ge \delta \psi (h_k)$ and the inequality $1 - e^{- s} \le s$, we get

$$\begin{aligned} \sum _{k} a_{k,j} \le \sum _{t\psi (h_k)\le 1} t \psi (h_k) +\sum _{t\psi (h_k)> 1} e^{- t \delta \psi (h_k)} \end{aligned}$$

(2.9)

By the lacunary property (0.7) we see that

$$\begin{aligned} \frac{t\psi (h_k)}{t\psi (h_{k+1})}\le \frac{1}{1+\delta }, \ \ \ \frac{e^{- t \delta \psi (h_{k+1})}}{e^{- t \delta \psi (h_k)}} =e^{t \delta (\psi (h_k)-\psi (h_{k+1})}\le {e^{- t \delta ^2\psi (h_k)}}. \end{aligned}$$

So the first term on the right hand side of (2.9) is bounded by a geometric series with ratio $(1+\delta )^{-1}$ and starting with 1. So it is smaller than $\frac{1+\delta }{\delta }$. The second term on the right hand side of (2.9) is bounded by a geometric series with ratio $e^{-\delta ^2}$ and starting with $e^{-\delta }$. So it is smaller than $\frac{e^{-\delta }}{1-e^{- {\delta ^2} }}$. We then conclude that

$$\begin{aligned} \sup _j \sum _k a_{k,j}\le 1+\delta ^{-1} +\frac{e^{-\delta }}{1-e^{-{\delta ^2}}} \le 1+\delta ^{-1}+\delta ^{-2}=:c_{\delta }. \end{aligned}$$

A similar estimate shows that

$$\begin{aligned} \sup _k \sum _j a_{k,j}\le c_\delta . \end{aligned}$$

Applying Lemma 2.1 to (2.8), we have

$$\begin{aligned} \left\| T_t \left[ |f - T_{t} (f) |^2 \right] \right\|\le & {} \left\| \sum _{k,j} |c_k|^2a_{k,j} \right\| ^\frac{1}{2} \left\| \sum _{k,j}|c_j|^ 2a_{k,j} \right\| ^\frac{1}{2}\\\le & {} (1+\delta ^{-1}+\delta ^{-2}) \left\| \sum _k |c_k|^2 \right\| . \end{aligned}$$

Taking the supremum over t, we get $\Vert f\Vert ^2_{BMO_c }\le c_\delta \left\| \sum _k |c_k|^2 \right\| .$ For the lower estimate, note that for any $f=\sum _kc_k\lambda _{h_k}$ with $\Vert \sum _k |c_k|^2\Vert <\infty$, we have

$$\begin{aligned} \Vert T_t \left[ |f-T_t(f)|^2 \right] \Vert\ge & {} \left\| (id\otimes \tau ) \left( T_t \left[ |f-T_{t} (f)|^2 \right] \right) \right\| \\= & {} \left\| \sum _{k,h_k\ne e} \left| \left[ 1-e^{-t\psi (h_k)} \right] c_k \right| ^2 \right\| . \end{aligned}$$

Since $1-e^{-t\psi (h_k)}$ tends to 1 uniformly in k as t tends to $\infty$, we get the lower estimate. Taking the adjoint, we prove the estimate for the BMO norms.

We now turn to the $H^1$-estimate (2.7). By duality, we may choose $b_k$ such that $\Vert \sum |b_k|^2\Vert =1$ and

$$\begin{aligned} \mathrm{tr} \left[ \left( \sum _k |c_k|^2 \right) ^{\frac{1}{2}} \right] =\mathrm{tr} \left( \sum c^{*}_{k} b_{k} \right) =\sup _{f,\varphi }(tr\otimes \tau ) ( f^{*} \varphi ), \end{aligned}$$

where the superum runs over all finite sum $\varphi =\sum _{k=1}^N b_k\lambda _{h_k}, f =\sum _{k=1}^N c_k\lambda _{h_k}$. We then have

$$\begin{aligned} (tr\otimes \tau ) (f^*\varphi )= & {} 4 ( tr\otimes \tau ) \left( \int _0^\infty \left( \frac{\partial }{\partial s} T_sf^* \right) \left( \frac{\partial }{\partial s} T_{s} \varphi \right) sds \right) \\\le & {} 4 (tr\otimes \tau )\left[ \left( \int _0^\infty \left| \frac{\partial }{\partial s} T_sf \right| ^2sds \right) ^\frac{1}{2} \right] \left\| \int _0^\infty \left| \frac{\partial }{\partial s} T_s\varphi \right| ^2sds \right\| ^\frac{1}{2}. \end{aligned}$$

In the last step of the inequalities above, we used Hölder’s inequality (1.2) for $x\in L^1(B(H\otimes L^2(0,\infty ))\otimes \hat{G} ),y \in B(H\otimes L^2(0,\infty ))\otimes \hat{G}$ with $x=\int _0^\infty \sqrt{s} e_{1,s}\otimes \frac{\partial }{\partial s}T_sf^* ds$ and $y=\int _0^\infty \sqrt{s} e_{s,1}\otimes \frac{\partial }{\partial s} T_s\varphi ds$. Combining the above estimates with Lemma 2.2, we obtain

$$\begin{aligned} \mathrm{tr} \left[ \left( \sum _k |c_k|^2 \right) ^\frac{1}{2} \right] \le 4 \left( 1+\frac{2}{\delta }\right) ^\frac{1}{2} ( \tau \otimes \mathrm{tr}) \left[ \left( \int _0^\infty \left| \frac{\partial }{\partial s} T_sf \right| ^2sds \right) ^\frac{1}{2} \right] . \end{aligned}$$

This proves one direction of (2.7), the other direction follows by taking tr on the both sides of (2.3). $\square$

Remark 2.4

Lust-Piquard and Pisier’s work [26] is the first one studying noncommutative Paley’s inequalities. The second part of this paper generalize Lust-Piquard and Pisier’s work to the group von Neumann algebras of ordered groups. An interesting point of Theorem 2.3 is that it gives interpretations of the row (and column) version of the noncommutative Paley’s inequality separately. Lust-Piquard and Pisier consider the the analytic $H^1$ norm which is equivalent to the Littlewood-Paley type $H^1$ norm (0.5) for scalar valued functions, but not for operator-valued functions [32]. Applying Theorem 2.3 to the Euclidean length on the integer group, we see that these two norms coincide for lacunary Fourier series with operator valued coefficients.

Given a $\psi$-lacunary sequence $(h_{k})_{k = 1}^{\infty }$ of elements in G, define the linear map T from $L^\infty (B(H),\ell _{ cr}^2)$ to BMO by

$$\begin{aligned} T[(c_k)_{k = 1}^{\infty }]=\sum _k c_k\lambda _{h_k}. \end{aligned}$$

Then T has a norm $c_\delta ^{\prime }$ from $L^\infty (B(H),\ell ^2_{cr})$ to BMO and norm 1 from $L^2(B(H),\ell _{cr}^2)$ to $L^2(B(H)\otimes \hat{G})$, where $c_{\delta }^{\prime }\le c_{\delta }^{\frac{1}{2}}$ from the proof of Theorem 2.3. Applying the interpolation result Lemma 1.2, we get

Corollary 2.5

Suppose that $(h_k)$ is a $\psi$-lacunary sequence for some conditionally negative definite $\psi$. We have that, for any $p>2$ and any f of the form $f=\sum _kc_k\lambda _ {h_k}$,

$$\begin{aligned} \Vert f\Vert _{L^p}\lesssim _{\delta } p\max \left\{ \left\| \left( \sum _k |c_k|^2 \right) ^\frac{1}{2} \right\| _{ p}, \left\| \left( \sum _k |c_k^*|^2 \right) ^\frac{1}{2} \right\| _{ p} \right\} . \end{aligned}$$

(2.10)

By duality, we get the following result: For any $1<p<2$,

$$\begin{aligned} \Vert (c_k)_k\Vert _{S^p(\ell _{cr}^2)} \lesssim \inf \{\Vert f\Vert _p: \hat{f}(h_k)=c_k\}. \end{aligned}$$

(2.11)

We will prove a column version of (2.11) in the next section.

Remark 2.6

Corollary 2.5 can also be obtained by combining the noncommutative $H^\infty$-calculus techniques developed in [19, p. 118], and the dilation theory proved in [46]. There is another approach via noncommutative Riesz transforms [22]. The order p in (2.10) is better than what is implied by these two approaches. For $\psi$ being the usual length function on the integer group, (2.10) holds with constants in the order of $\sqrt{p}$ [47] as $p\rightarrow \infty$, it is unclear whether this is the same for general $\psi$-lacunary sequences.

Remark 2.7

If $G={\mathbb {F}}_n$ and $\psi$ is the reduced word length, Haagerup proved in [13] that $\psi$ is a conditionally negative function. For this $\psi$, it is easy to verify that a set consisting of a $\psi$-lacunary sequence is the so-called Z(2) set [16, Definition 1.11]. So it is a $\Lambda (4)$ set by Harcharras’s work [16, Theorem 1.13]. This does not seem clear for Z(p) with $p>2$.

Remark 2.8

The sequence of free generators $\{g_i : i\in {\mathbb {N}}\}$ of ${\mathbb {F}}_\infty$ is a $\psi$-lacunary sequence for some $\psi$. Indeed, let $\pi$ be the group homomorphism from ${\mathbb {F}}_\infty$ to ${\mathbb {F}}_\infty$ sending $g_i$ to $g_{i}^{2^i}$. Then $\psi (h)=|\pi (h)|$ is a conditionally negative definite function. Because, the matrix

$$\begin{aligned} (|\pi (h_k^{-1}h_j)|)_{1\le k, j\le N} =(|(\pi (h_k))^{-1}\pi (h_j)|)_{1\le k, j\le N} \end{aligned}$$

is conditionally negative definite for any $h_k\in {\mathbb {F}}_\infty$ since the reduced word length is conditionally negative definite [13]. Note $\{g_i : i\in {\mathbb {N}}\}$ is obviously $\psi$-lacunary because $\psi (g_i)=2^i$.

Remark 2.9

One can extend (2.11) to the range $0<p\le 1$ as a Khintchine-type inequality

$$\begin{aligned} \Vert (c_k)_k\Vert _{S^p(\ell _{cr}^2)} \lesssim \left\| \sum _k c_k \lambda _{h_k} \right\| _{\frac{p}{2}}^2 \end{aligned}$$

(2.12)

by applying Pisier–Ricard’s theorem [45]. For the case $p=1$, one may follow Haagerup–Musat’s argument in [14] to get a better constant in (2.12).

Remark 2.10

The assumption of $\psi$ being conditionally negative definite is needed merely by the interpolation result Lemma 1.2. The arguments for other results of this section only need the assumptions (0.7), (0.8).

Remark 2.11

Let $P_t$ be the Poisson semigroup for bounded functions on the torus ${\mathbb {T}}$. As we pointed out before, the semigroup BMO associated with $P_t$ coincides with the classical BMO. Let $S_t=P_t\otimes id_{M_n}$ be its extension to the von Neumann algebra of bounded n by n matrix-valued functions on ${\mathbb {T}}$. The semigroup BMO$_c$ associated with $S_t$ coincides with the matrix-valued BMO$_{so}$ introduced in the literature (e.g. [32, 35]). Note that for the dyadic BMO$_{so}$-norm, the example of $f=\sum r_k\otimes e_{k,1}$, with $r_k=sgn \left[ \sin \left( 2^k\pi \theta \right) \right]$ being the k-th Rademacher function on ${\mathbb {T}}$ (which is identified by [0, 1]), shows that the estimate

$$\begin{aligned} \Vert f\Vert _{BMO^d_{so}}\lesssim {\sqrt{n} }\Vert f^*\Vert _{BMO^d_{so}} \end{aligned}$$

is optimal. The upper bound of the estimate is of the order at most $\sqrt{n}$ owing to the inequality

$$\begin{aligned} \left\| \sum _k|c_k|^2 \right\| \le \mathrm{tr}\left( \sum _k|c_k^*|^2 \right) \le n \left\| \sum _k|c_k^*|^2 \right\| \end{aligned}$$

which holds true for any sequence of n by n matrix $c_{k}$. There was no easy method to show that $\sqrt{n}$ is also optimal for the usual (non-dyadic) BMO$_{so}$ norm before the writing of this article. Note that the BMO by dyadic BMO trick does not help in producing such a concrete example. Theorem 2.3 provides such an example by taking $f=\sum _{0<k\le n} z^{2^k}\otimes e_{k,1}$.

4 Proof of Theorem 0.2

Throughout this section, we assume $(G, \le )$ is a countable (possibly non-abelian) discrete group with a bi-invariant total order. This is equivalent to say that G contains a normal subsemigroup $G_+$ such that, for $G_-=(G_+)^{-1}$,

$$\begin{aligned} G_+ \cup G_{-}=G, G_{+} \cap G_{-}=\{e\}. \end{aligned}$$

In this case, one has $G_+=\{g\in G : g\ge e\}$ and $x\le y$ if and only if $x^{-1}y\in G_+.$ It is well-known that the free groups have bi-invariant total orders (see Sect. 4). We use the notation $x<y$ if $x\le y$ and $x\ne y$.

For each $g\in G_+$, let $L_g=\{h: g\le h\le g^2\}$. For $E\subset G_+$, let N(E, g) be the number of elements of $E\cap L_g$, i.e. $N(E, g)=\#(L_g\cap E)$. Following Rudin’s terminology [48, Section 8.6], we say $E\subset G_+$ is lacunary, if there is a constant K such that

$$\begin{aligned} N(E)=\sup _{g\in G_+} N(E,g)\le K. \end{aligned}$$

For a general subset $E\subset G$, let $E_+=E\cap G_+, E_-=E-E_+$. We say E is lacunary if $N(E)=N(E_+)+N((E_-)^{-1})$ is finite. Please see Sect. 4 for examples of such E.

Let $({\mathcal {M}},tr)$ be a semifinite von Neumann algebra. Let ${\mathcal {N}}={\mathcal {M}}\overline{\otimes } \hat{G}$ with the trace $tr\otimes \tau$. For $f\in L^1({\mathcal {N}}), g\in G$, denote by $\hat{f}(g)=\tau [f (1\otimes \lambda _{g^{-1}})]$. It is clear that, for a finite sum $f=\sum _g c_g\lambda _g$, $\hat{f}(g)=c_g$.

Theorem 3.1

Assume that E is a lacunary subset of $G_+$. Then, for any sequence $(c_k)_k\subset L^1({\mathcal {M}})$, and any sequence $(g_k)_{k=1}^\infty \subseteq E$, we have

$$\begin{aligned}&\Vert (c_k)_{k = 1}^{\infty } \Vert _{L^1({\mathcal {M}},\ell ^2_{cr})} \nonumber \\&\quad \simeq \inf \left\{ (tr\otimes \tau ) (|f|) : f\in L^1({\mathcal {N}}), \hat{f}(g_k)=c_k, \hat{f}(g)=0, \forall g<e \right\} ,\qquad \quad \end{aligned}$$

(3.1)

Proof

By the convexity of $|\cdot |^2$ and the complete positivity of $\tau$, we have that, for any finite sequence $g_k\in G$,

$$\begin{aligned} \tau |\sum _k a_k\lambda _{g_k}|\le \left( \tau |\sum _k a_k\lambda _{g_k}|^2\right) ^\frac{1}{2} =\left( \sum _k |a_k|^2\right) ^\frac{1}{2}. \end{aligned}$$

So

$$\begin{aligned} \left\| \sum _k a_k\lambda _{g_k}\right\| _{1}=tr\left( \tau \left| \sum _k a_k\lambda _{g_k}\right| \right) \le tr\left( \sum _k |a_k|^2\right) ^\frac{1}{2}=\Vert (a_k)_{k = 1}^{\infty } \Vert _{L^1({\mathcal {M}},\ell ^2_{c})}. \end{aligned}$$

Taking the adjoints, we get

$$\begin{aligned} \left\| \sum _k b_k\lambda _{g_k}\right\| _{1}\le \Vert (b_k)_{k = 1}^{\infty } \Vert _{L^1({\mathcal {M}},\ell ^2_{r})}. \end{aligned}$$

Writing $c_k=a_k+b_k$, we get,

$$\begin{aligned} \left\| \sum _k c_k\lambda _{g_k}\right\| _{1}\le \Vert (c_k)_{k = 1}^{\infty } \Vert _{L^1({\mathcal {M}},\ell ^2_{cr})}. \end{aligned}$$

(3.2)

This shows that the right hand side of (3.1) is smaller.

We now prove the other direction of the equivalence. By approximation, without loss of generality, we may assume E is a finite set and ${\mathcal {N}}$ is a finite von Neumann algebra. For $f\in H^1({\mathcal {N}})$ and $\varepsilon >0$, by Lemma  1.3, there exist $y, z\in H^2({\mathcal {N}})$ such that $f=yz$ and $\Vert y\Vert _2\Vert z\Vert _2\le \Vert f\Vert _1+\varepsilon$.

Given an element $g_i \in E$ with $\widehat{f}(g_i)\ne 0$. Recall that $\hat{f}(g)=\tau (f\lambda _g^*)$, we have

$$\begin{aligned} \widehat{f}(g_i)= \sum _{e\le h\le g_i}\widehat{y}(h)\widehat{z}(h^{-1}g_i), \end{aligned}$$

(3.3)

since $\widehat{y}(g)=\widehat{z}(g)=0$ for all $g<e$.

Thus, the sum in (3.3) can be split into two parts;

$$\begin{aligned} \widehat{f}(g_i)=\sum _{e\le h\le g_i < h^2}\widehat{y}(h)\widehat{z}(h^{-1}g_i) +\sum _{e\le h\le h^{2}\le g_i}\widehat{y}(h)\widehat{z}(h^{-1}g_i). \end{aligned}$$

(3.4)

Let

$$\begin{aligned} Z_i=\sum _{e\le h\le h^{2}< g_i} \widehat{z}(h)\lambda _h =\sum _{e\le h\le g_i< h^2} \widehat{z}(h^{-1}g_i) \lambda _{h^{-1}g_i} . \end{aligned}$$

Similarly let

$$\begin{aligned} Y_i=\sum _{e\le h\le h^{2}\le g_i} \widehat{y}(h)\lambda _{h}. \end{aligned}$$

It follows that

$$\begin{aligned} Z_i\lambda _{g_i^{-1}} =\sum _{e\le h\le g_i < h^2} \widehat{z}(h^{-1}g_i)\lambda _{h^{-1}} \end{aligned}$$

and

$$\begin{aligned} \lambda _{g_i^{-1}} Y_i=\sum _{e\le h\le h^{2}\le g_i } \widehat{y}(h)\lambda _{g_i^{-1}h}. \end{aligned}$$

Let

$$\begin{aligned} A_i&:=\tau \left( yZ_i \lambda _{g_i^{-1}} \right) \end{aligned}$$

(3.5)

$$\begin{aligned} B_i&:=\tau \left( \lambda _{g_i^{-1}} Y_iz \right) . \end{aligned}$$

(3.6)

Then

$$\begin{aligned} \hat{f}(g_i)=A_i+B_i, \end{aligned}$$

(3.7)

because

$$\begin{aligned} A_i&=\tau \left( yZ_i \lambda _{g_i^{-1}} \right) \\ {}&=\tau \left[ \left( \sum _{g\ge e} \widehat{y}(g)\lambda _g \right) \left( \sum _{e\le h\le g_i< h^2} \widehat{z}(h^{-1}g_i)\lambda _{h^{-1}} \right) \right] \\ {}&= \tau \left[ \sum _{g\ge e}\sum _{e\le h\le g_i< h^2} \widehat{y}(g)\widehat{z}(h^{-1}g_i)\lambda _{gh^{-1}} \right] \\ {}&=\sum _{e\le h\le g_i< h^2} \widehat{y}(h)\widehat{z}(h^{-1}g_i) \end{aligned}$$

and

$$\begin{aligned} B_i= & {} {} \tau \left( \lambda _{g_i^{-1}} Y_iz \right) \\ {}= & {} {} \tau \left[ \left( \sum _{e\le h\le h^{2}\le g_i} \widehat{y}(h) \lambda _{g_i^{-1}h}\right) \left( \sum _{g\ge e} \widehat{z}(g)\lambda _g \right) \right] \\ {}= & {} {} \tau \left[ \left( \sum _{e\le h\le h^{2}\le g_i} \widehat{y}(h)\lambda _{g_i^{-1}h} \right) \left( \sum _{f\le g_i}\widehat{z}(f^{-1}g_i)\lambda _{f^{-1}g_i} \right) \right] \\ {}= & {} {} \tau \left[ \sum _{e\le h\le h^{2}\le g_i }\sum _{f\le g_i} \widehat{y}(h) \widehat{z}(f^{-1}g_{i})\lambda _{g_i^{-1}h}\lambda _{f^{-1}g_i} \right] \\ {}= & {} {} \sum _{e\le h\le h^{2}\le g_i } \widehat{y}(h)\widehat{z}(h^{-1}g_i). \end{aligned}$$

Applying the convexity of $\tau$ and Jensen’s inequality to (3.5), we have

$$\begin{aligned} \left\| (A_i)_{i = 1}^{n} \right\| _{L^1({\mathcal {M}}, \ell _r^2)}\le & {} {} (tr \otimes \tau ) \left[ \left( \sum _{i}| (yZ_i\lambda _{g_i^{-1}})^*|^2 \right) ^{\frac{1}{2}} \right] \\= & {} {} ( tr \otimes \tau ) \left[ \left( y \left( \sum _{i}Z_iZ_i \right) y^* \right) ^{\frac{1}{2}} \right] \\ {}\le & {} {} \left[ ( tr \otimes \tau ) (|y|^2) \right] ^{\frac{1}{2}} \left[ (tr \otimes \tau ) \left( \sum _{i}Z_iZ_i^* \right) \right] ^{\frac{1}{2}}\\= & {} {} \left[ ( tr \otimes \tau ) (|y|^2) \right] ^{\frac{1}{2}} \left[ \sum _{i}\sum _{e\le h\le g_i< h^2}\Vert \widehat{z}(h)\Vert _{L^2 ({\mathcal {M}})}^2 \right] ^{\frac{1}{2}}.\end{aligned}$$

On the other hand, we note that $e\le h\le g_i<h^2$ implies that $g_i\in L_{h}$. Since $N(E, g)\le K$, we get

$$\begin{aligned} \Vert (A_i)_{i = 1}^n\Vert _{L^1({\mathcal {M}}, \ell _r^2)}\le & {} {} \left[ (tr \otimes \tau ) (|y|^2) \right] ^{\frac{1}{2}} \left[ K\sum _{h}\Vert \widehat{z}(h)\Vert _{L^2({\mathcal {M}})}^2 \right] ^{\frac{1}{2}}\\= & {} {} K^\frac{1}{2} \Vert z\Vert _{L^2({\mathcal {N}})}\Vert y\Vert _{L^2({\mathcal {N}})} \le K^\frac{1}{2}(\Vert f\Vert _{L^1({\mathcal {N}})}+\varepsilon ). \end{aligned}$$

Now, we consider $(B_i)_i$.

$$\begin{aligned} \Vert (B_i)_{i = 1}^n\Vert _{L^1({\mathcal {M}}, \ell _c^2)}\le & {} {} (tr\otimes \tau ) \left[ \left( z^* \left( \sum _{i}|\lambda _{g_i^{-1}}Y_i|^2 \right) z \right) ^{\frac{1}{2}} \right] \\ {}\le & {} {} \left[ (tr \otimes \tau ) (|z|^2)) \right] ^{\frac{1}{2}} \left[ ( tr \otimes \tau ) \left( \sum _{i}|\lambda _{g_i^{-1}}Y_i|^2 \right) \right] ^{\frac{1}{2}}\\= & {} {} \Vert z\Vert _2 \left[ \sum _{i} (tr \otimes \tau ) (Y^*_iY_i) \right] ^{\frac{1}{2}}\\= & {} {} \Vert z\Vert _2 \left[ \sum _{i}\sum _{e\le h\le h^{2}\le g_i}\Vert \widehat{y}(h)\Vert _{L^2({\mathcal {M}})}^2 \right] ^{\frac{1}{2}}. \end{aligned}$$

Note that the condition $e\le h\le h^2\le g_i$ implies

$$\begin{aligned} h^{-1}g_i\le g_i\le hg_i<g_ih^{-1}g_i \end{aligned}$$

and

$$\begin{aligned} h' \le g_i\le h'^2 \end{aligned}$$

with $h'= {h^{-1} g_i}\ge e$ because “$\le$” is bi-invariant. We then get

$$\begin{aligned} \sum _{e\le h\le h^{2}\le g_i}\Vert \widehat{y}(h)\Vert _{L^2({\mathcal {M}})}^2\le \sum _{e\le h' \le g_i\le h'^2}\Vert \widehat{y}(h')\Vert _{L^2({\mathcal {M}})}^{2}. \end{aligned}$$

By the lacunary assumption $N(E)\le K$, we get

$$\begin{aligned} \Vert (B_i)_1^n\Vert _{L^1({\mathcal {M}}, \ell _c^2)}\le & {} \Vert z\Vert _2 \left[ \sum _{i}\sum _{e\le h'\le g_i\le h'^{2}}\Vert \widehat{y}(h')\Vert _{L^2({\mathcal {M}})}^2 \right] ^{\frac{1}{2}}\\\le & {} K^\frac{1}{2} \Vert z\Vert _{L^2({\mathcal {N}})}\Vert y\Vert _{L^2({\mathcal {N}})} \le K^\frac{1}{2}(\Vert f\Vert _{L^1({\mathcal {N}})}+\varepsilon ). \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert (\widehat{f}(g_i))_{i = 1}^n\Vert _{L^1({\mathcal {M}}, \ell _{cr}^2)}\le & {} \Vert (B_i)_{i =1}^n\Vert _{L^1({\mathcal {M}}, \ell _c^2)} +\Vert (A_i)_{i = 1}^n\Vert _{L^1({\mathcal {M}}, \ell _r^2)}\\\le & {} 2K^\frac{1}{2}(\Vert f\Vert _{L^1({\mathcal {N}})}+\varepsilon ), \end{aligned}$$

This completes the proof by letting $\varepsilon \rightarrow 0$. $\square$

Corollary 3.2

Assume that E is a lacunary subset of $G_+$. Then, for any sequence $(c_k)_k\subset L^p({\mathcal {M}})$, and any sequence $(g_k)_{k=1}^\infty \subseteq E$, we have

$$\begin{aligned} \Vert (c_k)_{k = 1}^{\infty } \Vert _{L^\infty ({\mathcal {M}},\ell ^2_{cr})}\simeq & {} \left\| \sum _{k=1}^\infty c_k\lambda _{g_k}\right\| _{BMO({\mathcal {N}})}, \end{aligned}$$

(3.8)

$$\begin{aligned} \Vert (c_k)_{k = 1}^{\infty } \Vert _{L^1({\mathcal {M}},\ell ^2_{cr})}\simeq & {} \left\| \sum _{k=1}^\infty c_k\lambda _{g_k}\right\| _{1}, \end{aligned}$$

(3.9)

$$\begin{aligned} \Vert (c_k)_{k = 1}^{\infty } \Vert _{L^p({\mathcal {M}},\ell ^2_{cr})}\simeq & {} \left\| \sum _{k=1}^\infty c_k\lambda _{g_k}\right\| _{p}, 1<p<\infty . \end{aligned}$$

(3.10)

$$\begin{aligned} \Vert (c_k)_{k = 1}^{\infty } \Vert _{L^p({\mathcal {M}},\ell ^2_{cr})}\simeq & {} \inf \left\{ \Vert f\Vert _p:f\in {L^p({\mathcal {N}})} ,\hat{f}(g_k)=c_k \right\} , 1\le p\le 2,\nonumber \\ \end{aligned}$$

(3.11)

Proof

The first equivalence follows from Theorem 3.1 and the duality. The second equivalence follows from Theorem 3.1 and the inequality (3.2). The third equivalence follows from the first two equivalences and the interpolation between $H^1({\mathcal {N}})$ and BMOA$({\mathcal {N}})$. For the forth equivalence, the case $p=2$ is trivial, the case $p=1$ follows from Theorem 3.1, the case $1<p<2$ follows from the interpolation between $H^1({\mathcal {N}})$ and $H^2({\mathcal {N}})$. $\square$

Corollary 3.3

For any sequence $(g_i)_{i=1}^\infty$ in a lacunary subset $E\in G$

$$\begin{aligned} \left\| \sum _{i=1}^\infty \lambda _{g_i}\otimes c_{g_i} \right\| _p\simeq & {} \Vert (c_{g_i})_{i = 1}^\infty \Vert _{L^p({\mathcal {M}}, \ell _{cr}^2)}, \quad 0<p<\infty . \end{aligned}$$

(3.12)

$$\begin{aligned} \Vert (c_{g_i})_{i = 1}^{\infty } \Vert _{L^p({\mathcal {M}},\ell ^2_{cr})}\simeq & {} \inf \left\{ \Vert f\Vert _p: f\in {L^p({\mathcal {N}})} , \hat{f}(g_i)=c_{g_i} \right\} , 1<p<\infty .\nonumber \\ \end{aligned}$$

(3.13)

Proof

Note $g_i\in G_-$ implies $g_i^{-1}\in G_+$. Taking adjoints, we see that all the equivalences in Corollary 3.2 also hold for lacunary subsets $E\subset G_-$ instead of $G_+$.

We write $E=E_+\cup E_-$ with $E_+=\{g\in E, g\ge e\}\subset G_+$ and $E_-=\{g\in E, g< e\}\subset G_-$. By (1.9) and (3.10), we have, for $1<p<\infty$,

$$\begin{aligned} \left\| \sum _{g_i\in E} c_{g_i}\otimes \lambda _{g_i}\right\| _{p}\simeq & {} \left\| \sum _{g_i\in E_+} c_{g_i}\otimes \lambda _{g_i}\right\| _p +\left\| \sum _{g_i\in E_-} c_{g_i}\otimes \lambda _{g_i}\right\| _p\\\simeq & {} \Vert (c_{g_i})_{g_i\in E_+}\Vert _{L^p({\mathcal {M}}, \ell _{cr}^2)} +\Vert (c_{g_i})_{g_i\in E_-}\Vert _{L^p({\mathcal {M}}, \ell _{cr}^2)}\\= & {} \Vert (c_{g_i})_{g_i\in E}\Vert _{L^p({\mathcal {M}}, \ell _{cr}^2)}. \end{aligned}$$

This proves (3.12) for the case $1<p<\infty$. The case where $0<p\le 1$ follows from the case $1<p<\infty$ and [45, Corollary 2.2] and [7, Theorem 2.6].

(3.12) implies that the right hand side of (3.13) is dominated by its left hand side up to a constant. We now prove the other direction of (3.13). Given $f=\sum _g \hat{f}(g)\otimes \lambda _g\in L^p({\mathcal {N}}), p\ge 2$, we have

$$\begin{aligned} \left( \sum _g |\hat{f}(g)|^2\right) ^\frac{1}{2}=(\tau |f|^2)^\frac{1}{2} \le (\tau |f|^p)^{\frac{1}{p}}. \end{aligned}$$

So, if $\hat{f}({g_i})=c_{g_i}$,

$$\begin{aligned} \Vert (c_{g_i})_{g_i\in E}\Vert ^p_{L^p({\mathcal {M}}, \ell _{c}^2)}\!=\! tr \left( \sum _{g_i} |c_{g_i}|^2\right) ^\frac{p}{2} \!\le tr \left( \sum _g |\hat{f}(g)|^2\right) ^\frac{p}{2} \!\le tr \otimes \tau (|f|^p)\!=\! \Vert f\Vert ^p_{p}. \end{aligned}$$

Taking adjoints, we get $\Vert (c_{g_i})_{g_i\in E} \Vert ^p_{L^p({\mathcal {M}}, \ell _{r}^2)}\le \Vert f\Vert ^p_{p}$. Therefore,

$$\begin{aligned} \Vert (c_{g_i})_{g_i\in E}\Vert ^p_{L^p({\mathcal {M}}, \ell _{cr}^2)}\le \Vert f\Vert ^p_{p}. \end{aligned}$$

Given $f=\sum _g \hat{f}(g)\otimes \lambda _g\in L^p({\mathcal {N}}), 1<p< 2$ with $\hat{f}_{g_i}=c_{g_i}$, we have, by (3.11) and its adjoint version,

$$\begin{aligned} \Vert (c_{g_i})_{g_i\in E}\Vert _{L^p({\mathcal {M}}, \ell _{cr}^2)}= & {} \Vert (c_{g_i})_{g_i\in E_+}\Vert _{L^p({\mathcal {M}}, \ell _{cr}^2)} +\Vert (c_{g_i})_{g_i\in E_-}\Vert _{L^p({\mathcal {M}}, \ell _{cr}^2)}\\\lesssim & {} \left\| \sum _{g\ge e } \hat{f}(g) \otimes \lambda _{g}\right\| _p +\left\| \sum _{g <e} \hat{f}({g })\otimes \lambda _{g }\right\| _p\\\simeq & {} \Vert f\Vert _{p} \end{aligned}$$

In the last step, we have applied (1.9). This completes the proof of (3.13). $\square$

Remark 3.4

It would be interesting to see whether Pisier–Ricard’s argument [45] can push Theorem 3.1 to the case where $p<1$.

5 The Case of Free Groups

Let $G={\mathbb {F}}_2$ be the nonabelian group with two free generators a, b. Denote by |g| the reduced word length of $g\in {\mathbb {F}}_2$. Every $g \in {\mathbb {F}}_2$ can be uniquely expressed as

$$\begin{aligned} g=a^{j_1}b^{k_1}\cdots a^{j_{N}}b^{k_{N}} \end{aligned}$$

(4.1)

with $j_i,k_i\in {\mathbb {Z}}$ and $j_i \ne 0$ for $1<i\le N$ and $k_i\ne 0$ for $1\le i< N.$ For $0<q\le 2$, set the q-length of g to be

$$\begin{aligned} \Vert g\Vert _q=\sum _i |j_i| ^q+\sum _{i}|k_i|^q. \end{aligned}$$

(4.2)

Then, $\psi : {\mathbb {F}}_{2} \rightarrow {\mathbb {R}}$ defined by $\psi (g)= \Vert g\Vert _q$ is a conditionally negative definite function for all $0<q\le 2$. When $q=1$, $\Vert g\Vert _q$ is the reduced word length function. The property of it being conditional negative definite was studied in [13] for $q=1$ and in [6, Corollary 1] for $0<q\le 2$. All results contained in Sect. 2, 3 apply to this $\psi$. In particular, all $\Vert \cdot \Vert _q$-lacunary sequences are completely unconditional in $L^p(\hat{\mathbb {F}}_2)$ for all $0<p<\infty$. However, this is not clear for $p=\infty$.

Given a conditionally negative definite length $\psi$ with $\mathrm{\mathrm{ker}}(\psi ) =\{e\}$, we say a subset $A \subseteq G$ is a (respectively complete) $\psi$ Paley-set, if there exists a constant $C_A$ such that

$$\begin{aligned} \left\| \sum _{h_k\in A} c_k\lambda _{h_k} \right\| _{BMO(\psi )} \le C_A \max \left\{ \left\| \sum _{h_k\in A} |c_k|^2 \right\| ^\frac{1}{2}, \left\| \sum _{h_k\in A} |c_k^*|^2 \right\| ^\frac{1}{2} \right\} , \end{aligned}$$

(4.3)

for any choice of finitely many $c_k\in {\mathbb {C}}$ (respectively K(H)). We say A is a Paley-set if it is a $\psi$ Paley-set for some conditionally negative definite length $\psi$ with $\mathrm{ker} (\psi ) =\{e\}$. This definition coincides with the classical “Paley”-set, when $G={\mathbb {Z}}$, and $\psi$ is the word length on ${\mathbb {Z}}$. In that case, every Paley set is a Sidon set. One may wonder to what extent this is still true. The concept of Sidon sets had been studied in the noncommutative setting for a long time, and was recently re-investigated by Pisier [41, 42] and by Wang [50]. Pisier defines noncommutative completely Sidon sets using the full $C^*$-algebras of discrete groups and proves the stability of his completely Sidon sets by taking finite unions. An interesting feature of Pisier’s definition is that only non-amenable groups can have infinite completely Sidon sets. The authors wish to consider a weaker definition in the hope of covering the case of lacunary sequence studied in this article.

We say a subset $A\in G$ is a (completely) unconditional Sidon set, if $\{\lambda _h : h\in A\}$ is (completely) unconditional in the reduced $C^*$-algebra of the group, i.e. there exists a constant $C_A$ such that

$$\begin{aligned} \left\| \sum _{h_k\in A} \varepsilon _k c_k\lambda _{h_k} \right\| \le C_A \left\| \sum _{h_k\in A} c_k\lambda _{h_k} \right\| , \end{aligned}$$

for any choice $\varepsilon _k=\pm 1$, $c_k\in {\mathbb {C}}$ (respectively K(H)). In the case that $G={{\mathbb {F}}}_2$ and $\psi$ being the reduced word length (or q-length defined in (4.2)), every length-lacunary set is a Paley set and a completely $\Lambda (p)$ set for all $2<p<\infty$ as shown in this article. The question is as follows:

Question Suppose that $(h_{k})_{k = 1}^{\infty }$ is a length-lacunary sequence of elements in ${\mathbb {F}}_{2}$, e.g. $\frac{|h_{k+1}|}{|h_k|}>2$. Is $\{h_k\}$ a (completely) unconditional Sidon set? In other words, does there exist a constant C such that

$$\begin{aligned} \left\| \sum _{k} \varepsilon _k c_k\lambda _{h_k} \right\| \le C \left\| \sum _{k} c_k\lambda _{h_k} \right\| , \end{aligned}$$

for any choice $\varepsilon _k=\pm 1$ and $c_k\in {\mathbb {C}}$ (respectively K(H))?

The transference method used in the work [10] is powerful for the study of harmonic analysis on the quantum tori. A similar method applies to the free group case. For $g\in {\mathbb {F}}_2$ in the form of (4.1), let

$$\begin{aligned} |g|_z= \left| \sum _{i=1}^N j_i \right| ^2+ \left| \sum _{i=1}^N k_i \right| ^2. \end{aligned}$$

Then

$$\begin{aligned} \psi _z:g\mapsto |g|_z \end{aligned}$$

(4.4)

is another conditionally negative definite function on ${\mathbb {F}}_2,$² and the unbounded linear operator $L_z:~\lambda _g\mapsto \psi _z\lambda _g$ generates a symmetric Markov semigroup on the free group von Neumann algebra $\hat{\mathbb {F}}_2$. For $(z_1,z_2) \in {\mathbb {T}}^2$, let $\pi _z$ be the $*$-homomorphism on $\hat{\mathbb {F}}_2$ such that

$$\begin{aligned} \pi _z(\lambda _a)=z_1\lambda _a,\quad \pi _z(\lambda _b)=z_2\lambda _b. \end{aligned}$$

Given $f\in \hat{\mathbb {F}}_2$, viewing $\pi _z(f)$ as an operator valued function on ${\mathbb {T}}^2$, one can see that

$$\begin{aligned} \pi _z^{-1}(\Delta \otimes id)\pi _z(f) =L_z(f), \end{aligned}$$

(4.5)

with $\Delta$ being the Laplacian on ${\mathbb {T}}^2$. This identity allows one to transfer classical results to free groups with $L_z$ taking the role of the Laplacian, including the corresponding Paley’s inequality proved in this article. The disadvantage is that this transference method cannot produce any helpful information on the large subgroup $ker (\psi _z)$. We will show that the second part of this paper implies a Paley’s theory on $ker (\psi _z)$.

Let us first recall a bi-invariant order on free groups ${\mathbb {F}}_2$. For notational convenience, we denote the free generators by a, b. We define the ring $\Lambda = {\mathbb {Z}}[ A, B]$ to be the ring of formal power series in the non-commuting variables A and B. Let $\mu$ be the group homomorphism from ${\mathbb {F}}_2$ to the group generated by $\{1+A,1+B\}$ in $\Lambda$ such that:

$$\begin{aligned}&\mu (a)=1+ A, \ \mu (a^{-1})=1-A+A^2-A^3+\cdots , \\&\mu (b)=1+B,\ \mu (b^{-1})=1-B+B^2-B^3+\cdots . \end{aligned}$$

Then $\mu$ is injective. Denote by “$\le$” the dictionary order on $\Lambda$ assuming $0\le B\le A$. To be precise, write the element of $\Lambda$ in a standard form, with lower degree terms preceding higher degree terms, and within a given degree, list the terms in the sequence according to the dictionary ordering assuming $0\le B\le A$. Compare two elements of $\Lambda$ by writing them both in standard form and order them according to the natural ordering of the coefficients in the first term at which they differ. We then formally define the ordering on the free group ${\mathbb {F}}_2$ by setting

$$\begin{aligned} g\le h~\text{ in }~{\mathbb {F}}_2{\ \ \mathrm{if}\ \ } \mu (g)\le \mu (h)~\text{ in }~\Lambda . \end{aligned}$$

This biinvariant order was introduced by Vinogradov [11, 49]. But the corresponding normal semigroup is not finitely generated [17].

Let $J_A(g)$ (respectively $J_B(g)$) be the coefficient of the A term (respectively B term) in $\mu (g)$; and $J_{AB}(g)$ (respectively $J_{BA}(g))$ be the coefficient of the AB term (respectively BA term) in $\mu (g)$. More generally, for any word X of A, B, denote by $J_X(g)$ the coefficient of the X term in $\mu (g)$. Note that $J_A(g)=J_A(a^{J_A(g)})$, and $J_B(g)=J_B(b^{J_B(g)})$. For $g\in {\mathbb {F}}_2$ in the form of (4.1), that is

$$\begin{aligned} g=a^{j_1}b^{k_1}\cdots a^{j_{N}}b^{k_{N}} \end{aligned}$$

(4.6)

with $j_i,k_i\in {\mathbb {Z}}$ and $j_i \ne 0$ for $1<i\le N$ and $k_i\ne 0$ for $1\le i< N,$ we get by direct computations,

$$\begin{aligned} J_A(g)&= \sum _{s=1}^N j_s, \quad J_B(g) =\sum _{s=1}^N k_s, \end{aligned}$$

(4.7)

$$\begin{aligned} J_{AB}(g)&=\sum _{1\le s\le t \le N} j_s k_t, \quad J_{BA}(g) =\sum _{1\le t<s \le N} j_s k_t. \end{aligned}$$

(4.8)

From (4.7), (4.8), we see that

$$\begin{aligned} J_{AB}(g)+J_{BA}(g)=J_A(g)J_B(g). \end{aligned}$$

Using that $\mu$ is a group homomorphism, we have

$$\begin{aligned}&J_A(gh)=J_A(g)+J_A(h), \end{aligned}$$

(4.9)

$$\begin{aligned}&J_{AB}(gh)=J_A(g)J_B(h)+J_{AB} (g)+J_{AB}(h), \end{aligned}$$

(4.10)

Let

$$\begin{aligned} {\mathbb {F}}_2^0= & {} \text {ker} (\psi _z) =\{g\in {\mathbb {F}}_2 : J_A(g)=J_B(g)=0\},\\ {\mathbb {F}}_2^{00}= & {} \{g\in {\mathbb {F}}^0_2 : J_{AB}(g)=0\}=\{g\in {\mathbb {F}}^0_2 :J_{AB}(g)=J_{BA}(g)=0\}. \end{aligned}$$

Then, ${\mathbb {F}}_2^0, {\mathbb {F}}_2^{00}$ are subgroups because of (4.9), (4.10), and ${\mathbb {F}}_2^0= \text {ker} (\psi _z)$ with $\psi _z$ defined in (4.4). For $g\in {\mathbb {F}}_2^0$, $g>e$ if $J_{AB}(g)>0$ since $J_{AA}(g)=0$. Recall that we say a sequence of $\ell _n\ne 0\in {\mathbb {Z}}$ is lacunary if there exists $\delta >1$ such that $\inf _n\frac{\ell _{n+1}}{\ell _{n}}\ge \delta .$ We then get the following property by definition.

Proposition 4.1

Given a sequence $g_n\in {\mathbb {F}}_2$, then $E=\{g_n : n\in {\mathbb {N}}\}$ is a lacunary subset of ${\mathbb {F}}_2$ if any of the following holds:

• The sequence $J_A(g_n)\in {\mathbb {Z}}$ is lacunary.

• $J_A(g_n)=0$ for all n and the sequence $J_B(g_n) \in {\mathbb {Z}}$ is lacunary.

• $J_A(g_n)=J_B(g_n)=0$ for all n, and $J_{AB}(g_n)$ is lacunary.

For instance, $\{a^{2^i}b^{k_i}\in {\mathbb {F}}_2: i, k_i\in {\mathbb {N}}_+\}$ and $\{a^{2^k}b^{2^k}a^{-2^k} b^{-2^k} : k\in {\mathbb {N}}\}$ are lacunary subsets of ${\mathbb {F}}_2$.

Remark 4.2

Corollary 3.3 implies that the sets E given in Proposition 4.1 are all completely $\Lambda (p)$ sets [16].

Corollary 4.3

Suppose $(g_k)_k\in {\mathbb {F}}^0_2$ is a sequence with $(J_{AB}(g_k))_k\in {\mathbb {Z}}$ lacunary. Then for any $(c_k)_{k}$ with elements in $S^p(H)$, we have

$$\begin{aligned} \Vert (c_k) \Vert ^p_{S^p(\ell _{cr}^2)}\simeq ( \mathrm {tr}\otimes \tau ) \left( \left| \sum _kc_k\otimes \lambda _{g_k} \right| ^p \right) \end{aligned}$$

(4.11)

for all $0<p<\infty$. Moreover, for $p=1$, we have

$$\begin{aligned} \Vert (c_k) \Vert _{S^1(\ell _{cr}^2)} \simeq \inf \left\{ (tr\otimes \tau ) \left( \left| \sum _{J_{AB}(g)\ge 0} \hat{f}(g)\otimes \lambda _{g} \right| +\left| \sum _{J_{AB}(g)<0} \hat{f}(g)\otimes \lambda _{g} \right| \right) \right\} \end{aligned}$$

(4.12)

Here, the infimum runs over all $f\in S^1(H)\otimes L^1( \hat{{\mathbb {F}}_2})$ with $\hat{f}(g_k)=c_k$.

Proof

(4.11) follows from Corollary 3.3. For (4.12), we only need to prove the relation “$\lesssim$” as the other direction is trivial. Since ${\mathbb {F}}_2^{0}$ and ${\mathbb {F}}_2^{00}$ are subgroups, the projection $P_0$ (and $P_{00}$) onto $L^1(\hat{\mathbb {F}_2^0})$ (and $L^1(\hat{\mathbb {F}_2^{00}})$) is completely contractive. Given $f\in S^1(H)\otimes L^1( \hat{{\mathbb {F}}_2})$ with $\hat{f}(g_k)=c_k$, let $y=P_{0}f-P_{00}f$. Then, we have $\hat{y}(g_k)=c_k$. By Theorem 3.1, we have

$$\begin{aligned} \begin{aligned}&\Vert (c_k)_{k = 1}^{n} \Vert _{S^1(\ell _{cr}^2)}\\ {}&\quad \lesssim ( tr \otimes \tau ) \left[ \left| \sum _{g\ge e} \hat{y}(g)\otimes \lambda _{g} \right| \right] +(tr \otimes \tau ) \left[ \left| \sum _{g<e} \hat{y}(g)\otimes \lambda _{g} \right| \right] \\ {}&\quad = (tr \otimes \tau ) \left[ \left| \sum _{g\in {\mathbb {F}}_2^0, J_{AB}(g)>0} \hat{f}(g)\otimes \lambda _{g} \right| \right] +(tr \otimes \tau ) \left[ \left| \sum _{g\in {\mathbb {F}}_2^0,J_{AB}(g)<0} \hat{f}(g)\otimes \lambda _{g}\right| \right] \\ {}&\quad \le (tr \otimes \tau ) \left[ \left| \sum _{g\in {\mathbb {F}}_2^0, J_{AB}(g)\ge 0} \hat{f}(g)\otimes \lambda _{g} \right| +|P_{00}f| + \left| \sum _{g\in {\mathbb {F}}_2^0,J_{AB}(g)<0} \hat{f}(g)\otimes \lambda _{g} \right| \right] \\ {}&\quad \le 2 (tr \otimes \tau ) \left[ \left| \sum _{g\in {\mathbb {F}}_2^0, J_{AB}(g)\ge 0} \hat{f}(g)\otimes \lambda _{g} \right| + \left| \sum _{g\in {\mathbb {F}}_2^0,J_{AB}(g)<0} \hat{f}(g)\otimes \lambda _{g}\right| \right] \\ {}&\quad \le 2 ( tr \otimes \tau ) \left[ \left| \sum _{g\in {\mathbb {F}}_2, J_{AB}(g)\ge 0} \hat{f}(g)\otimes \lambda _{g} \right| +\left| \sum _{g\in {\mathbb {F}}_2,J_{AB}(g)<0} \hat{f}(g)\otimes \lambda _{g} \right| \right] . \end{aligned} \end{aligned}$$

$\square$

Remark 4.4

The associated positive cone of any total left order (including the one introduced above) on free groups is NOT represented by a regular language [17]. This increases the mystery of the associated noncommutative Hardy spaces (norms). Corollary 4.3 shows that there are more transparent alternatives (e.g. (4.12) ) of the noncommutative real $H^1$-norm that may be used to formulate the corresponding Paley’s inequalities.

Remark 4.5

Interested readers are invited to prove a similar theory by computing $J_{AAB}(g)$.