# A Hörmander-Mikhlin multiplier theory for free groups and amalgamated free
products of von Neumann algebras

Tao Mei  Department of Mathematics, Baylor University, Waco, TX USA
<EMAIL_ADDRESS>,  Éric Ricard  Normandie Univ, UNICAEN, CNRS, LMNO, 14000
Caen, France<EMAIL_ADDRESS>and  Quanhua Xu  Laboratoire de
Mathématiques, Université de Bourgogne Franche-Comté, 25030 Besançon Cedex,
France<EMAIL_ADDRESS>
###### Abstract.

We establish a platform to transfer $L_{p}$-completely bounded maps on tensor
products of von Neumann algebras to $L_{p}$-completely bounded maps on the
corresponding amalgamated free products. As a consequence, we obtain a
Hörmander-Mikhlin multiplier theory for free products of groups. Let
$\mathbb{F}_{\infty}$ be a free group on infinite generators
$\\{g_{1},g_{2},\cdots\\}$. Given $d\geq 1$ and a bounded symbol $m$ on
$\mathbb{Z}^{d}$ satisfying the classical Hörmander-Mikhlin condition, the
linear map
$M_{m}:\mathbb{C}[\mathbb{F}_{\infty}]\to\mathbb{C}[\mathbb{F}_{\infty}]$
defined by $\lambda(g)\mapsto m(k_{1},\cdots,k_{d})\lambda(g)$ for
$g=g_{i_{1}}^{k_{1}}\cdots g_{i_{n}}^{k_{n}}\in\mathbb{F}_{\infty}$ in reduced
form (with $k_{l}=0$ in $m(k_{1},\cdots,k_{d})$ for $l>n$), extends to a
complete bounded map on $L_{p}(\widehat{\mathbb{F}}_{\infty})$ for all
$1<p<\infty$, where $\widehat{\mathbb{F}}_{\infty}$ is the group von Neumann
algebra of $\mathbb{F}_{\infty}$. A similar result holds for any free product
of discrete groups.

2000 Mathematics Subject Classification: Primary: 46L07, 46L50. Secondary:
46L52, 46L54.

Key words: Free groups, free products of groups, amalgamated free products of
von Neumann algebras, noncommutative $L_{p}$-spaces, Hörmander-Mikhlin
multipliers, free Fourier multipliers, completely bounded maps

##  1\. Introduction

Fourier multipliers are by far the most important operators in analysis. One
can say that classical harmonic analysis has been developed around Fourier
multipliers. The main task there is to find criteria for the boundedness of
Fourier multipliers on various function spaces, notably on $L_{p}$-spaces.
However, it is in general impossible to characterize their $L_{p}$-boundedness
for finite $p\neq 2$. In this regard, the most fundamental result is the
celebrated Hörmander-Mikhlin multiplier theorem which asserts that if $m$ is a
$C^{[\frac{d}{2}]+1}$ function on $\mathbb{R}^{d}\backslash\\{0\\}$ such that

$\sup_{0\leq|\alpha|\leq[\frac{d}{2}]+1}\,\sup_{\xi\in\mathbb{R}^{d}\backslash\\{0\\}}|\xi|^{|\alpha|}|\partial^{\alpha}m(\xi)|<\infty,$

where $\alpha=(\alpha_{1},\cdots,\alpha_{d})$ denotes a multi-index of
nonnegative integers and $|\alpha|=\alpha_{1}+\cdots+\alpha_{d}$, then the
associated Fourier multiplier $T_{m}$, defined via Fourier transform by
$\widehat{T_{m}(f)}=m\widehat{f}$, is a bounded map on
$L_{p}({\mathbb{R}}^{d})$ for all $1<p<\infty$. A similar statement holds for
the periodic case, i.e., for ${\mathbb{T}}^{d}$ instead of ${\mathbb{R}}^{d}$.
More precisely, if $m:{\mathbb{Z}}^{d}\to{\mathbb{C}}$ satisfies

(1) $\|m\|_{{\rm
HM}}=\sup_{0\leq|\alpha|\leq[\frac{d}{2}]+1}\,\sup_{k\in{\mathbb{Z}}^{d}}|k|^{|\alpha|}|\partial^{\alpha}m(k)|<\infty,$

where
$\partial^{\alpha}=\partial_{k_{1}}^{\alpha_{1}}\cdots\partial_{k_{d}}^{\alpha_{d}}$
denotes the partial discrete difference operator of order
$\alpha=(\alpha_{1},\cdots,\alpha_{d})$, then the associated Fourier
multiplier $T_{m}$ is a bounded map on $L_{p}({\mathbb{T}}^{d})$ for all
$1<p<\infty$.

In noncommutative harmonic analysis and operator algebras, the study Fourier
multipliers on general groups is of paramount importance. These multipliers
are also called Herz-Schur multipliers in the literature. The underlying
$L_{p}$-spaces are then the noncommutative $L_{p}$-spaces on the group von
Neumann algebras. This line of investigation finds one of its origins in the
study of approximation properties of operator algebras inaugurated by
Haagerup’s pioneering work [6] on Fourier multipliers on free groups in which
he solved the longstanding problem on Grothendieck’s approximation property
for the reduced C*-algebra of a free group: this algebra has the completely
bounded approximation property. Later, together with De Cannière [2] and
Cowling [5], Haagerup studied completely bounded multipliers on the Fourier
algebras of locally compact groups.

One of the remarkable recent achievements in the interplay between multipliers
and noncommutative $L_{p}$-spaces is the result of Lafforgue and de la Salle
[19] asserting that the noncommutative $L_{p}$-space with $p>4$ associated to
the von Neumann algebra of $SL_{3}({\mathbb{Z}})$ fails the completely bounded
approximation property. Haagerup and coauthors [9, 10, 8] extended their
result to connected simple Lie groups; later, de Laat and de la Salle [18]
discovered that $L_{p}$ multipliers are very tightly connected with Banach
space geometry and group representations. Other striking illustrations of this
interplay are Junge and Parcet’s work [16] on the Maurey-Rosenthal
factorization for noncommutative $L_{p}$-spaces which relies on Schur
multiplier estimates. Multipliers are also intimately related to functional
calculus in von Neumann algebras as shown by the resolution of Krein’s
celebrated Lipschitz continuity problem in noncommutative $L_{p}$-spaces by
Potapov and Sukochev [32], see also [3] where the crucial use of Fourier
multipliers is more apparent.

In a recent series of articles [7, 11], Haagerup and co-authors obtained a
beautiful handy characterization of the completely bounded radial multipliers
on the von Neumann algebras (i.e. $L_{\infty}$) of free groups and $p$-adic
groups, and, more generally, on homogeneous trees and free products; see also
Wysoczański’s previous work [35]. All this has motivated many other
publications of the same nature, in particular, building on Ozawa’s work [26],
Mei and de la Salle [22] obtained a similar characterization for hyperbolic
groups. Nevertheless, little has been understood for non-radial
multipliers111A better understanding of non-radial multipliers seem to be in
order to answer some fundamental open questions, such as the construction of
Schauder basis, for the free group reduced $C^{*}$-algebras and the
corresponding noncommutative $L_{p}$-spaces., nor for $L_{p}$-spaces with
$2<p<\infty$. Here, the most challenging problem is to find a Hörmander-
Mikhlin type criterion for Fourier multipliers on $L_{p}$-spaces to be
completely bounded.

The recent papers [14, 15, 27] provide a first advance towards this direction.
The work [23] by the first and second named authors is more related to the
present article, it introduced the free Hilbert transforms on free groups and
free products of von Neumann algebras and showed their $L_{p}$-boundedness for
all $1<p<\infty$. We pursue this line of investigation by going considerably
beyond, and will exhibit a large family of $L_{p}$ multipliers in the free
setting.

To proceed further, we need some notation and refer to the next section for
all unexplained notions. Let $\Gamma$ be a discrete group and $\lambda$ its
left regular representation on $\ell_{2}(\Gamma)$. The group von Neumann
algebra $vN(\Gamma)$ is the von Neumann algebra generated by
$\lambda(\Gamma)$, it is also equal to the weak* closure of the group algebra
${\mathbb{C}}[\Gamma]$ that consists of all polynomials in $\lambda$:

${\mathbb{C}}[\Gamma]=\big{\\{}\sum_{g\in\Gamma}\alpha(g)\lambda(g):\alpha(g)\in{\mathbb{C}}\big{\\}}.$

Following notation in quantum groups, we will denote $vN(\Gamma)$ by
$\widehat{\Gamma}$ in this article. $\widehat{\Gamma}$ is equipped with its
canonical tracial state $\tau$ defined by
$\tau(x)=\langle\delta_{e},\,x\delta_{e}\rangle$ for any
$x\in\widehat{\Gamma}$, where $\delta_{e}$ denotes the Dirac mass at the
identity $e$ of $\Gamma$. $L_{p}(\widehat{\Gamma})$ is the noncommutative
$L_{p}$-space based on $(\widehat{\Gamma},\,\tau)$, it is equipped with the
natural operator space structure introduced by Pisier.

For a complex function $m$ on $\Gamma$, the associated Fourier multiplier
$T_{m}$ is a linear map on ${\mathbb{C}}[\Gamma]$ determined by
$\lambda(g)\mapsto m(g)\lambda(g)$. We call $m$ a (completely) bounded
$L_{p}$-multiplier on $\Gamma$ if $T_{m}$ extends to a (completely) bounded
map on $L_{p}(\widehat{\Gamma})$. Our main aim is to find sufficient
conditions for $m$ to be such a multiplier. Note that the periodic Hörmander-
Mikhlin theorem recalled at the beginning concerns the case
$\Gamma={\mathbb{Z}}^{d}$ in which
$\widehat{\Gamma}=L_{\infty}({\mathbb{T}}^{d})$, ${\mathbb{T}}$ being the unit
circle equipped with normalized Haar measure. In this case, every function $m$
on ${\mathbb{Z}}^{d}$ satisfying (1) is a completely bounded
$L_{p}$-multiplier on ${\mathbb{Z}}^{d}$.

Throughout the article, $\mathbb{F}_{\infty}$ will denote a free group on
infinite generators $\\{g_{1},g_{2},\cdots\\}$. An element
$g\in\mathbb{F}_{\infty}$ different from the identity $e$ is written as a
reduced word on $\\{g_{1},g_{2},\cdots\\}$:

(2) $g=g_{i_{1}}^{k_{1}}g_{i_{2}}^{k_{2}}\cdots g_{i_{n}}^{k_{n}}\;\text{ with
}\;i_{1}\neq i_{2}\neq\cdots\neq
i_{n},\;k_{j}\in{\mathbb{Z}}\setminus\\{0\\},\;1\leq j\leq n.$

Given a complex function $m$ on ${\mathbb{Z}}^{d}$ and a fixed
$d\in{\mathbb{N}}$, we define
$M_{m}:{\mathbb{C}}[\mathbb{F}_{\infty}]\to{\mathbb{C}}[\mathbb{F}_{\infty}]$
by

$\displaystyle
M_{m}(\lambda(g))=\left\\{\begin{array}[]{lcl}m(k_{1},\cdots,k_{d})\,\lambda(g)&\textrm{
if}&d\leq n,\\\ m(k_{1},\cdots k_{n},0,\cdots,0)\,\lambda(g)&\textrm{
if}&d>n\end{array}\right.$

for every $g$ as in (2). Below is our first main theorem. It is contained in
[24] for $d=1$.

######  Theorem 1.1.

Let $d\in{\mathbb{N}}$ and $1<p<\infty$. If $m$ is a completely bounded
$L_{p}$-multiplier on ${\mathbb{Z}}^{d}$, then $M_{m}$ extends to a completely
bounded map on $L_{p}(\widehat{\mathbb{F}}_{\infty})$. In particular, the
conclusion holds if $m$ satisfies (1).

To achieve Theorem 1.1, we establish a new platform to transfer the
$L_{p}$-complete boundedness of Fourier multipliers on tori to Fourier
multipliers on free groups. The key tool is a group of unitary actions on
$L_{2}(\widehat{}\mathbb{F}_{\infty})$ and its complete boundedness on
$L_{p}(\widehat{}\mathbb{F}_{\infty})$ for all $1<p<\infty$. Given
$z=(z_{j,i})_{1\leq j\leq d,\,1\leq
i<\infty}\in{\mathbb{T}}^{d}\times{\mathbb{T}}^{\infty}$, let

$\alpha^{Ld}_{z}(\lambda(g))=z_{1,i_{1}}^{k_{1}}z_{2,i_{2}}^{k_{2}}\cdots
z_{d,i_{d}}^{k_{d}}\lambda(g)$

for $g$ as in (2) (with $k_{\ell}=0$ if $\ell>n$). It is obvious that
$\alpha^{Ld}$ extends to a unitary action of the group
${\mathbb{T}}^{d}\times{\mathbb{T}}^{\infty}$ on
$L_{2}(\widehat{}\mathbb{F}_{\infty})$.

######  Theorem 1.2.

The above action $\alpha^{Ld}$ of
${\mathbb{T}}^{d}\times{\mathbb{T}}^{\infty}$ on
$L_{2}(\widehat{}\mathbb{F}_{\infty})$ extends to a uniformly completely
bounded action on $L_{p}(\widehat{}\mathbb{F}_{\infty})$ for every
$1<p<\infty$ with constant depending only on $d$ and $p$.

Theorem 1.1 will follow from Theorem 1.2 and a standard transference argument.
Our initial proof of Theorem 1.2 is very long and technical. We have then
luckily found a new and shorter argument; the new one depends, in a crucial
way, on an analogous result on amalgamated free products of von Neumann
algebras. The new approach has a more advantage that it allows us to easily
extend Theorems 1.1 and 1.2 to free products of general discrete groups too.

In this context, let $(\mathcal{A}_{i},\tau_{i})_{i\in I}$ be a family of
finite von Neumann algebras equipped with normal faithful tracial states. Let
$\mathcal{B}$ be a common von Neumann subalgebra of the $\mathcal{A}_{i}$’s
and $\mathbb{E}_{i}:\mathcal{A}_{i}\to\mathcal{B}$ be the corresponding trace
preserving conditional expectation. We denote by $(\mathcal{A},\tau)=*_{i\in
I,\mathcal{B}}(\mathcal{A}_{i},\tau_{i})$ the amalgamated free product of the
$(\mathcal{A}_{i},\tau_{i})$’s over $\mathcal{B}$. Let $\mathbb{E}$ be the
conditional expectation from $\mathcal{A}$ to $\mathcal{B}$. As usual, for
$X\subset L_{p}(\mathcal{A})$ we write $\mathring{X}$ for the set
$\\{x-\mathbb{E}(x):x\in X\\}$.

Assume that we are given maps $T_{j,i}:L_{p}(\mathcal{A}_{i})\to
L_{p}(\mathcal{A}_{i})$ with $1\leq j\leq d$ and $i\in I$ satisfying the
following conditions

  * $\bullet$

$T_{j,i}$ is $\mathcal{B}$-bimodular, that is, $T_{k,i}(axb)=aT_{j,i}(x)b$ for
$a,b\in\mathcal{B}$ and $x\in L_{p}(\mathcal{A}_{i})$;

  * $\bullet$

$T_{j,i}\big{(}\widering{L_{p}(\mathcal{A}_{i})}\big{)}\subset\widering{L_{p}(\mathcal{A}_{i})}$.

We can define a map $T^{Ld}$ as follows. $T^{Ld}(b)=b$ for $b\in\mathcal{B}$
and

$\displaystyle
T^{Ld}(a)=\left\\{\begin{array}[]{lcl}T_{1,i_{1}}(a_{1})\otimes\cdots\otimes
T_{d,i_{d}}(a_{d})\otimes a_{d+1}\otimes\cdots\otimes
a_{n}&\textrm{if}&d<n,\\\ T_{1,i_{1}}(a_{1})\otimes\cdots\otimes
T_{n,i_{n}}(a_{n})&\textrm{if}&d\geq n\end{array}\right.$

for $a=a_{1}\otimes\cdots\otimes
a_{n}\in\mathring{\mathcal{A}}_{i_{1}}\otimes_{\mathcal{B}}\cdots\otimes_{\mathcal{B}}\mathring{\mathcal{A}}_{i_{n}}$
with $i_{1}\neq i_{2}\neq\cdots\neq i_{n}$.

The following theorem and its proof contain the main novelty of the article,
it gives a surprisingly simple condition to ensure the complete boundedness of
the map $T^{Ld}$ on $L_{p}(\mathcal{A})$.

######  Theorem 1.3.

Let $1<p<\infty$. Assume that the $T_{j,i}$’s extend to completely bounded
maps on $L_{p}(\mathcal{A}_{i})$ and on $L_{2}(\mathcal{A}_{i})$ uniformly in
$i,j$. Then $T^{Ld}$ extends to a completely bounded map on
$L_{p}(\mathcal{A})$ with

$\|T^{Ld}\|_{{\rm
cb}(L_{p}(\mathcal{A}))}\lesssim_{p,d}\prod_{j=1}^{d}\sup_{i\in
I}\Big{(}\|T_{j,i}\|_{{\rm cb}(L_{p}(\mathcal{A}_{i}))}+\|T_{j,i}\|_{{\rm
cb}(L_{2}(\mathcal{A}_{i}))}\Big{)}.$

It is clear that the assumption above is necessary for the validity of the
conclusion. Thus the theorem gives a characterization of the complete
boundedness of $T^{Ld}$ on $L_{p}(\mathcal{A})$. Another aspect to be
emphasized is the fact that $p$ is a single fixed index. One of the main
ingredients in our argument is a length reduction formula for the
$L_{p}$-norms associated with amalgamated free products that we state as
Theorem 4.6. This formula is interesting for its won right, it was previously
proved in [17, Theorem B] for homogeneous free polynomials.

The article is organized as follows. After a brief introduction to necessary
preliminaries, we give in section 3 some norms on Hilbert
$\mathcal{B}$-modules that are the modular extensions of the usual column and
row noncommutative $L_{p}$-norms; we show that bounded modular maps extend to
these modular noncommutative $L_{p}$-spaces. The results in this section
provide crucial tools for section 4 that is the core of the article and
contains our major ideas. Section 4 gives the proof of Theorem 1.3 that is
done through a reduction formula for polynomials in a free product of von
Neumann algebras, the latter is proved in its turn with the help of an
intermediate result (Theorem 4.5) that is a special case of Theorem 1.3 where
$d=1$, $p>2$ and every $T_{i}$ is a $*$-representation on $\mathcal{A}_{i}$
leaving $\mathcal{B}$ invariant and commuting with $\mathbb{E}_{i}$ for all
$i\in I$. Theorems 1.1 and 1.2 easily follow from the results in section 4;
moreover, the arguments can be extended to the free products of general (non
abelian) discrete groups. We do all this in section 5. This section also
contains some more paraproducts which might be interesting in free analysis.
We end the article with an appendix on the endpoint $p=\infty$.

We will use the following convention through the article: $A\lesssim B$ (resp.
$A\lesssim_{p}B$) means that $A\leq cB$ (resp. $A\leq c_{p}B$) for some
absolute positive constant $c$ (resp. a positive constant $c_{p}$ depending
only on $p$). $A\simeq B$ or $A\simeq_{p}B$ means that these inequalities as
well as their inverses hold.

##  2\. Preliminaries

This section presents some preliminaries on noncommutative $L_{p}$-spaces and
amalgamated free products.

Murray and von Neumann’s work [25] demonstrates von Neumann algebras as a
natural framework to do noncommutative analysis. The elements in a von Neumann
algerba $\mathcal{M}$ can be integrated over the equipped trace $\tau$ and
measured by the associated $L_{p}$-norms. In the sequel, $\mathcal{M}$ will
denote a von Neumann algebra with a normal faithful semifinite trace $\tau$.
For each $1\leq p\leq\infty,$ let $L_{p}(\mathcal{M})$ be the noncommutative
$L_{p}$-space associated with the pair $(\mathcal{M},\tau)$. Note that
$L_{\infty}(\mathcal{M})=\mathcal{M}$ with the operator norm. We refer to [31]
for details and more historical references.

All $L_{p}$-spaces in this article will be equipped with their natural
operator space structure introduced by Pisier in [29, 30]. We also refer to
[30] for operator space theory. In fact, what we will need from the latter
theory is only [29, Lemma 1.7] that asserts a linear map
$T:L_{p}(\mathcal{M})\to L_{p}(\mathcal{M})$ is completely bounded iff ${\rm
Id}_{S_{p}}\otimes T:L_{p}(B(\ell_{2})\overline{\otimes}\mathcal{M})\to
L_{p}(B(\ell_{2})\overline{\otimes}\mathcal{M})$ is bounded; in this case, the
completely bounded norm of $T$ is equal to the usual norm of ${\rm
Id}_{S_{p}}\otimes T$ and denoted by $\|T\|_{{\rm cb}(L_{p}(\mathcal{M}))}$ or
simply $\|T\|_{{\rm cb}}$ when no ambiguity is possible. Note that
$L_{2}(\mathcal{M})$ is an operator Hilbert space and every bounded map on it
is automatically completely bounded with the same norm.

We will often be concerned with group von Neumann algebras. Let $\Gamma$ be a
discrete group. Recall the definition of its left regular representation
$\lambda$: for any $g\in\Gamma$, $\lambda(g)$ is just the left translation by
$g$ on $\ell_{2}(\Gamma)$, i.e., $\lambda(g)(f)(h)=f(g^{-1}h)$ for any
$h\in\Gamma$ and $f\in\ell_{2}(\Gamma)$. Thus
$\lambda(g)(\delta_{h})=\delta_{gh}$, where $\delta_{h}$ is the Dirac mass at
$h$. ${\mathbb{C}}[\Gamma]$ denotes the linear span of $\lambda(\Gamma)$. Then
the group von Neumann algebra $\widehat{\Gamma}$ is the weak* closure of
${\mathbb{C}}[\Gamma]$ in $B(\ell_{2}(\Gamma))$. The canonical trace of
$\widehat{\Gamma}$ is the vector state induced by $\delta_{e}$ with $e$ the
identity of $\Gamma$. When $\Gamma$ is abelian (e.g.
$\Gamma={\mathbb{Z}}^{d}$) $L_{p}(\widehat{\Gamma})$ coincides with the usual
$L_{p}$-space on the dual group of $\Gamma$.

We turn to the second part on amalgamated free products. Let
$(\mathcal{A}_{i},\tau_{i})_{i\in I}$ be a family of finite von Neumann
algebras equipped with normal faithful normalized traces. Let $\mathcal{B}$ be
a common von Neumann subalgebra of $\mathcal{A}_{i}$ for all $i\in I$, and
$\mathbb{E}_{i}:\mathcal{A}_{i}\to\mathcal{B}$ the trace preserving
conditional expectation. Denote
$(\mathcal{A},\tau)=*_{i,\mathcal{B}}(\mathcal{A}_{i},\tau_{i})$, the
amalgamated free product of $(\mathcal{A}_{i},\tau_{i})_{i\in I}$ over
$\mathcal{B}$. We will briefly recall the construction to fix notation.

For any $x\in\mathcal{A}_{i}$, we denote by $\mathring{x}=x-\mathbb{E}_{i}x$
and $\mathring{\mathcal{A}}_{i}=\\{\mathring{x}:x\in\mathcal{A}_{i}\\}$, which
yields a natural decomposition
$\mathcal{A}_{i}=\mathcal{B}\oplus\mathring{\mathcal{A}}_{i}$. We use the
multi-index notation: $\underline{i}=(i_{1},\cdots,i_{n})\in I^{n}$. The space

$\mathcal{W}=\mathcal{B}\oplus\bigoplus_{n\geq
1}\bigoplus_{\begin{subarray}{c}(i_{1},\cdots,i_{n})\in I^{n}\\\ i_{1}\neq
i_{2}\neq\cdots\neq
i_{n}\end{subarray}}\mathring{\mathcal{A}}_{i_{1}}\otimes_{\mathcal{B}}\cdots\otimes_{\mathcal{B}}\mathring{\mathcal{A}}_{i_{n}}=\bigoplus_{n\geq
0}\bigoplus_{\begin{subarray}{c}(i_{1},\cdots,i_{n})\in I^{n}\\\ i_{1}\neq
i_{2}\neq\cdots\neq i_{n}\end{subarray}}\mathcal{W}_{\underline{i}}$

is a $*$-algebra by using concatenation and centering with respect to
$\mathcal{B}$. The natural projection $\mathbb{E}$ from $\mathcal{W}$ onto
$\mathcal{B}$ is a conditional expectation. $\tau_{i}\circ\mathbb{E}$ is a
trace on $\mathcal{W}$ that is independent of $i$ and will be denoted by
$\tau$. Then $(\mathcal{A},\tau)$ is the finite von Neumann algebra obtained
by the GNS construction from $(\mathcal{W},\tau)$. Thus $\mathcal{W}$ is weak*
dense in $\mathcal{A}$ and dense in $L_{p}(\mathcal{A})$ for $p<\infty$.
Moreover, $\mathbb{E}$ extends to a trace preserving conditional expectation
from $\mathcal{A}$ onto $\mathcal{B}$. As $\mathbb{E}$ restricts to
$\mathbb{E}_{i}$ on $\mathcal{A}_{i}$, from now on we skip the index $i$ for
the conditional expectations.

For $n\geq 0$ we denote by $P_{n}$ the natural projection

$P_{n}:\mathcal{W}\to\mathcal{W}_{n}=\bigoplus_{\begin{subarray}{c}(i_{1},\cdots,i_{n})\in
I^{n}\\\ i_{1}\neq i_{2}\cdots\neq
i_{n}\end{subarray}}\mathring{\mathcal{A}}_{i_{1}}\otimes_{\mathcal{B}}\cdots\otimes_{\mathcal{B}}\mathring{\mathcal{A}}_{i_{n}}$

and $P_{\geq n}={\rm Id}-(P_{1}+\cdots+P_{n-1}).$ Note that $P_{n}$ extends to
a completely bounded projection on $L_{p}(\mathcal{A})$ with cb-norm $\leq
2n+1$ for all $1\leq p\leq\infty$ (see [34]).

For $k\in I$ let

$\displaystyle\mathcal{L}_{k}=\bigoplus_{\underline{i}:i_{1}=k}\mathcal{W}_{\underline{i}}\;\textrm{
and }\;\mathcal{R}_{k}=\mathcal{L}_{k}^{*}.$

We denote the associated orthogonal projections on $\mathcal{W}$ by $L_{k}$
and $R_{k}$. Given a family $\varepsilon=(\varepsilon_{i})_{i\in\\{0\\}\cup
I}$ of elements in $\mathcal{B}$, and $x\in\mathcal{W}$, we define

(5) ${\mathcal{H}}_{\varepsilon}(x)=\varepsilon_{0}\,\mathbb{E}(x)+\sum_{k\in
I}\varepsilon_{k}L_{k}(x)\;\text{ and }\;{\mathcal{H}}_{\varepsilon}^{{\rm
op}}(x)=\mathbb{E}(x)\varepsilon_{0}^{*}+\sum_{k\in
I}R_{k}(x)\varepsilon_{k}^{*}.$

The following is the heart of [23], we state it as a lemma for later
reference.

######  Lemma 2.1.

Let $1<p<\infty$ and $\varepsilon=(\varepsilon_{i})_{i\in\\{0\\}\cup I}$ be a
family of unitaries in the center $\mathcal{Z}(\mathcal{B})$ of $\mathcal{B}$.
Then ${\mathcal{H}}_{\varepsilon}$ and ${\mathcal{H}}_{\varepsilon}^{{\rm
op}}$ extend to completely bounded maps on $L_{p}(\mathcal{A})$. Morover,

$\|{\mathcal{H}}_{\varepsilon}(x)\|_{p}\simeq_{p}\|x\|_{p}\simeq_{p}\|{\mathcal{H}}^{{\rm
op}}_{\varepsilon}(x)\|_{p}\,,\quad x\in L_{p}(\mathcal{A}).$

##  3\. Norms on modules

This section is a preparatory part for the next one. Here we will introduce
the modular versions of the usual row and column $p$-operator spaces and show
the corresponding boundedness results. This is quite tedious but we have to
cope it since they will be the key tools for the proofs of our main results on
amalgamated free products in the next section. Our references for Hilbert
C*-modules are [20, 28].

Let $\mathcal{B}$ be a von Neumann subalgebra of a semifinite von Neumann
algebra $(\mathcal{M},\tau)$ such that the restriction of $\tau$ to
$\mathcal{B}$ is semifinite too. We need to introduce several norms on tensor
products related to $L_{p}$-modules as in [17].

Let $E$ be a right Hilbert $\mathcal{B}$-module with $\mathcal{B}$-inner
product $\langle\cdot,\,\cdot\rangle$ (or $\langle\cdot,\,\cdot\rangle_{E}$ if
we deal with several modules). $E$ is equipped with the norm induced by its
inner product. However, we do not assume that $E$ is complete. A typical
example is, given an index set $\mathcal{I}$, the module

$\mathcal{C}_{\mathcal{I}}(\mathcal{B})=\big{\\{}(x_{\alpha})_{\alpha\in\mathcal{I}}\subset\mathcal{B}\;:\;\sup_{S\subset\mathcal{I}\textrm{
finite}}\big{\|}\sum_{\alpha\in
S}x_{\alpha}^{*}x_{\alpha}\big{\|}_{\mathcal{B}}<\infty\big{\\}}$

with inner product $\langle x,\,y\rangle=\lim_{S}\sum_{\alpha\in
S}x_{\alpha}^{*}y_{\alpha}$ which is well defined as a weak* limit over finite
subsets of $\mathcal{I}$ for inclusion. By [20, 28], a general module $E$ can
be embedded into a self-dual Hilbert module and there exist an index set
$\mathcal{I}$ and a right $\mathcal{B}$-module map
$u=(u_{\alpha})_{\alpha\in\mathcal{I}}:E\to\mathcal{C}_{\mathcal{I}}(\mathcal{B})$
so that $\langle x,\,y\rangle=\langle u(x),\,u(y)\rangle.$

Given $2\leq p<\infty$, we introduce a norm on the amalgamated tensor product
$E\otimes_{\mathcal{B}}L_{p}(\mathcal{M})$ as follows:

$\|x\|_{E^{c}\otimes_{p}L_{p}(\mathcal{M})}=\Big{\|}\Big{(}\sum_{i,j=1}^{n}a_{i}^{*}\langle
x_{i},x_{j}\rangle a_{j}\Big{)}^{1/2}\Big{\|}_{L_{p}(\mathcal{M})}$

for $x=\sum_{i=1}^{n}x_{i}\otimes a_{i}\in
E\otimes_{\mathcal{B}}L_{p}(\mathcal{M})$. Equipped with this norm,
$E\otimes_{\mathcal{B}}L_{p}(\mathcal{M})$ is denoted by
$E^{c}\otimes_{p}L_{p}(\mathcal{M})$ (the superscript $c$ refers to column).
In [17], this norm is denoted by $\big{\|}\sum_{i}|x_{i}\rangle
a_{i}\big{\|}_{p}.$ As explained in section 3 there, via the above concrete
embedding of $E$ into $\mathcal{C}_{\mathcal{I}}(\mathcal{B})$, the map

$u\otimes{\rm
Id}_{L_{p}(\mathcal{M})}:E^{c}\otimes_{\mathcal{B}}L_{p}(\mathcal{M})\to\mathcal{C}_{\mathcal{I}}(\mathcal{B})\otimes_{\mathcal{B}}L_{p}(\mathcal{M})\subset
L_{p}\big{(}B(\ell_{2}(\mathcal{I}))\overline{\otimes}\mathcal{M})\big{)}$

is then an isometry that allows us to view
$E^{c}\otimes_{p}L_{p}(\mathcal{M})$ as a subspace of the column subspace of
the last space. This also fully justifies that
$\|\cdot\|_{E^{c}\otimes_{p}L_{p}(\mathcal{M})}$ is indeed a norm.

Similarly, given $F$ a left Hilbert $\mathcal{B}$-module, we define a norm on
$L_{p}(\mathcal{M})\otimes_{\mathcal{B}}F$ by

$\|y\|_{L_{p}(\mathcal{M})\otimes_{p}F^{r}}=\Big{\|}\Big{(}\sum_{i,j=1}^{n}b_{i}\langle
y_{i},\,y_{j}\rangle b_{j}^{*}\Big{)}^{1/2}\Big{\|}_{L_{p}(\mathcal{M})}$

for $y=\sum_{i=1}^{n}b_{i}\otimes y_{i}$. This norm is denoted by
$\big{\|}\sum_{i}b_{i}\langle y_{i}|\,\big{\|}_{p}$ in [17]. As above,
$L_{p}(\mathcal{M})\otimes_{p}F^{r}$ can be identified with a subspace of the
row subspace of
$L_{p}\big{(}B(\ell_{2}(\mathcal{J}))\overline{\otimes}\mathcal{M}\big{)}$ for
some index set $\mathcal{J}$.

We can gather the two definitions together and introduce a norm on
$E\otimes_{\mathcal{B}}L_{p}(\mathcal{M})\otimes_{\mathcal{B}}F$; the
resulting space is denoted by
$E^{c}\otimes_{p}L_{p}(\mathcal{M})\otimes_{p}F^{r}$, it isometrically embeds
into
$L_{p}\big{(}B(\ell_{2}(\mathcal{J},\mathcal{I}))\overline{\otimes}\mathcal{M}\big{)}$.
This is independent of the choice of the sets $\mathcal{I},\mathcal{J}$.

We will need extra notions when we assume that $E$ and $F$ are also bimodules
(i.e., $\mathcal{B}\subset\mathcal{L}_{\mathcal{B}}(E)$, the algebra of
adjointable right $\mathcal{B}$-modular maps on $E$, and
$\mathcal{B}\subset\,_{\mathcal{B}}\mathcal{L}(F)$, the algebra of adjointable
left $\mathcal{B}$-modular maps on $F$). Given $E^{\prime}$ another right
Hilbert $\mathcal{B}$-module, we can consider
$E^{\prime}\otimes_{\mathcal{B}}E$ as a right $\mathcal{B}$-module with the
internal inner product; consequently, we have the space
$(E^{\prime}\otimes_{\mathcal{B}}E)^{c}\otimes_{p}L_{p}(\mathcal{M})$. Note
that the norm of
$(E^{\prime}\otimes_{\mathcal{B}}E)^{c}\otimes_{p}L_{p}(\mathcal{M})$
coincides with that of $E^{\prime
c}\otimes_{p}(E^{c}\otimes_{p}L_{p}(\mathcal{M}))$ when viewing
$E^{c}\otimes_{p}L_{p}(\mathcal{M})$ in the column subspace of
$L_{p}\big{(}B(\ell_{2}(\mathcal{I}))\overline{\otimes}\mathcal{M}\big{)}$ as
above. Similar constructions apply to the row case too. It is clear that all
these operations are naturally associative.

######  Remark 3.1.

We will often use without any reference that the above norms are injective. We
mean that if $E^{\prime}\subset E$, $F^{\prime}\subset F$ are submodules and
$(\mathcal{M}^{\prime},\tau)\subset(\mathcal{M},\tau)$ is a semifinite von
Neumann subalgebra, then $E^{\prime
c}\otimes_{p}L_{p}(\mathcal{M}^{\prime})\otimes_{p}F^{\prime r}$ isometrically
sits in $E^{c}\otimes_{p}L_{p}(\mathcal{M})\otimes_{p}F^{r}$.

######  Proposition 3.2.

Assume that $T:E\to E$ is a bounded right $\mathcal{B}$-modular map. Then

$\big{\|}T\otimes{\rm
Id}_{L_{p}(\mathcal{M})}:E^{c}\otimes_{p}L_{p}(\mathcal{M})\to
E^{c}\otimes_{p}L_{p}(\mathcal{M})\big{\|}\leq\|T\|.$

If additionally $E$ is equipped with a left $\mathcal{B}$-action and $T$ is
$\mathcal{B}$-bimodular, then for any right Hilbert $\mathcal{B}$-module
$E^{\prime}$

$\big{\|}{\rm Id}_{E^{\prime}}\otimes T\otimes{\rm
Id}_{L_{p}(\mathcal{M})}\big{\|}_{B\big{(}(E^{\prime}\otimes_{\mathcal{B}}E)^{c}\otimes_{p}L_{p}(\mathcal{M})\big{)}}\leq\|T\|.$

Similar statements hold for left modules too.

###### Proof.

By [28, Theorem 2.8], we have $\langle T(x),\,T(x)\rangle\leq\|T\|^{2}\langle
x,\,x\rangle$ for $x\in E$. Let $x_{i}\in E$ and $b_{i}\in\mathcal{B}$ for
$1\leq i\leq n$. Then

$\langle\sum_{i}T(x_{i})b_{i},\,\sum_{i}T(x_{i})b_{i}\rangle\leq\|T\|^{2}\langle\sum_{i}x_{i}b_{i},\,\sum_{i}x_{i}b_{i}\rangle,$

so we can reinterpret this inequality in the matrix algebra
$\mathbb{M}_{n}(\mathcal{B})$ as

$0\leq\big{(}\langle
T(x_{i}),\,T(x_{j})\rangle\big{)}_{i,j}\leq\|T\|^{2}\big{(}\langle
x_{i},\,x_{j}\rangle\big{)}_{i,j}.$

The latter immediately implies

$\Big{(}\sum_{i,j}a_{i}^{*}\langle T(x_{i}),T(x_{j})\rangle
a_{j}\Big{)}^{1/2}\leq\|T\|^{2}\Big{(}\sum_{i,j}a_{i}^{*}\langle
x_{i},x_{j}\rangle a_{j}\Big{)}^{1/2}$

for any $a_{i}\in L_{p}(\mathcal{M})$. This yields the first assertion.

The second follows from the same argument by noticing that ${\rm
Id}_{E^{\prime}}\otimes T$ is bounded on the right Hilbert
$\mathcal{B}$-module $E^{\prime}\otimes_{\mathcal{B}}E$ with norm $\|T\|$.
Indeed, for finite families $x_{i}\in E$ and $y_{i}\in E^{\prime}$ with $1\leq
i\leq n$

$\langle\sum_{i}y_{i}\otimes T(x_{i}),\,\sum_{i}y_{i}\otimes
T(x_{i})\rangle_{E^{\prime}\otimes_{\mathcal{B}}E}=\sum_{i,j}\langle
T(x_{i}),\langle y_{i},\,y_{j}\rangle_{E^{\prime}}T(x_{j})\rangle_{E}.$

As a positive element in $\mathbb{M}_{n}(\mathcal{B})$, the matrix
$\big{(}\langle y_{i},\,y_{j}\rangle_{E^{\prime}}\big{)}_{i,j}$ can be written
as

$\big{(}\langle
y_{i},\,y_{j}\rangle_{E^{\prime}}\big{)}_{i,j}=\sum_{k}\big{(}b_{k,i}^{*}\,b_{k,j}\big{)}_{i,j}\;\text{
with }\;b_{k,j}\in\mathcal{B}.$

Thus we deduce

$\displaystyle\langle\sum_{i}y_{i}\otimes T(x_{i}),\,\sum_{i}y_{i}\otimes
T(x_{i})\rangle_{E^{\prime}\otimes_{\mathcal{B}}E}$
$\displaystyle\leq\|T\|^{2}\sum_{k}\langle\sum_{i}b_{k,i}x_{i},\,\sum_{i}b_{k,i}x_{i}\rangle_{E}$
$\displaystyle=\|T\|^{2}\langle\sum_{i}y_{i}\otimes
x_{i},\,\sum_{i}y_{i}\otimes x_{i}\rangle_{E^{\prime}\otimes_{\mathcal{B}}E}.$

This is the desired boundedness of ${\rm Id}_{E^{\prime}}\otimes T$ on
$E^{\prime}\otimes_{\mathcal{B}}E$. Thus the proposition is proved. ∎

Thanks to the identifications recalled at the beginning of this section, the
previous proposition immediately implies the following

######  Remark 3.3.

If $T:E\to E$ is a bounded right $\mathcal{B}$-modular map, then for any left
$\mathcal{B}$-module $F$

$\big{\|}T\otimes{\rm Id}_{L_{p}(\mathcal{M})}\otimes{\rm
Id}_{F}:E^{c}\otimes_{p}L_{p}(\mathcal{M})\otimes_{p}F^{r}\to
E^{c}\otimes_{p}L_{p}(\mathcal{M})\otimes_{p}F^{r}\big{\|}\leq\|T\|.$

A similar statement holds for bimodules, namely, ${\rm Id}_{E^{\prime}}\otimes
T\otimes{\rm Id}_{L_{p}(\mathcal{M})}\otimes{\rm Id}_{F}$ is bounded on
$(E^{\prime}\otimes_{\mathcal{B}}E)^{c}\otimes_{p}L_{p}(\mathcal{M})\otimes_{p}F^{r}$
with norm less than or equal to $\|T\|$ if additionally $E$ is equipped with a
left $\mathcal{B}$-action and $T$ is $\mathcal{B}$-bimodular.

######  Remark 3.4.

Proposition 3.2 is the modular version of the fact that the row and column
$p$-operator spaces are homogeneous (i.e., when $\mathcal{B}=\mathbb{C}$ and
$E$ and $F$ are just Hilbert spaces).

A typical situation to which we will apply the previous results is the case
where $E=F=\mathcal{A}$ with $\mathcal{A}$ a finite von Neumann algebra
containing $\mathcal{B}$. The right inner product on $\mathcal{A}$ is $\langle
x,\,y\rangle=\mathbb{E}(x^{*}y)$ and the left $\langle\langle
x,\,y\rangle\rangle=\mathbb{E}(xy^{*})$, $\mathbb{E}$ being the trace
preserving conditional expectation from $\mathcal{A}$ onto $\mathcal{B}$. In
this case, we have
$\mathcal{A}^{c}\otimes_{2}L_{2}(\mathcal{B})=L_{2}(\mathcal{A})$
isometrically. Thus if $T:\mathcal{A}\to\mathcal{A}$ is right
$\mathcal{B}$-modular and bounded for $\langle\cdot,\,\cdot\rangle$, it
automatically extends to a map $\widetilde{T}:L_{2}(\mathcal{A})\to
L_{2}(\mathcal{A})$ with $\|\widetilde{T}\|_{B(L_{2}(\mathcal{A}))}\leq\|T\|$.
Since $\|\langle x,\,x\rangle\|_{\mathcal{B}}=\sup_{b\in
L_{2}(\mathcal{B}),\|b\|_{2}\leq 1}\|xb\|_{2}$ for $x\in\mathcal{A}$, we
actually have $\|\widetilde{T}\|_{B(L_{2}(\mathcal{A}))}=\|T\|$. We may still
write $T$ instead of $\widetilde{T}$.

We state this fact as a lemma for later use.

######  Lemma 3.5.

Let $(\mathcal{A},\tau)$ be a finite von Neumann and
$\mathcal{B}\subset\mathcal{A}$ a von Neumann subalgebra with the associated
conditional expectation $\mathbb{E}$. Assume that $\mathcal{A}$ is equipped
with the right $\mathcal{B}$-module inner product $\langle
x,\,y\rangle=\mathbb{E}(x^{*}y)$. Then any bounded right $\mathcal{B}$-modular
map $T:\mathcal{A}\to A$ extends to a bounded map on $L_{2}(\mathcal{A})$ with
$\|T\|_{B(L_{2}(\mathcal{A}))}=\|T\|$.

If $T:\mathcal{A}\to\mathcal{A}$ is completely positive and leaves
$\mathcal{B}$ invariant, then it satisfies the above conditions. If
additionally $\mathbb{E}\circ T\leq\mathbb{E}$, then $T$ also extends to a
completely bounded map on $L_{p}(\mathcal{A})$.

We will need the following

######  Proposition 3.6.

Let $T:L_{p}(\mathcal{M})\to L_{p}(\mathcal{M})$ be a completely bounded map.
Then for any right Hilbert $\mathcal{B}$-module $E$ and left Hilbert
$\mathcal{B}$-module $F$ we have

$\big{\|}{\rm Id}_{E}\otimes T\otimes{\rm Id}_{F}\big{\|}_{{\rm
cb}(E^{c}\otimes_{p}L_{p}(\mathcal{M})\otimes_{p}F^{r})}\leq\|T\|_{{\rm cb}}.$

###### Proof.

This is a direct consequence of the fact that identifying
$E^{c}\otimes_{p}L_{p}(\mathcal{M})\otimes F^{r}$ with a subspace of
$L_{p}\big{(}B(\ell_{2}(\mathcal{J},\mathcal{I}))\overline{\otimes}\mathcal{M}\big{)}$,
then ${\rm Id}_{E}\otimes T\otimes{\rm Id}_{F}$ acts like ${\rm Id}\otimes T$.
∎

##  4\. Multipliers on amalgamated free products

This section is the core of the article and contains our major novelty. The
principal result is Theorem 4.9 that will imply Theorem 1.3 by iterations.
Throughout the section, $(\mathcal{A}_{i},\tau_{i})_{i\in I}$ will denote a
family of finite von Neumann algebras containing $\mathcal{B}$ as a common
subalgebra. We will use notation introduced in section 2 on amalgamated free
products, in particular, $(\mathcal{A},\tau)=*_{i\in
I,\mathcal{B}}(\mathcal{A}_{i},\tau_{i})$.

###  4.1. An intermediate result

Given a family $\pi=(\pi_{i})_{i\in I}$ of $*$-representations
$\pi_{i}:\mathcal{A}_{i}\to\mathcal{A}_{i}$ such that $\pi_{i}(b)=b$ for all
$b\in\mathcal{B}$ and $\mathbb{E}\circ\pi_{i}=\mathbb{E}$, we introduce a
linear map $T_{\pi}$ on $\mathcal{W}$ by $T_{\pi}(b)=b$ for $b\in\mathcal{B}$
and

$T_{\pi}(a_{1}\otimes\cdots\otimes a_{n})=\pi_{i_{1}}(a_{1})\otimes
a_{2}\otimes\cdots\otimes a_{n}$

for $n\geq 1$ and $a_{1}\otimes\cdots\otimes
a_{n}\in\mathcal{W}_{\underline{i}}$ where
$\underline{i}=(i_{1},\cdots,i_{n})$ and $i_{1}\neq\cdots\neq i_{n}$. Define
$T_{\pi}^{{\rm op}}$ on $\mathcal{W}$ as $T_{\pi}^{{\rm
op}}(x)=T_{\pi}(x^{*})^{*}$. Note that both $T_{\pi}$ and $T_{\pi}^{{\rm op}}$
commute with the projections $P_{n}$.

We aim to show that $T_{\pi}$ extends to a bounded map on $L_{p}(\mathcal{A})$
for $1<p<\infty$. To this end, we introduce some paraproducts on
$\mathcal{W}\times\mathcal{W}$ in the manner of [23]: for $x,y\in\mathcal{W}$

$\displaystyle x\overset{1,0}{{\dagger}}y$
$\displaystyle=xy-\mathbb{E}_{\varepsilon}[H_{\varepsilon}^{{\rm op}}(xH^{{\rm
op}}_{\varepsilon}(y))],$ $\displaystyle x\overset{0,1}{{\dagger}}y$
$\displaystyle=xy-\mathbb{E}_{\varepsilon}[H_{\varepsilon}(H_{\varepsilon}(x)y)],$
$\displaystyle x\overset{1,1}{{\dagger}}y$
$\displaystyle=xy-x\overset{1,0}{{\dagger}}y-x\overset{0,1}{{\dagger}}y,$

where $H_{\varepsilon}$ and $H^{{\rm op}}_{\varepsilon}$ are the free Hilbert
transform defined in (5) and $\mathbb{E}_{\varepsilon}$ is the conditional
expectation over all possible choices of symmetric independent signs
$\varepsilon=(\varepsilon_{i})$. Note that when $x$ and $y$ are elementary
tensors, then $x\overset{1,0}{{\dagger}}y$ collects in $xy$ the parts that do
not end in the same algebra as $y$, that is, all letters in $y$ must have been
simplified. Similarly, $x\overset{0,1}{{\dagger}}y$ collects in $xy$ the parts
that do not start in the same algebra as $x$.

We will need some elementary free algebraic facts.

######  Lemma 4.1.

Let $g\in\mathcal{W}_{l}$, $h\in\mathcal{W}_{n}$ with $l,n\geq 0$.

  1. i)

If $l>n$, then $T_{\pi}(gh)=T_{\pi}(g)h$.

  2. ii)

If $n>l$, then $T_{\pi}^{{\rm op}}(gh)=gT_{\pi}^{{\rm op}}(h)$.

  3. iii)

If $l=n$, then $P_{\geq 2}[T_{\pi}(gh)]=P_{\geq 2}[T_{\pi}(g)h]$ and $P_{\geq
2}[T_{\pi}^{{\rm op}}(gh)]=P_{\geq 2}[gT_{\pi}^{{\rm op}}(h)]$.

###### Proof.

One can assume that $g$ and $h$ are elementary tensors. The first item is then
clear as the first letter of $g$ cannot be cancelled if $l>n$. The second is
obtained by passing to adjoints. Similarly, if $n=l$, $P_{\geq 2}(gh)$ is a
sum of elementary tensors that all start with the first letter of $g$ and end
with the last letter of $h$, up to a multiplication by an element of
$\mathcal{B}$. ∎

The following Cotlar type formula immediately follows from the previous lemma.

######  Lemma 4.2.

For $g,h\in\mathcal{W}$ we have

(6) $P_{\geq 2}\big{[}T_{\pi}(g)T_{\pi}^{{\rm op}}(h)\big{]}=P_{\geq
2}\big{[}T_{\pi}(gT_{\pi}^{{\rm op}}(h))+T_{\pi}^{{\rm
op}}(T_{\pi}(g)h)-T_{\pi}T_{\pi}^{{\rm op}}(gh)\big{]}.$

###### Proof.

By linearity, it suffices to show the formula for $g\in\mathcal{W}_{l}$,
$h\in\mathcal{W}_{n}$. Then it remains to check it by using Lemma 4.1
according to the different cases. We omit the details. ∎

######  Lemma 4.3.

Let $g=g_{1}\otimes\cdots\otimes g_{l}\in\mathcal{W}_{l}$,
$h=h_{n}\otimes\cdots\otimes h_{1}\in\mathcal{W}_{n}$ with $l,n\geq 0$, and
let $g^{\prime}=g_{2}\otimes\cdots\otimes g_{l}$ and
$h^{\prime}=h_{n}\otimes\cdots\otimes h_{2}$.

  1. i)

If $n>l$, then $P_{1}(gh)=\delta_{n,l+1}\mathbb{E}(gh^{\prime})h_{1}$.

  2. ii)

If $l>n$, then $P_{1}(gh)=\delta_{n+1,l}\,g_{1}\mathbb{E}(g^{\prime}h)$.

  3. iii)

If $l=n$, then $P_{1}(gh)=P_{1}(g_{1}\mathbb{E}(g^{\prime}h^{\prime})h_{1})$.

###### Proof.

This proof is easy. Let us verify only i). If $n>l+1$, then both $P_{1}(gh)$
and $\mathbb{E}(gh^{\prime})h_{1}$ vanish. The case $n=l+1$ is checked by
induction on $l$ and $n$ thanks to the following simplification formula:

$gh=x\otimes\widering{g_{n}h_{l}}\otimes y+x\,\mathbb{E}(g_{l}h_{n})y,$

where $x=g_{1}\otimes\cdots\otimes g_{l-1}$ and $y=h_{n-1}\otimes\cdots\otimes
h_{1}$. It then follows that $P_{1}(gh)=P_{1}(x\,\mathbb{E}(g_{l}h_{n})y)$. ∎

The following is a complement to the Cotlar formula (6).

######  Lemma 4.4.

For $g,h\in\mathcal{W}$ we have

(7) $\displaystyle P_{1}\big{(}T_{\pi}(g)T_{\pi}^{{\rm
op}}(h)\big{)}=T_{\pi}\big{[}P_{1}\big{(}g\overset{1,0}{{\dagger}}T_{\pi}^{{\rm
op}}(h)\big{)}\big{]}+T_{\pi}^{{\rm
op}}\big{[}P_{1}\big{(}T_{\pi}(g)\overset{0,1}{{\dagger}}h\big{)}\big{]}+T_{\pi}\big{[}P_{1}\big{(}g\overset{1,1}{{\dagger}}h\big{)}\big{]}.$

###### Proof.

We will check the formula according to the three cases in Lemma 4.3 and use
notation there.

For i), we can assume $n=l+1$, otherwise all terms are 0. Then

$P_{1}\big{(}T_{\pi}(g)T_{\pi}^{{\rm
op}}(h)\big{)}=\mathbb{E}\big{(}T_{\pi}(g)h^{\prime}\big{)}\,\pi_{j_{1}}(h_{1}).$

On the other hand, for elementary tensors
$x\in\mathcal{W}_{(i_{1},\cdots,i_{l})}$ and
$y\in\mathcal{W}_{(j_{n},\cdots,j_{1})}$, we have
$x\overset{1,0}{{\dagger}}y=0$ as the last letter of $y$ in $xy$ is not
cancelled. To deal with $x\overset{0,1}{{\dagger}}y$, note that

$H_{\varepsilon}[P_{1}(H_{\varepsilon}(x)y)]=\varepsilon_{i_{1}}\varepsilon_{j_{1}}P_{1}(xy).$

It thus follows that $P_{1}(x\overset{0,1}{{\dagger}}y)=P_{1}(xy)$ since
$\mathbb{E}(xy^{\prime})=0$ unless $i_{1}=j_{2}$ but then $j_{1}\neq j_{2}$
and $\mathbb{E}_{\varepsilon}(\varepsilon_{i_{1}}\varepsilon_{j_{1}})=0$.
Hence the right hand side of (7) is exactly $T_{\pi}^{{\rm
op}}\big{[}P_{1}\big{(}T_{\pi}(g)h\big{)}\big{]}=\mathbb{E}\big{(}T_{\pi}(g)h^{\prime}\big{)}\pi_{j_{1}}(h_{1})$.

The case ii) is obtained from i) by passing to adjoints.

For iii), let $x\in\mathcal{W}_{(i_{1},\cdots,i_{n})}$ and
$y\in\mathcal{W}_{(j_{n},\cdots,j_{1})}$ be elementary tensors. Then
$P_{1}(xy)=0$ unless $i_{k}=j_{k}$ for all $1\leq k\leq n$. Hence we can
assume that $i_{1}=j_{1}$. Then noting that

$H_{\varepsilon}[P_{1}(H_{\varepsilon}(x)y)]=\varepsilon_{i_{1}}^{2}P_{1}(xy)=P_{1}(xy),$

we deduce $P_{1}(x\overset{0,1}{{\dagger}}y)=0$; by symmetry,
$P_{1}(x\overset{1,0}{{\dagger}}y)=0$ too. Thus the right hand side of (7)
becomes

$T_{\pi}\big{[}P_{1}\big{(}g\overset{1,1}{{\dagger}}h\big{)}\big{]}=T_{\pi}\big{[}P_{1}\big{(}g_{1}\mathbb{E}(g^{\prime}h^{\prime})h_{1}\big{)}\big{]}=\pi_{i_{1}}\big{[}P_{1}\big{(}g_{1}\mathbb{E}(g^{\prime}h^{\prime})h_{1}\big{)}\big{]}.$

However, the left hand side is

$P_{1}\big{[}\pi_{i_{1}}(g_{1})\mathbb{E}(g^{\prime}h^{\prime})\pi_{i_{1}}(h_{1})\big{]}=P_{1}\big{[}\pi_{i_{1}}\big{(}g_{1}\mathbb{E}(g^{\prime}h^{\prime})h_{1}\big{)}\big{]}=\pi_{i_{1}}\big{[}P_{1}\big{(}g_{1}\mathbb{E}(g^{\prime}h^{\prime})h_{1}\big{)}\big{]}$

for $\pi_{i_{1}}$ is a $*$-representation, leaves the elements of
$\mathcal{B}$ invariant and $P_{1}\pi_{i_{1}}=\pi_{i_{1}}P_{1}$. (7) is thus
proved in the case $l=n$ too. ∎

The following is an intermediate result to Theorem 4.9, it will be the key for
the reduction formula in Theorem 4.6 below.

######  Theorem 4.5.

The map $T_{\pi}$ extends to a completely bounded map on $L_{p}(\mathcal{A})$
for all $2\leq p<\infty$ with cb-norm majorized by a constant depending only
on $p$.

###### Proof.

As the $\pi_{i}$’s are trace preserving and leave $\mathcal{B}$ invariant,
$T_{\pi}$ is an isometry on $\mathcal{W}$ for the $L_{2}(\mathcal{A})$-norm,
thus it extends to an isometry on $L_{2}(\mathcal{A})$. We need only to prove
the boundedness of $T_{\pi}$ since the complete boundedness will then be
automatic thanks to the usual trick of replacing $\mathcal{B}$ by the matrix
algebra $\mathbb{M}_{n}(\mathcal{B})$.

As in [23], we show that the $L_{p}$-boundedness of $T_{\pi}$ implies its
$L_{2p}$-boundedness for $2\leq p<\infty$. Starting with $p=2$, using
iteration and interpolation, we will deduce the assertion for the full range
$2<p<\infty$. In the following, we will denote by $\gamma_{p}$ the norm of
$T_{\pi}$ and $T_{\pi}^{{\rm op}}$ on $\mathcal{W}$ equipped with the
$L_{p}$-norm.

For $x\in\mathcal{W}$ we write

$\displaystyle T_{\pi}(x)T_{\pi}(x)^{*}$
$\displaystyle=T_{\pi}(x)T_{\pi}^{{\rm op}}(x^{*})$ $\displaystyle=P_{\geq
2}[T_{\pi}(x)T_{\pi}^{{\rm op}}(x^{*})]+P_{1}[T_{\pi}(x)T_{\pi}^{{\rm
op}}(x^{*})]+\mathbb{E}[T_{\pi}(x)T_{\pi}^{{\rm op}}(x^{*})].$

By (6) and the fact that $P_{\geq 2}$ has norm less than 5 on
$L_{p}(\mathcal{A})$, we have

$\big{\|}P_{\geq 2}[T_{\pi}(x)T_{\pi}^{{\rm op}}(x^{*})]\big{\|}_{p}\leq
5\,\big{[}2\gamma_{p}\|x\|_{2p}\|T_{\pi}(x)\|_{2p}+\gamma_{p}^{2}\|x\|_{2p}^{2}\big{]}.$

On the other hand, by [23, Proposition 3.14] or Lemma 2.1, the paraproducts
$\overset{i,j}{{\dagger}}$ are bounded from $L_{2p}(\mathcal{A})\times
L_{2p}(\mathcal{A})$ to $L_{p}(\mathcal{A})$ with norm less than $\eta_{p}$.
Thus using (7) and the fact that $P_{1}$ has norm less than 3 on
$L_{p}(\mathcal{A})$, we get

$\big{\|}P_{1}[T_{\pi}(x)T_{\pi}^{{\rm op}}(x^{*})]\big{\|}_{p}\leq
3\eta_{p}\gamma_{p}\big{[}2\|x\|_{2p}\|T_{\pi}(x)\|_{2p}+\|x\|_{2p}^{2}\big{]}.$

Clearly,

$\big{\|}\mathbb{E}[T_{\pi}(x)T_{\pi}^{op}(x^{*})]\big{\|}_{p}=\|\mathbb{E}(xx^{*})\|_{p}\leq\|x\|_{p}^{2}.$

So combining all the estimates yields

$\|T_{\pi}(x)\|_{2p}^{2}\leq\gamma_{p}\big{(}10+6\eta_{p})\|x\|_{2p}\|T_{\pi}(x)\|_{2p}+(1+3\eta_{p}\gamma_{p}+5\gamma_{p}^{2})\|x\|_{p}^{2}.$

It thus follows that

$\|T_{\pi}(x)\|_{2p}\leq C\gamma_{p}\eta_{p}\|x\|_{2p}$

for some absolute constant $C$, whence $\gamma_{2p}\leq C\gamma_{p}\eta_{p}$.
This finishes the proof. ∎

###  4.2. A length reduction formula

We show here how to recursively estimate the $L_{p}$-norm of an element in
$\mathcal{W}$ in the spirit of [17]. The space $\mathcal{W}$ is naturally a
right Hilbert $\mathcal{B}$-module with inner product $\langle
x,\,y\rangle=\mathbb{E}(x^{*}y)$ and also a left Hilbert $\mathcal{B}$-module
with $\langle\langle x,\,y\rangle\rangle=\mathbb{E}(xy^{*})$. The same holds
for $\mathcal{W}_{1}$ and $\mathring{\mathcal{W}}$ as submodules. A typical
element in $\mathcal{W}$ can be written as a finite sum

(8) $x=x_{0}+x_{1}+\sum_{i,\alpha}a_{i}(\alpha)\otimes
b_{i}(\alpha){\mathop{=}^{\rm def}}x_{0}+x_{1}+z,$

where $x_{0}\in\mathcal{B}$, $x_{1}\in\mathcal{W}_{1}$ and
$a_{i}(\alpha)\in\mathring{\mathcal{A}_{i}}$ and
$b_{i}(\alpha)\in\mathring{\mathcal{W}}$ with $L_{i}(b_{i}(\alpha))=0$.

The following result extends the main result of [17] on homogeneous
polynomials to any polynomials, which is the key tool for the argument in the
next subsection.

######  Theorem 4.6.

With the notation above, for $2\leq p<\infty$, we have $($considering
$z\in\mathcal{W}_{1}\otimes\mathring{\mathcal{W}})$

(9)
$\|x\|_{p}\approx_{p}\|x_{0}\|_{p}+\|x_{1}\|_{p}+\|z\|_{\mathcal{W}_{1}^{c}\otimes_{p}L_{p}(\mathcal{A})}+\|z\|_{L_{p}(\mathcal{A})\otimes_{p}\mathring{\mathcal{W}}^{r}}.$

###### Proof.

By the boundedness of the projections $P_{0}$ and $P_{1}$, it suffices to
prove the estimate for $z$. To this end, we consider two copies of
$\mathcal{A}$, and put a superscript to distinguish them. We use associativity
of the free product to write

$\mathcal{A}^{(1)}*_{\mathcal{B}}\mathcal{A}^{(2)}=\big{(}*_{i\in
I,\,\mathcal{B}}\mathcal{A}_{i}^{(1)}\big{)}*_{\mathcal{B}}\big{(}*_{i\in
I,\,\mathcal{B}}\mathcal{A}_{i}^{(2)}\big{)}=*_{i\in
I,\,\mathcal{B}}\big{(}\mathcal{A}_{i}^{(1)}*_{\mathcal{B}}\mathcal{A}_{i}^{(2)}\big{)}{\mathop{=}^{\rm
def}}*_{i\in I,\,\mathcal{B}}\widetilde{\mathcal{A}}_{i}.$

Since the traces are compatible,
$\|x\|_{L_{p}(\mathcal{A})}=\|x^{(1)}\|_{L_{p}(\mathcal{A}^{(1)}*_{\mathcal{B}}\mathcal{A}^{(2)})}$
for every $x\in L_{p}(\mathcal{A})$.

For each $i$, we define the swap map $\pi_{i}$ on
$\widetilde{\mathcal{A}}_{i}$ by

$\displaystyle a_{1}^{(1)}\otimes a_{2}^{(2)}\otimes
a_{3}^{(1)}\otimes\cdots\mapsto a_{1}^{(2)}\otimes a_{2}^{(1)}\otimes
a_{3}^{(2)}\otimes\cdots$ $\displaystyle a_{1}^{(2)}\otimes a_{2}^{(1)}\otimes
a_{3}^{(2)}\otimes\cdots\mapsto a_{1}^{(1)}\otimes a_{2}^{(2)}\otimes
a_{3}^{(1)}\otimes\cdots$

for $\mathcal{B}$-centered elements $a_{k}^{(j)}$ and by $\pi_{i}(b)=b$ for
$b\in\mathcal{B}$. It is clear that $\pi_{i}$ is a $*$-representation,
$\mathbb{E}$-preserving and leaves the elements of $\mathcal{B}$ invariant.
Thus, noticing that $\pi_{i}^{2}={\rm Id}_{\widetilde{\mathcal{A}}_{i}}$, we
use Theorem 4.5 to get

$\Big{\|}\sum_{i,\alpha}a_{i}(\alpha)\otimes
b_{i}(\alpha)\Big{\|}_{L_{p}(\mathcal{A})}\simeq_{p}\Big{\|}\sum_{i,\alpha}a_{i}(\alpha)^{(2)}\otimes
b_{i}(\alpha)^{(1)}\Big{\|}_{L_{p}(\mathcal{A}^{(1)}*_{\mathcal{B}}\mathcal{A}^{(2)})}.$

The element $\sum_{i,\alpha}a_{i}(\alpha)^{(2)}\otimes b_{i}(\alpha)^{(1)}$ is
homogeneous of degree 2 with respect to the length of
$\mathcal{A}^{(1)}*_{\mathcal{B}}\mathcal{A}^{(2)}$. Thus by [17, Theorem B]

$\displaystyle\Big{\|}\sum_{i,\alpha}a_{i}(\alpha)^{(2)}\otimes
b_{i}(\alpha)^{(1)}\Big{\|}_{L_{p}(\mathcal{A}^{(1)}*_{\mathcal{B}}\mathcal{A}^{(2)})}\simeq\Big{\|}\sum_{i,\alpha}|a_{i}(\alpha)^{(2)}\rangle
b_{i}^{(1)}(\alpha)\Big{\|}_{p}+\Big{\|}\sum_{i,\alpha}a_{i}(\alpha)^{(2)}\langle
b_{i}^{(1)}(\alpha)|\Big{\|}_{p}\,.$

Since taking copies clearly does not change norms, we have

$\displaystyle\Big{\|}\sum_{i,\alpha}|a_{i}(\alpha)^{(2)}\rangle
b_{i}^{(1)}(\alpha)\Big{\|}_{p}$
$\displaystyle=\Big{\|}\sum_{i,\alpha}a_{i}(\alpha)^{(2)}\otimes
b_{i}^{(1)}(\alpha)\Big{\|}_{(\mathcal{W}_{1}^{(2)})^{c}\otimes_{p}L_{p}(\mathcal{A}^{(1)}*_{\mathcal{B}}\mathcal{A}^{(2)})}$
$\displaystyle=\Big{\|}\sum_{i,\alpha}a_{i}(\alpha)\otimes
b_{i}(\alpha)\Big{\|}_{\mathcal{W}_{1}^{c}\otimes_{p}L_{p}(\mathcal{A})}\,.$

Similarly,

$\Big{\|}\sum_{i,\alpha}a_{i}(\alpha)^{(2)}\langle
b_{i}^{(1)}(\alpha)|\Big{\|}_{p}=\Big{\|}\sum_{i,\alpha}a_{i}(\alpha)\otimes
b_{i}(\alpha)\Big{\|}_{L_{p}(\mathcal{A})\otimes_{p}{\mathring{\mathcal{W}}}^{r}}.$

This concludes the proof of the theorem. ∎

######  Remark 4.7.

We could also have used [17, Theorem C] to make some terms more explicit.
Namely,

$\|x_{1}\|_{p}\simeq\big{\|}\big{(}\mathbb{E}(x_{1}^{*}x_{1})\big{)}^{1/2}\big{\|}_{p}+\Big{(}\sum_{i}\|L_{i}(x_{1})\|_{p}^{p}\Big{)}^{1/p}+\big{\|}\big{(}\mathbb{E}(x_{1}x_{1}^{*})\big{)}^{1/2}\big{\|}_{p},$

and

$\displaystyle\|z\|_{L_{p}(\mathcal{A})\otimes_{p}\mathring{\mathcal{W}}^{r}}$
$\displaystyle\simeq_{p}\big{\|}z\big{\|}_{\mathcal{W}_{1}^{c}\otimes\mathring{\mathcal{W}}^{r}}+\big{\|}\big{(}\mathbb{E}(zz^{*})\big{)}^{1/2}\big{\|}_{p}$
$\displaystyle\quad+\Big{(}\sum_{i}\big{\|}\big{(}\sum_{\alpha,\beta}a_{i}(\alpha)\mathbb{E}(b_{i}(\alpha)b_{i}(\beta)^{*})a_{i}(\beta)\big{)}^{1/2}\big{\|}_{p}^{p}\Big{)}^{1/p}.$

Here $\mathcal{W}_{1}^{c}\otimes\mathring{\mathcal{W}}^{r}$ has to be
understood as
$\mathcal{W}_{1}^{c}\otimes_{p}L_{p}(\mathcal{B})\otimes_{p}\mathring{\mathcal{W}}^{r}$.

######  Remark 4.8.

It is now rather easy to get an analogue of [17, Theorem C]. The norm of the
last term $\|z\|_{\mathcal{W}_{1}^{c}\otimes_{p}L_{p}(\mathcal{A})}$
corresponds to that of an element in $S_{p}\otimes_{p}L_{p}(\mathcal{A})$ and
we can formally iterate the argument. Thus, we can write the norm of $x\in
P_{\geq k}\big{(}L_{p}(\mathcal{A})\big{)}$ as a sum of $2k+1$ norms. For
simplicity assume $x\in P_{\geq k}(\mathcal{W})$, they are given by

$\displaystyle\|x\|_{\mathcal{W}_{l}^{c}\otimes\mathcal{W}^{r}},\quad 0\leq
l\leq k,$
$\displaystyle\|x\|_{\mathcal{W}_{l}^{c}\otimes_{p}(\oplus_{p}\mathring{L_{p}(\mathcal{A}_{i}))}\otimes_{p}\mathcal{W}^{r}},\quad
0\leq l\leq k-2$
$\displaystyle\|x\|_{\mathcal{W}_{k-1}^{c}\otimes_{p}L_{p}(\mathcal{A})}.$

The last one being recursive. We leave the details to the interested reader.

###  4.3. Maps of the $d$-th letters and the proof of Theorem 1.3

This subsection contains our principal result that is the key step of the
proof of Theorem 1.3. Fix $1<p<\infty$. Given a family of maps
$T_{i}:L_{p}(\mathcal{A}_{i})\to L_{p}(\mathcal{A}_{i})$ we will define an
associated map of the $d$-th letters of reduced words in $\mathcal{W}$ for
$d\geq 1$. The minimal assumption required for the $T_{i}$’s is the following

  * (H1)

$T_{i}$ is $\mathcal{B}$-bimodular and
$T_{i}\big{(}\widering{L_{q}(\mathcal{A}_{i})}\big{)}\subset\widering{L_{q}(\mathcal{A}_{i})}$
for $q=2$ and $q=p$.

  * (H2)

$T_{i}:L_{q}(\mathcal{A}_{i})\to L_{q}(\mathcal{A}_{i})$ is completely bounded
and

${\rm cb}_{q}=\sup_{i\in I}\|T_{i}\|_{{\rm
cb}(L_{q}(\mathcal{A}_{i}))}<\infty\;\;\text{ for }q=2\text{ and }q=p.$

Note that the cb-norm of $T_{i}$ on $L_{2}(\mathcal{A}_{i})$ coincides with
its usual norm for $L_{2}(\mathcal{A}_{i})$ is a homogeneous operator space.
On the other hand, it is obvious that

${\rm cb}_{q}=\big{\|}\oplus_{i}T_{i}:L_{q}(\oplus_{i}\mathcal{A}_{i})\to
L_{q}(\oplus_{i}\mathcal{A}_{i})\big{\|}_{\rm cb}\,.$

Now we define a linear map $T^{(d)}$ on $\mathcal{W}$ by $T^{(d)}(b)=b$ for
$b\in\mathcal{B}$ and

$T^{(d)}(a_{1}\otimes\cdots\otimes
a_{n})=\left\\{\begin{array}[]{lcl}a_{1}\otimes\cdots\otimes a_{d-1}\otimes
T_{i_{d}}(a_{d})\otimes a_{d+1}\otimes\cdots\otimes a_{n}&\textrm{if}&d\leq
n,\\\ a_{1}\otimes\cdots\otimes a_{n}&\textrm{if}&d>n\end{array}\right.$

for $n\geq 1$ and $a_{1}\otimes\cdots\otimes
a_{n}\in\mathcal{W}_{\underline{i}}$ with $\underline{i}=(i_{1},\cdots,i_{n})$
and $i_{1}\neq\cdots\neq i_{n}$. Note that the range of $T^{(d)}$ is not
inside $\mathcal{W}$ but clearly in $L_{p}(\mathcal{A})$.

######  Theorem 4.9.

Under the hypotheses (H1) and (H2), $T^{(d)}$ extends to a completely bounded
map on $L_{p}(\mathcal{A})$ with

$\|T^{(d)}\|_{{\rm cb}(L_{p}(\mathcal{A}))}\lesssim_{p,d}{\rm cb}_{2}+{\rm
cb}_{p}.$

###### Proof.

We start the proof by a crucial observation related to Lemma 3.5. We view
$\mathcal{A}_{i}$ as a right Hilbert $\mathcal{B}$-module with the inner
product $\langle x,\,y\rangle=\mathbb{E}(x^{*}y)$. We claim that
$\mathbb{E}(T_{i}(x)^{*}T_{i}(y))$ belongs to $\mathcal{B}$ for any
$x,y\in\mathcal{A}$. By polarization, we can assume $x=y$. Then for $b\in
L_{2}(\mathcal{A}_{i})$, by the modularity of $T_{i}$,

$\tau[b^{*}\mathbb{E}(T_{i}(x)^{*}T_{i}(x))b]=\tau[\mathbb{E}(T_{i}(xb)^{*}T_{i}(xb))]=\|T_{i}(xb)\|_{2}\leq\|T_{i}\|\,\|xb\|_{2}\,.$

This implies the claim, as well as
$\mathbb{E}(T_{i}(x)^{*}T_{i}(x))\leq\|T_{i}\|\langle x,\,x\rangle$. Thus
$T_{i}(\mathcal{A}_{i})$ is a $\mathcal{B}$-bimodule. A similar statement
holds when $\mathcal{A}_{i}$ is viewed as a left $\mathcal{B}$-module with the
inner product $\langle\langle x,\,y\rangle\rangle=\mathbb{E}(xy^{*})$.

We have stated the results in section 3 for maps $T:E\to E$ but they clearly
remain true if $T:E\to E^{\prime}$. We can thus apply them to the restriction
of $T_{i}$ to $\mathcal{A}_{i}$ with range $T_{i}(\mathcal{A}_{i})$. The same
holds for direct sums.

An immediate consequence is the boundedness of $T^{(d)}$ on $\mathcal{W}$
equipped with the $L_{2}$-norm, this follows from the orthogonality of the
$\mathcal{W}_{n}$’s and Proposition 3.2 combined with the above observation.
So the theorem holds for $p=2$. On the other hand, by duality, we need only to
consider the case $2<p<\infty$ that will be assumed in the remainder of the
proof.

Before going into the core of the proof, we point out that the result of
Theorem 4.6 can be applied to $T^{(d)}(x)$ when $x\in\mathcal{W}$ using an
obvious approximation argument. One just need to adapt correctly the modules,
replacing $\mathcal{W}_{1}^{c}$ by $T^{(1)}(\mathcal{W}_{1})^{c}$ if $d=1$ and
$\mathring{\mathcal{W}}^{r}$ by ${T^{(d-1)}(\mathring{\mathcal{W}})}^{r}$ if
$d\geq 1$.

We continue the proof by induction on $d$. The main part is the initial step:
$d=1$. By the usual argument of tensoring with the matrix algebras
$\mathbb{M}_{n}$, it suffices to prove the boundedness of $T^{(1)}$. We will
apply Theorem 4.6. Let $x=x_{0}+x_{1}+z\in\mathcal{W}$ as (8).

To deal with $\|T^{(1)}(x_{1})\|_{p}$, we use the Khintchine inequality from
Remark 4.7. We have that $T^{(1)}$ is bounded on
$L_{p}(\oplus_{i}\mathcal{A}_{i})$ with norm majorized by ${\rm cb}_{p}$, that
is,

$\Big{(}\sum_{i}\|L_{i}(T^{(1)}(x_{1}))\|_{p}^{p}\Big{)}^{1/p}=\Big{(}\sum_{i}\|T_{i}(L_{i}(x_{1}))\|_{p}^{p}\Big{)}^{1/p}\leq{\rm
cb}_{p}\Big{(}\sum_{i}\|x_{1}\|_{p}^{p}\Big{)}^{1/p}\,.$

On the other hand, thanks to the previous observation, $\oplus_{i}T_{i}$ can
be viewed as a bounded modular map on $\mathcal{W}_{1}$ with respect to both
inner products $\langle\cdot,\,\cdot\rangle$ and
$\langle\langle\cdot,\,\cdot\rangle\rangle$ with norm bounded by ${\rm
cb}_{2}$. Thus by Proposition 3.2,

$\big{\|}\big{(}\mathbb{E}[T^{(1)}(x_{1})^{*}T^{(1)}(x_{1})]\big{)}^{1/2}\big{\|}_{p}=\big{\|}\big{(}\mathbb{E}[([\oplus_{i}T_{i}](x_{1}))^{*}[\oplus_{i}T_{i}](x_{1})]\big{)}^{1/2}\big{\|}_{p}\leq{\rm
cb}_{2}\big{\|}\big{(}\mathbb{E}(x_{1}^{*}x_{1})\big{)}^{1/2}\big{\|}_{p}$

and similarly for the second inner product. Hence,

$\|T^{(1)}(x_{1})\|_{p}\lesssim({\rm cb}_{2}+{\rm cb}_{p})\|x_{1}\|_{p}.$

For the remaining part $z$, note that

$T^{(1)}(z)=\sum_{i,\alpha}[\oplus_{j}T_{j}](a_{i}(\alpha))\otimes
b_{i}(\alpha).$

Thus by the observation that $\oplus_{i}T_{i}:\mathcal{W}_{1}\to
T^{(1)}(\mathcal{W}_{1})$ is $\mathcal{B}$-bimodular and Proposition 3.2, we
again have

$\Big{\|}\sum_{i,\alpha}T_{i}(a_{i}(\alpha))\otimes
b_{i}(\alpha)\Big{\|}_{{T^{(1)}(\mathcal{W}_{1})^{c}}\otimes_{p}L_{p}(\mathcal{A})}\leq{\rm
cb}_{2}\Big{\|}\sum_{i,\alpha}a_{i}(\alpha)\otimes
b_{i}(\alpha)\Big{\|}_{{\mathcal{W}_{1}^{c}}\otimes_{p}L_{p}(\mathcal{A})}.$

For the other norm of $z$, we use Proposition 3.6 to obtain

$\Big{\|}\sum_{i,\alpha}T_{i}(a_{i}(\alpha))\otimes
b_{i}(\alpha)\Big{\|}_{L_{p}(\mathcal{A})\otimes_{p}\mathring{\mathcal{W}}^{r}}\leq{\rm
cb}_{p}\Big{\|}\sum_{i,\alpha}a_{i}(\alpha)\otimes
b_{i}(\alpha)\Big{\|}_{L_{p}(\mathcal{A})\otimes_{p}\mathring{\mathcal{W}}^{r}}.$

Thus $T^{(1)}$ extends to a (completely) bounded map on $L_{p}(\mathcal{A})$.

Now assume that $d\geq 2$ and $T^{(d-1)}$ is completely bounded on
$\mathcal{W}$ for the $L_{p}$-norm. For $x=x_{0}+x_{1}+z$ as above, we have

$T^{(d)}(x_{0}+x_{1})=x_{0}+x_{1}\;\text{ and
}\;T^{(d)}(z)=\sum_{i,\alpha}a_{i}(\alpha)\otimes T^{(d-1)}(b_{i}(\alpha)).$

Using the boundedness of $T^{(d-1)}$ on $L_{2}(\mathcal{A})$ and Proposition
3.2, we have

$\Big{\|}T^{(d)}\Big{(}\sum_{i,\alpha}a_{i}(\alpha)\otimes
b_{i}(\alpha)\Big{)}\Big{\|}_{L_{p}(\mathcal{A})\otimes_{p}{T^{(d-1)}(\mathring{\mathcal{W}})^{r}}}\leq{\rm
cb}_{2}\Big{\|}\sum_{i,\alpha}a_{i}(\alpha)\otimes
b_{i}(\alpha)\Big{\|}_{L_{p}(\mathcal{A})\otimes_{p}\mathring{\mathcal{W}}^{r}}.$

Similarly, the complete boundedness of $T^{(d-1)}$ on $L_{p}(\mathcal{A})$ and
Proposition 3.6 imply

$\Big{\|}T^{(d)}\Big{(}\sum_{i,\alpha}a_{i}(\alpha)\otimes
b_{i}(\alpha)\Big{)}\Big{\|}_{\mathcal{W}_{1}^{c}\otimes_{p}L_{p}(\mathcal{A})}\lesssim_{p,d}{\rm
cb}_{p}\Big{\|}\sum_{i,\alpha}a_{i}(\alpha)\otimes
b_{i}(\alpha)\Big{\|}_{\mathcal{W}_{1}^{c}\otimes_{p}L_{p}(\mathcal{A})}.$

This concludes the induction, and the proof of the theorem too. ∎

Theorem 1.3 immediately follows from Theorem 4.9.

###### Proof of Theorem 1.3.

For $1\leq k\leq d$ let $T_{k}^{(k)}$ be the map $T^{(k)}$ in Theorem 4.9
associated to the family $(T_{k,i})_{i\in I}$. Then

$T^{Ld}=T_{1}^{(1)}T_{2}^{(2)}\cdots T_{d}^{(d)}.$

This yields the assertion.∎

We extend Lemma 2.1 to the Hilbert transform of the $d$-th letters in the
spirit of Theorem 4.9 and Theorem 1.3. Let
$\varepsilon=(\varepsilon_{i})_{i\in I}$ be a family of elements in the unit
ball of $\mathcal{Z}(\mathcal{B})$. Let $T_{i}$ be the left multiplication map
on $L_{q}(\mathcal{A}_{i})$ by $\varepsilon_{i}$. Clearly, the $T_{i}$’s
satisfy the hypotheses (H1) and (H2). Denote the corresponding $T^{(d)}$ by
$\mathcal{H}_{\varepsilon}^{(d)}$. If $d=1$, this coincides with the free
Hilbert transform in (5). More generally, given $d$ let
$\varepsilon=(\varepsilon_{j,i})_{1\leq j\leq d,i\in I}$ be a family in the
unit ball of $\mathcal{Z}(\mathcal{B})$. The corresponding map $T^{Ld}$ as in
Theorem 1.3 is denoted by $\mathcal{H}_{\varepsilon}^{Ld}$.

The following is a particular case of Theorem 1.3, it extends [23, Theorem
4.7] to the amalgamated free product case.

######  Corollary 4.10.

Both $\mathcal{H}_{\varepsilon}^{(d)}$ and $\mathcal{H}_{\varepsilon}^{Ld}$
extend to completely bounded maps on $L_{p}(\mathcal{A})$ for all $1<p<\infty$
with cb-norms controlled by constants depending only on $p$ and $d$.

We conclude this section with the boundedness of some paraproducts that
generalize those introduced at the beginning of subsection 4.1. These
paraproducts are of independent interest in free analysis.

Let $\varepsilon=(\varepsilon_{j,i})_{j\geq 1,i\in I}$ be an independent
family of symmetric random variables with values $\pm 1$. Let
$\mathcal{H}_{\varepsilon}^{Ld}$ be the map associated to
$(\varepsilon_{j,i})_{1\leq j\leq d,i\in I}$ as in the previous corollary, and
let $\mathcal{H}_{\varepsilon}^{Ld,{\rm
op}}(x)=[\mathcal{H}_{\varepsilon}^{Ld}(x^{*})]^{*}$. We use the convention
that $\mathcal{H}_{\varepsilon}^{L0}={\rm Id}$. For $j,k\geq 0$ and
$x,y\in\mathcal{W}$, define

$x\overset{j,k}{\top}y=\mathbb{E}_{\varepsilon}\,\mathbb{E}_{\varepsilon^{\prime}}\,\mathcal{H}_{\varepsilon}^{Lj}\,\mathcal{H}_{\varepsilon^{\prime}}^{Lk,{\rm
op}}[\mathcal{H}_{\varepsilon}^{Lj}(x){\mathcal{H}}_{\varepsilon^{\prime}}^{Lk,{\rm
op}}(y)],$

where $\mathbb{E}_{\varepsilon}$ denotes the underlying expectation and
$\varepsilon^{\prime}$ is an independent copy of $\varepsilon$. This
paraproduct is easily understood for elementary tensors $x$ and $y$:
$x\overset{j,k}{\top}y$ then collects all those terms in the development of
$xy$ into elementary tensors whose first $j$ letters come from the same
algebras of the first $j$ letters of $x$, and whose last $k$ letters from the
same algebras of the last $k$ letters of $y$.

The previous corollary implies the following

######  Corollary 4.11.

The paraproduct $\overset{j,k}{\top}$ extends to a bounded bilinear map from
$L_{2p}(\mathcal{A})\times L_{2p}(\mathcal{A})$ to $L_{p}(\mathcal{A})$ for
all $1<p<\infty$ with norm majorized by a constant depending only on $p,j$ and
$k$.

######  Remark 4.12.

The reader familiar with Haagerup noncommutative $L_{p}$-spaces can extend,
with necessary modifications, all results of this section to the type III
case, that is, to amalgamated free products of von Neumann algebras equipped
with faithful normal states instead of traces.

##  5\. Multipliers on free products of groups

In this section we will first prove Theorems 1.1 and 1.2, then consider
Fourier multipliers on free products of general discrete groups. Recall that
$\widehat{\Gamma}$ denotes the group von Neumann algebra of a discrete group
$\Gamma$ generated by the left regular representation $\lambda$.

###  5.1. Proofs of Theorems 1.1 and 1.2

We start by the results on the free group $\mathbb{F}_{\infty}$.

###### Proof of Theorem 1.2.

We will apply Theorem 1.3 to the special case where
$\mathcal{A}_{i}=\widehat{\mathbb{Z}}=L_{\infty}({\mathbb{T}})$ for all
$i\in\mathbb{N}$. Fix a family $z=(z_{j,i})_{1\leq j\leq d,\,1\leq i<\infty}$
of complex numbers with modulus 1. Define $T_{j,i}$ to be the measure
preserving $*$-representation on $\mathcal{A}_{i}$ given by
$T_{j,i}(\lambda(n))=z_{j,i}^{n}\lambda(n)$ for any $n\in{\mathbb{Z}}$, or
equivalently in terms of the generator $\zeta\in L_{\infty}({\mathbb{T}})$,
$T_{j,i}(\zeta^{n})=z_{j,i}^{n}\zeta^{n}$. $T_{j,i}$ extends to a complete
isometry on $L_{p}({\mathbb{T}})$ for $1\leq p<\infty$. The corresponding map
$T^{Ld}$ is exactly the map $\alpha_{z}^{Ld}$ in Theorem 1.2. Thus Theorem 1.3
implies that $\alpha_{z}^{Ld}$ is completely bounded on
$L_{p}(\widehat{}\mathbb{F}_{\infty})$ for $1<p<\infty$, whence Theorem 1.2. ∎

###### Proof of Theorem 1.1.

We will use $\alpha_{z}^{Ld}$ in the previous proof for the special case where
$z_{j,i}=z_{j}$ for all $i$, and write $\alpha_{z}=\alpha_{z}^{Ld}$ for
$z\in{\mathbb{T}}^{d}$. Thus $\alpha$ is a uniformly completely bounded action
of ${\mathbb{T}}^{d}$ on $L_{p}(\widehat{}\mathbb{F}_{\infty})$. We then
easily deduce Theorem 1.1 by the standard transference argument as presented
in [4]. Let us give the details.

Let $m$ be a Hörmander-Mikhlin multiplier on ${\mathbb{Z}}^{d}$, that is, $m$
satisfies (1). Then the associated Fourier multiplier $T_{m}$ on
$L_{p}({\mathbb{T}}^{d})$ is completely bounded with cb-norm majorized by
$C_{p,d}\|m\|_{{\rm HM}}$. This follows from [1] for $d=1$ and [21, 37] for
$d\geq 2$ since the Schatten $p$-class $S_{p}$ is a UMD space. Note that valid
for general UMD spaces, the results in these papers require more regularity on
$m$ than the condition (1), that is, the partial discrete derivations should
run to all orders up to $d$ instead of $[\frac{d}{2}]+1$ in (1). However,
using the arguments of [36, section 4.1], we can show that when the UMD space
in consideration is a noncommutative $L_{p}(\mathcal{M})$, we can go down
again to the classical order $[\frac{d}{2}]+1$. Hence

$\big{\|}T_{m}\otimes{\rm Id}_{S_{p}}:L_{p}({\mathbb{T}}^{d};S_{p})\to
L_{p}({\mathbb{T}}^{d};S_{p})\big{\|}_{\rm cb}\lesssim_{p,d}\|m\|_{{\rm
HM}}\,.$

The Schatten $p$-class $S_{p}$ here can be replaced by $L_{p}(\mathcal{M})$
for any QWEP $\mathcal{M}$, in particular, by
$L_{p}(\widehat{\mathbb{F}}_{\infty})$. Thus

$\big{\|}T_{m}\otimes{\rm
Id}_{L_{p}(\widehat{\mathbb{F}}_{\infty})}:L_{p}({\mathbb{T}}^{d};L_{p}(\widehat{\mathbb{F}}_{\infty}))\to
L_{p}({\mathbb{T}}^{d};L_{p}(\widehat{\mathbb{F}}_{\infty}))\big{\|}_{\rm
cb}\lesssim_{p,d}\|m\|_{{\rm HM}}\,.$

Now given $x\in L_{p}(\widehat{\mathbb{F}}_{\infty})$ define $f\in
L_{p}({\mathbb{T}}^{d};L_{p}(\widehat{\mathbb{F}}_{\infty}))$ by
$f(z)=\alpha_{z}(x)$ for $z\in{\mathbb{T}}^{d}$. Then by Theorem 1.2

$\|f(z)\|_{L_{p}(\widehat{\mathbb{F}}_{\infty})}\simeq_{p,d}\|x\|_{L_{p}(\widehat{\mathbb{F}}_{\infty})},\quad
z\in{\mathbb{T}}^{d}.$

Clearly, we have the intertwining identity:

$[T_{m}\otimes{\rm
Id}_{L_{p}(\widehat{\mathbb{F}}_{\infty})}](f)(z)=\alpha_{z}(M_{m}(x)),\quad
z\in{\mathbb{T}}^{d}.$

Thus we deduce

$\displaystyle\big{\|}M_{m}(x)\big{\|}^{p}_{L_{p}(\widehat{\mathbb{F}}_{\infty})}$
$\displaystyle\lesssim_{p,d}\int_{{\mathbb{T}}^{d}}\big{\|}\alpha_{z}(M_{m}(x))\big{\|}^{p}_{L_{p}(\widehat{\mathbb{F}}_{\infty})}dz$
$\displaystyle=\big{\|}[T_{m}\otimes{\rm
Id}_{L_{p}(\widehat{\mathbb{F}}_{\infty})}](f)\big{\|}^{p}_{L_{p}({\mathbb{T}}^{d};L_{p}(\widehat{\mathbb{F}}_{\infty}))}$
$\displaystyle\lesssim_{p,d}\|m\|^{p}_{{\rm
HM}}\,\big{\|}f\big{\|}^{p}_{L_{p}({\mathbb{T}}^{d};L_{p}(\widehat{\mathbb{F}}_{\infty}))}$
$\displaystyle\lesssim_{p,d}\|m\|^{p}_{{\rm
HM}}\,\big{\|}x\big{\|}_{L_{p}(\widehat{\mathbb{F}}_{\infty})}\,.$

Therefore, $M_{m}$ is bounded on $L_{p}(\widehat{\mathbb{F}}_{\infty})$ with
norm controlled by $C_{p,d}\|m\|_{{\rm HM}}$. The complete boundedness follows
from the usual argument of tensoring with $S_{p}$. ∎

We end this subsection with some examples of Fourier multipliers on the free
group. The free Hilbert transform of [23] is a typical example of Fourier
multipliers studied in this article. Theorem 1.1 allows us to exhibit plenty
of examples of Fourier multipliers on the free group. We give here just some
typical ones.

######  Example 5.1.

Let $\mathcal{A}_{i}=L_{\infty}({\mathbb{T}})$ for all $i\in\mathbb{N}$ and
$z=(z_{j,i})_{1\leq j\leq d,i\in I}$ be a family of complex numbers of modulus
1. Then the corresponding transforms ${\mathcal{H}}_{z}^{(d)}$ and
${\mathcal{H}}_{z}^{Ld}$ in Corollary 4.10 are completely bounded on
$L_{p}(\widehat{}\mathbb{F}_{\infty})$ for all $1<p<\infty$.

######  Example 5.2.

We give two more examples of similar nature: For $1\leq j\leq d$ let $T_{j,i}$
be the map on $\mathcal{A}_{i}=L_{\infty}({\mathbb{T}})$ defined by
$T_{j,i}(\zeta^{n})=z_{j,i}^{{\rm sgn}(n)}\zeta^{n}$ for
$\zeta\in{\mathbb{T}}$ and $n\in{\mathbb{Z}}$. Clearly, $T_{j,i}$ is
completely bounded on $L_{p}({\mathbb{T}})$ for all $1<p<\infty$. Let
$\widetilde{\mathcal{H}}_{z}^{Ld}$ be the corresponding map $T^{Ld}$ in
Theorem 1.3 and $\widetilde{\mathcal{H}}_{z}^{(d)}$ the map $T^{(d)}$ in
Theorem 4.9 associated to $(T_{d,i})_{i\in{\mathbb{N}}}$. Again,
$\widetilde{\mathcal{H}}_{z}^{Ld}$ and $\widetilde{\mathcal{H}}_{z}^{(d)}$ are
completely bounded on $L_{p}(\widehat{}\mathbb{F}_{\infty})$ for all
$1<p<\infty$.

######  Example 5.3.

The Riesz transforms $R_{j}$, $1\leq j\leq d$, on $L_{p}({\mathbb{T}}^{d})$
are the Fourier multipliers of symbols

$m_{j}(k)=\frac{k_{j}}{\|k\|}\;\text{ for
}k=(k_{1},\cdots,k_{d})\in{\mathbb{Z}}^{d},\;\;\|k\|=\sqrt{k_{1}^{2}+\cdots+k_{d}^{2}}.$

It is classical that $R_{j}$ is completely bounded on
$L_{p}({\mathbb{T}}^{d})$ for $1<p<\infty$. The corresponding multipliers
$M_{m_{j}}$ in Theorem 1.1 are denoted by $R_{j}^{Ld}$ and may be called the
free Riesz transforms of the first $d$ letters on $\mathbb{F}_{\infty}$.
$R_{j}^{L1}$ is just the free Hilbert transform of [23]. $R_{j}^{Ld}$, $1\leq
j\leq d$, are completely bounded on $L_{p}(\widehat{}\mathbb{F}_{\infty})$ for
$1<p<\infty$.

######  Example 5.4.

Our final example is given by the classical Littlewood-Paley multiplier. It is
well known that the following Littlewood-Paley multiplier

$m=\sum_{i=0}^{\infty}\varepsilon_{i}\mathds{1}_{\\{k\in{\mathbb{Z}}^{d}:2^{i}-1\leq|k|<2^{i+1}\\}}\;\text{
with }\;\varepsilon_{i}=\pm 1,\;\;|k|=|k_{1}|+\cdots+|k_{d}|$

is a completely bounded Fourier $L_{p}$-multiplier on ${\mathbb{Z}}^{d}$ for
all $1<p<\infty$. The corresponding multiplier $M_{m}$, denoted by
$LP_{c}^{Ld}$, is completely bounded on
$L_{p}(\widehat{}\mathbb{F}_{\infty})$. One can equally consider the more
commonly used Littlewood-Paley multiplier:

$m=\sum_{(i_{1},\cdots,i_{d})\in{\mathbb{Z}}_{+}^{d}}\varepsilon_{i}\mathds{1}_{R_{(i_{1},\cdots,i_{d})}}\,,$

where $R_{(i_{1},\cdots,i_{d})}=I_{i_{1}}\times\cdots\times I_{i_{d}}$ and
$I_{j}=\\{k\in{\mathbb{Z}}:2^{j}-1\leq|k|<2^{j+1}\\}$. It gives rise to a
completely bounded multiplier on $L_{p}(\widehat{}\mathbb{F}_{\infty})$ too.

###  5.2. More paraproducts

We make the paraproducts in Corollary 4.11 more precise in the case of free
groups. Let
$z=(z_{j,i})_{j\in{\mathbb{N}},\,i\in{\mathbb{N}}}\in{\mathbb{T}}^{\mathbb{N}}\times{\mathbb{T}}^{\mathbb{N}}$.
Let $\alpha_{z}^{Ld}$ be the map associated to $(z_{j,i})_{1\leq j\leq
d,i\in{\mathbb{N}}}$ in Theorem 1.2. Note that

$\alpha_{z}^{Ld}=T_{z}^{(1)}\cdots T_{z}^{(d)},$

where $T_{z}^{(j)}(\lambda(g))=z_{j,i_{j}}^{k_{j}}\lambda(g)$ (with $k_{j}=0$
for $j>n$) for $g=g_{i_{1}}^{k_{1}}\cdots
g_{i_{n}}^{k_{n}}\in\mathbb{F}_{\infty}$ in reduced form. Put

$\alpha_{z}^{Ld,{\rm op}}(x)=[\alpha_{\overline{z}}^{Ld}(x^{*})]^{*}\;\text{
and }\;T_{z}^{(j),{\rm op}}(x)=[T_{\overline{z}}^{(j)}(x^{*})]^{*}.$

Let $\varepsilon=(\varepsilon_{j,i})_{j\in{\mathbb{N}},i\in{\mathbb{N}}}$ be
an independent family of symmetric signs. Recall that we use
$\mathcal{H}_{\varepsilon}^{(j)}$ and $\mathcal{H}_{\varepsilon}^{(j),{\rm
op}}$ to denote the free Hilbert transforms on the (last) $j$-th letters
considered in Corollary 4.10,

$H_{\varepsilon}^{(j)}(\lambda_{g})=\varepsilon_{j,i_{j}}\lambda_{g}\;\text{
and }\;H_{\varepsilon}^{(j),{\rm
op}}(\lambda_{g})=\varepsilon_{j,i_{m+1-j}}\lambda_{g}.$

We use again the convention that
$T_{z}^{0}=H_{\varepsilon}^{0}=\alpha_{z}^{L0}={\rm Id}$. For $j,k\geq 0$ and
$x,y\in\mathcal{{\mathbb{C}}}[\mathbb{F}_{\infty}]$, define

$\displaystyle x\,\overset{j,k}{\top}\,y$
$\displaystyle=\mathbb{E}_{z}\,\mathbb{E}_{z^{\prime}}\,\alpha_{z}^{Lj}\,\alpha_{z^{\prime}}^{Lk,{\rm
op}}\big{[}\alpha_{z}^{Lj}(x)\,\alpha_{z^{\prime}}^{Lk,{\rm op}}(y)\big{]},$
$\displaystyle x\,\overset{j+,k}{\top}\,y$
$\displaystyle=\mathbb{E}_{\varepsilon}\,H_{\varepsilon}^{(j+1)}\big{[}H_{\varepsilon}^{(j+1)}(x)\overset{j,k}{\top}y\big{]},$
$\displaystyle x\,\overset{j,k+}{\top}\,y$
$\displaystyle=\mathbb{E}_{\varepsilon}\,H_{\varepsilon}^{(k+1),{\rm
op}}\big{[}x\overset{j,k}{\top}H_{\varepsilon}^{(k+1),{\rm op}}(y)\big{]},$

where $\mathbb{E}_{z}$ and $E_{\varepsilon}$ denote the expectations on $z$
and $\varepsilon$, respectively.

As in the setting of free products of von Neumann algebras, we can easily
interpret these paraproducts for $x=\lambda(g)$ and $y=\lambda(h)$. To this
end, we say that the first $j$ blocks of $g$ survive in $gh$ if the first $j$
blocks of $gh$ and $g$ are exactly the same, and that the $j$-th block of $g$
marks in $gh$ if the $j$-th blocks of $gh$ and $g$ are powers of a same
generator. Replacing “first” by “last” (i.e., counting the letters of a
reduced word in the reverse order), we get similar notions.

Thus for $g,h\in\mathbb{F}_{\infty}$

  * •

$\lambda(g)\overset{j,k}{\top}\lambda(h)=\lambda(gh)$ if the first $j$ blocks
of $g$ and the last $k$ blocks of $h$ survive in $gh$,
$\lambda(g)\overset{j,k}{\top}\lambda(h)=0$ otherwise;

  * •

$\lambda(g)\overset{j+,k}{\top}\lambda(h)=\lambda(gh)$ if the first $j$ blocks
of $g$ and the last $k$ blocks of $h$ survive in $gh$, and in addition, the
$(j+1)$-th block of $g$ marks in $gh$,
$\lambda(g)\overset{j+,k}{\top}\lambda(h)=0$ otherwise.

Our first approach to Theorem 1.2 heavily relies on the boundedness of the
above paraproducts and several variants of them. Now this boundedness
immediately follows from Theorem 1.2.

######  Proposition 5.5.

All the above paraproducts extend to bounded bilinear maps from
$L_{2p}(\widehat{}\mathbb{F}_{\infty})\times
L_{2p}(\widehat{}\mathbb{F}_{\infty})$ to
$L_{p}(\widehat{}\mathbb{F}_{\infty})$ for all $1<p<\infty$ with norm
majorized by constants depending only on $p,j$ and $k$.

###  5.3. Extension to free products of groups

In the proof of Theorem 1.1 one can easily replace $\mathbb{Z}$ by any abelian
discrete group $\Gamma$. However, to go beyond the abelian case, one needs
extra efforts. In this subsection, $\Gamma$ will denote a general discrete
group. Let $\Gamma_{\infty}=\Gamma^{*{\mathbb{N}}}$ be the infinite free power
of $\Gamma$. Each $g\in\Gamma_{\infty}\setminus\\{e\\}$ is written as a
reduced word:

$g=g_{1}g_{2}\cdots g_{n}$

with $g_{j}\neq e$ belonging to the $i_{j}$-th copy of $\Gamma$ in
$\Gamma_{\infty}$ and $i_{1}\neq i_{2}\neq\cdots\neq i_{n}$.

We begin by extending Theorem 1.2 to this general setting. Define a linear map
$\alpha^{Ld}:\mathbb{C}[\Gamma_{\infty}]\to\mathbb{C}[\Gamma^{d}\times\Gamma_{\infty}]$
as follows: for $g=g_{1}\cdots g_{n}\in\Gamma_{\infty}$ as above in reduced
form,

$\displaystyle\alpha^{Ld}(\lambda(g))=\lambda_{\Gamma^{d}}(g_{1},\cdots,g_{d})\otimes\lambda(g)$

with $g_{\ell}=e$ in $\lambda_{\Gamma^{d}}(g_{1},\cdots,g_{d})$ if $\ell>n$.
Here we have denoted by $\lambda_{\Gamma^{d}}$ the left regular representation
of $\Gamma^{d}$ to avoid ambiguity ($\lambda$ being that of
$\Gamma_{\infty}$).

######  Theorem 5.6.

Let $d\in{\mathbb{N}}$ and $1<p<\infty$. Then the map $\alpha^{Ld}$ extends to
a completely isomorphic embedding of $L_{p}(\widehat{\Gamma}_{\infty})$ into
$L_{p}(\widehat{\Gamma}^{d}\,\overline{\otimes}\,\widehat{\Gamma}_{\infty})$.

###### Proof.

We use an argument similar to that of Theorem 4.6. Let
$G=*_{l=1,\cdots,d+1}\Gamma$ and $G_{\infty}=*_{i\geq 1}G$. To avoid confusion
we use $\Gamma_{i,l}$ to denote the $l$-th copy of $\Gamma$ in the $i$-th copy
of $G$ in $G_{\infty}$.

First, let $\pi:\widehat{G}\to\widehat{G}$ be the $*$-representation given by
the cyclic permutation of the copies that sends the $l$-th copy of $\Gamma$ to
the $l+1\hskip-4.26773pt\pmod{d+1}$-th copy. Consider the map $T^{Ld}$ on
$L_{p}(\widehat{G}_{\infty})$ given by Theorem 1.3 associated to
$T_{k}=\pi^{k},1\leq k\leq d$, as well as its inverse associated to
$T^{-1}_{k}=\pi^{-k},1\leq k\leq d$. By Theorem 1.3, $T^{Ld}$ is a complete
isomorphism on $L_{p}(\widehat{G}_{\infty})$. We will need the restriction of
$T^{Ld}$ to a copy of $L_{p}(\widehat{\Gamma}_{\infty})$ in
$L_{p}(\widehat{G}_{\infty})$ and will make its presentation more precise.

Let us identify $\Gamma_{\infty}$ with $*_{i\geq 1}\Gamma_{i,1}$. For an
element in $g\in\Gamma_{i,1}$, we denote its copy in $\Gamma_{i,l}$ by
$g^{(l)}$. Thus, for $g=g_{1}\cdots g_{n}\in\Gamma_{\infty}$ in reduced form,
we have

(10) $\displaystyle T^{Ld}(\lambda(g))=\lambda(g_{1}^{(2)}\cdots
g_{d}^{(d+1)}g_{d+1}^{(1)}\cdots g_{n}^{(1)}).$

As we have explained, (10) defines a complete isomorphic embedding of
$L_{p}(\widehat{\Gamma}_{\infty})$ into $L_{p}(\widehat{G}_{\infty})$.

Next, we consider the group morphism $\phi$ from $G_{\infty}$ onto
$\Gamma^{d}$ such that for all $i$ and $g\in\Gamma_{i,k}\subset G_{\infty}$

$\phi(g)=e\;\text{ if }k=1\;\text{ and
}\;\phi(g)=(\underbrace{e,\cdots,e}_{k-2},g,e,\cdots,e)\;\text{ if }2\leq
k\leq d+1.$

Let $U=\lambda_{\Gamma^{d}}\circ\phi$. Then $U$ is a unitary representation of
$G_{\infty}$ on $\ell_{2}(\Gamma^{d})$. Applying the Fell absorption principle
to $U$, we get a completely isometric embedding of
$L_{p}(\widehat{G}_{\infty})$ into
$L_{p}(\widehat{\Gamma^{d}}\,\overline{\otimes}\,\widehat{G}_{\infty})$:

(11) $\displaystyle\Pi_{\phi}:\sum_{g\in
G_{\infty}}c(g)\lambda(g)\mapsto\sum_{g\in
G_{\infty}}c(g)\lambda_{\Gamma^{d}}(\phi(g))\otimes\lambda(g).$

By (10) and (11), we see that

$\alpha^{Ld}(x)=\Pi_{\phi}\cdot T^{Ld}(x),\quad
x\in\mathbb{C}[\Gamma_{\infty}].$

Thus $\alpha^{Ld}$ extends to a completely isomorphic embedding on
$L_{p}(\widehat{\Gamma}_{\infty})$. ∎

######  Remark 5.7.

There is an alternate proof to Theorem 5.6. First, one may apply Theorem 4.6
and Remark 4.7 for
$\mathcal{A}_{i}=\widehat{\Gamma^{d}}\,\overline{\otimes}\,\widehat{\Gamma}$
for all $i\in{\mathbb{N}}$, and
$\mathcal{B}=\widehat{\Gamma^{d}}\,\otimes\mathbb{C}$, and check that

$\|\alpha^{L1}(x)\|_{L_{p}(\widehat{\Gamma^{d}}\,\overline{\otimes}\,\widehat{\Gamma}_{\infty})}\simeq_{p}\|x\|_{L_{p}(\widehat{\Gamma}_{\infty})}$

for any $x\in{\mathbb{C}}[\Gamma_{\infty}]$. Next, one may apply Theorem 4.6
repeatedly to get

$\displaystyle\|\alpha^{Ld}(x)\|_{L_{p}(\widehat{\Gamma^{d}}\,\overline{\otimes}\,\widehat{\Gamma}_{\infty})}\simeq_{p}\|\alpha^{L(d-1)}(x)\|_{L_{p}(\widehat{\Gamma^{d}}\,\overline{\otimes}\,\widehat{\Gamma}_{\infty})}\simeq_{p}\cdots\simeq_{p}\|x\|_{L_{p}(\widehat{\Gamma}_{\infty})}.$

This alternate proof gives a better constant.

As in the free group case, the previous theorem, together with transference,
easily implies a multiplier result on $\Gamma_{\infty}$. Given a bounded
function $m$ on ${\Gamma}^{d}$, define a linear map $M_{m}$ on
$L_{2}(\hat{\Gamma}_{\infty})$ by

$M_{m}(\lambda(g))=m(g_{1},g_{2},\cdots,g_{d})\lambda(g)$

with $g_{\ell}=e$ in $m(g_{1},g_{2},\cdots,g_{d})$ if $\ell>n$ for every
$g\in\Gamma_{\infty}$ in reduced form. Note that $\widehat{\Gamma}$ is QWEP
iff $\Gamma$ is hyperlinear (cf. [33]).

######  Theorem 5.8.

Assume that $\widehat{\Gamma}$ is QWEP. Let $d\in{\mathbb{N}}$ and
$1<p<\infty$. If the Fourier multiplier $T_{m}$ is completely bounded on
$L_{p}(\widehat{\Gamma^{d}})$, then $M_{m}$ extends to a completely bounded
map on $L_{p}(\widehat{\Gamma}_{\infty})$.

###### Proof.

This proof is similar to that of Theorem 1.1. As $\widehat{\Gamma}$ is QWEP,
so is $\widehat{\Gamma}_{\infty}$. Thus $T_{m}\otimes{\rm
Id}_{L_{p}(\widehat{\Gamma}_{\infty})}$ is completely bounded on
$L_{p}(\widehat{\Gamma^{d}}\,\overline{\otimes}\,\widehat{\Gamma}_{\infty})$
by Junge’s noncommutative Fubini theorem [12]. Let
$x\in{\mathbb{C}}[\Gamma_{\infty}]$. Using the action $\alpha^{Ld}$ in Theorem
5.6, we have

$\|x\|_{L_{p}(\widehat{\Gamma}_{\infty})}\simeq_{d,p}\|\alpha^{Ld}(x)\|_{L_{p}(\widehat{\Gamma^{d}}\,\overline{\otimes}\,\widehat{\Gamma}_{\infty})}\,.$

It remains to use the interviewing formula

$T_{m}\otimes{\rm Id}[\alpha^{Ld}(x)]=a^{Ld}[M_{m}(x)]$

to conclude as in the proof of Theorem 1.1. ∎

In the same line, we conclude this subsection by stating another application.
Consider a family of discrete groups $\Gamma_{i}$, $i\in I$, and its free
product $\Gamma_{\infty}=*_{i\in I}\Gamma_{i}$; consider also a family of
finite von Neumann algebras $(\mathcal{M}_{i},\tau_{i}),i\in I$, and its von
Neumann tensor product $(\mathcal{M},\tau)=\overline{\otimes}_{i\in
I}(\mathcal{M}_{i},\tau_{i})$. Let
$\\{m_{i,g}\\}_{g\in\Gamma_{i}}\subset\mathcal{M}_{i}$ for all $i\in I$, and
let $M_{i}$ be the operator-valued Fourier multiplier on $\Gamma_{i}$:

$M_{i}(\lambda(g))=m_{i,g}\otimes\lambda(g),\quad g\in\Gamma_{i}.$

We construct a map $M^{Ld}$ similar to Theorem 5.8. Given $g=g_{1}\cdots
g_{n}\in\Gamma_{\infty}$ in reduced form, define

$M^{Ld}(\lambda(g))=\left\\{\begin{array}[]{ll}m_{i_{1},g_{1}}\otimes\cdots\otimes
m_{i_{d},g_{d}}\otimes\lambda(g)&\textrm{ if }n\geq d,\\\
m_{i_{1},g_{1}}\otimes\cdots\otimes m_{i_{n},g_{n}}\otimes\lambda(g)&\textrm{
if }n<d.\end{array}\right.$

######  Theorem 5.9.

Let $1<p<\infty$. Then $M^{Ld}$ extends to a completely bounded Fourier
multiplier from $L_{p}(\widehat{\Gamma}_{\infty})$ to
$L_{p}(\mathcal{M}\,\overline{\otimes}\,\widehat{\Gamma}_{\infty})$ iff the
family $\big{\\{}\big{\|}M_{i}\big{\|}_{{\rm
cb}\big{(}L_{p}(\widehat{\Gamma}_{i}),\,L_{p}(\mathcal{M}_{i}\,\overline{\otimes}\,\widehat{\Gamma}_{i})\big{)}}\big{\\}}_{i\in
I}$ is bounded. In this case, we have

$\big{\|}M^{Ld}\|_{{\rm
cb}\big{(}L_{p}(\widehat{\Gamma}_{\infty}),\,L_{p}(\mathcal{M}\,\overline{\otimes}\,\widehat{\Gamma}_{\infty})\big{)}}\simeq_{p,d}\sup_{i\in
I}\big{\|}M_{i}\big{\|}_{{\rm
cb}\big{(}L_{p}(\widehat{\Gamma}_{i}),\,L_{p}(\mathcal{M}_{i}\,\overline{\otimes}\,\widehat{\Gamma}_{i})\big{)}}.$

###### Proof.

We can easily adapt the proofs of Theorem 5.6 to the present setting, so we
omit the details. ∎

##  Appendix A Endpoint boundedness of free Hilbert transforms

Our arguments rely on the $L_{p}$-boundedness ($1<p<\infty$) of the free
Hilbert transforms $H_{\varepsilon}$ in Lemma 2.1 and their variants
$H_{\varepsilon}^{(j)}$ in Corollary 4.10. We will discuss their bounds on
homogeneous polynomials when $p=\infty$, since they cannot be bounded in full
generality at the end point. It is also natural to ask if one can get an
$L_{\infty}$-BMO boundedness for the BMO spaces studied in [13].

###  A.1. Bounds on homogeneous polynomials

We work in the setting of amalgamated free products of von Neumann algebras as
in section 4. Let $H_{\varepsilon}^{(j)}$ be the maps introduced in Corollary
4.10 with $\varepsilon$ a family of signs.

The case $j=1$ of the following theorem follows from [17, Proposition 2.8].

######  Theorem A.1.

Let $d\geq 1$ and $1\leq j\leq d$. Then for any $x\in\mathcal{W}_{d}$ we have

$\|H_{\varepsilon}^{(j)}(x)\|_{\infty}\lesssim\min\big{\\{}\log(j+2),\,\log(d-j+2)\big{\\}}\|x\|_{\infty}.$

###### Proof.

For $z\in\mathbb{T}$, let $U_{z}$ be the unitary on $L_{2}({\mathcal{A}})$
sending $w\in{\mathcal{W}}_{n}$ to $z^{n}w$ for $n\geq 0$. Given
$x\in{\mathcal{W}}_{d},y\in L_{2}({\mathcal{A}})$ and $0\leq k\leq 2d$, let

$x\bot^{k}y=\mathbb{E}_{z}[z^{d-k}U_{\overline{z}}(xU_{z}(y))].$

It is easy to verify that

$xy=\sum_{k=0}^{2d}x\bot^{k}y\;\;\text{and
}\;\;\|x\bot^{k}y\|_{2}\leq\|x\|_{\infty}\|y\|_{2}.$

Moreover, we easily show

$H_{\varepsilon}^{(j)}(x)\bot^{k}y=\left\\{\begin{array}[]{lcl}H_{\varepsilon}^{(j)}(x\bot^{k}y)&\textrm{if}&k\leq
2(d-j),\\\
x\bot^{k}H_{\varepsilon}^{(d+1-j)}(y)&\textrm{if}&k>2(d-j).\end{array}\right.$

Thus

$H_{\varepsilon}^{(j)}(x)y=\sum_{0\leq k\leq
2(d-j)}H_{\varepsilon}^{(j)}(x\bot^{k}y)+\sum_{2(d-j)<k\leq
2d}x\bot^{k}H_{\varepsilon}^{(d+1-j)}(y).$

On the other hand, using the elementary estimate

$\big{\|}\sum_{k=0}^{n}z^{k}\big{\|}_{L_{1}({\mathbb{T}})}\simeq\log(n+2),$

we get

$\big{\|}\sum_{k=0}^{n}x\bot^{k}y\big{\|}_{2}+\big{\|}\sum_{k=n}^{2d}x\bot^{k}y\big{\|}_{2}\lesssim\min\big{\\{}\log(n+2),\,\log(2d-n+2)\big{\\}}\|x\|_{\infty}\|y\|_{2}.$

Therefore, since $H_{\varepsilon}^{(j)}$ is isometric on $L_{2}(\mathcal{A})$,
we deduce

$\displaystyle\big{\|}H_{\varepsilon}^{(j)}(x)y\big{\|}_{2}$
$\displaystyle\leq\big{\|}\sum_{k=0}^{2(d-j)}x\bot^{k}y\big{\|}_{2}+\big{\|}\sum_{k=2(d-j)+1}^{2d}x\bot^{k}H_{\varepsilon}^{(d+1-j)}(y)\big{\|}_{2}$
$\displaystyle\lesssim\min\big{\\{}\log(j+2),\,\log(d-j+2)\big{\\}}\|x\|_{\infty}\|y\|_{2},$

whence the desired estimate on $\|H_{\varepsilon}^{(j)}(x)\|_{\infty}$. ∎

###  A.2. Failure of the $L_{\infty}$-BMO boundedness

Let us restrict ourselves to the free group case. Recall that the Poisson
semigroup $(S_{t})_{t\geq 0}$ on $\mathbb{F}_{\infty}$ is the normal unital
completely positive semigroup given by

$S_{t}(\lambda(g))=e^{-t|g|}\lambda(g),\quad g\in\mathbb{F}_{\infty}.$

We also recall the definitions of various ${\rm BMO}$-spaces according to
[13]. As usual we denote by $L_{p}^{0}(\widehat{}\mathbb{F}_{\infty})$ the
subspace of centered elements (i.e., elements with vanishing trace) in
$L_{p}(\widehat{}\mathbb{F}_{\infty})$. Define

$\displaystyle{\rm BMO}^{c}(S)$ $\displaystyle=\big{\\{}x\in
L^{0}_{2}(\widehat{}\mathbb{F}_{\infty}):\|x\|_{{\rm
BMO}_{S}}<\infty\big{\\}},$ $\displaystyle{\rm bmo}^{c}(S)$
$\displaystyle=\big{\\{}x\in
L^{0}_{2}(\widehat{}\mathbb{F}_{\infty}):\|x\|_{{\rm
bmo}_{S}}<\infty\big{\\}},$

where

$\displaystyle\|x\|_{{\rm BMO}^{c}(S)}$
$\displaystyle=\sup_{t>0}\big{\|}S_{t}\big{[}|x-S_{t}(x)|^{2}\big{]}\big{\|}_{\infty}^{1/2},$
$\displaystyle\|x\|_{{\rm bmo}^{c}(S)}$
$\displaystyle=\sup_{t>0}\big{\|}S_{t}(|x|^{2})-|S_{t}(x)|^{2})\big{\|}_{\infty}^{1/2}.$

Similarly, we define the row versions ${\rm BMO}^{r}(S)$ and ${\rm
bmo}^{r}(S)$ by passing to adjoints. One of the main results of [13] states
that the intersection space ${\rm BMO}^{c}(S)\cap{\rm BMO}^{r}(S)$ behaves
well with complex interpolation, i.e., it replaces $L_{\infty}$ as an endpoint
space in the complex interpolation scale
$\big{\\{}L_{p}(\widehat{}\mathbb{F}_{\infty})\big{\\}}_{p>1}$.

Let $H_{\varepsilon}$ denote the free Hilbert transform of the first letters
associated to a sequence of signs, see Lemma 2.1. It is easy to see that
$\|H_{\varepsilon}(x)\|_{{\rm bmo}^{c}(S)}=\|x\|_{{\rm bmo}^{c}(S)}$ for
$x\in\mathbb{C}[\mathbb{F}_{\infty}]$. We will explain why one cannot hope the
boundedness of $H_{\varepsilon}$ from
$L_{\infty}(\widehat{}\mathbb{F}_{\infty})=\widehat{}\mathbb{F}_{\infty}$ to
any of ${\rm BMO}^{r}(S)$, ${\rm BMO}^{c}(S)$ or ${\rm bmo}^{r}(S)$.

######  Lemma A.2.

Let $a,b\in\mathbb{F}_{\infty}$ be two free elements. Let $z$ be a finite sum
$z=\sum_{k\geq 1}c_{k}\lambda_{a^{k}}$ and $z_{n}=z\lambda(b^{n})$. Then

  1. i)

$\displaystyle\lim_{n\to\infty}\|z_{n}\|_{{\rm bmo}^{r}(S)}=\|z\|_{\infty}$,

  2. ii)

$\displaystyle(e^{-1}-e^{-2})\|z\|_{\infty}\leq\limsup_{n\to\infty}\|x_{n}\|_{{\rm
BMO}^{\alpha}(S)}\leq 2\|z\|_{\infty},\;\alpha\in\\{c,r\\}.$

###### Proof.

i) It is clear that

$\|z_{n}\|_{{\rm bmo}^{r}(S)}\leq\|z_{n}\|_{\infty}=\|z\|_{\infty}.$

For a fixed $t>0$, we have $\lim\|S_{t}(z_{n})\|_{\infty}=0$. Thus,

$\limsup_{n\to\infty}\|z_{n}\|_{{\rm
bmo}^{r}(S)}^{2}\geq\sup_{t}\big{\|}S_{t}(z_{n}z_{n}^{*})\big{\|}_{\infty}=\|z\|_{\infty}^{2}.$

ii) The upper bound is clear. For the lower, we use the Kadison inequality to
ensure

$\big{\|}S_{t}|z_{n}^{*}-S_{t}(z_{n}^{*})|^{2}\big{\|}_{\infty},\;\big{\|}S_{t}|z_{n}-S_{t}(z_{n})|^{2}\big{\|}_{\infty}\geq\big{\|}(S_{t}-S_{2t})z_{n}\big{\|}_{\infty}^{2}.$

But in $\widetilde{}\mathbb{F}_{\infty}$,

$\lim_{n\to\infty}\big{(}(S_{\frac{1}{n}}-S_{\frac{2}{n}})z_{n}\big{)}\lambda_{b^{-n}}=(e^{-1}-e^{-2})z.$

This finishes the proof. ∎

As $H_{\varepsilon}(z_{n})=H_{\varepsilon}(z)\lambda_{b^{n}}$, we get

######  Corollary A.3.

The transform $H_{\varepsilon}$ is unbounded from
$L_{\infty}(\widehat{}\mathbb{F}_{\infty})$ to any of ${\rm BMO}^{r}(S)$,
${\rm BMO}^{c}(S)$ and ${\rm bmo}^{r}(S)$.

Acknowledgements. The first author is partially supported by NSF award
DMS-1700171. The second and third authors are partially supported by the
French ANR project No. ANR-19-CE40-0002, and the third author is also
partially supported by the Natural Science Foundation of China (No.12031004).

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