# Stability analysis of fixed point of fractional-order coupled map lattices

Sachin Bhalekar<EMAIL_ADDRESS><EMAIL_ADDRESS>(Corresponding Author) Prashant M. Gade<EMAIL_ADDRESS>School of
Mathematics and Statistics, University of Hyderabad, Hyderabad, 500046 India
Department of Physics, RTM Nagpur University, Nagpur

###### Abstract

We study the stability of synchronized fixed-point state for linear
fractional-order coupled map lattice(CML). We observe that the eigenvalues of
the connectivity matrix determine the stability as for integer-order CML.
These eigenvalues can be determined exactly in certain cases. We find exact
bounds in one-dimensional lattice with translationally invariant coupling
using the theory of circulant matrices. This can be extended to any finite
dimension. Similar analysis can be carried out for the synchronized fixed
point of nonlinear coupled fractional maps where eigenvalues of the Jacobian
matrix play the same role. The analysis is generic and demonstrates that the
eigenvalues of connectivity matrix play a pivotal role in stability analysis
of synchronized fixed point even in coupled fractional maps.

††journal:

##  1 Introduction

Fractional dynamics extends the dynamical systems to systems with memory and
studies in fractional order differential equations have exploded in the recent
past. In integer-order systems, dynamical systems theory has been enriched by
studies in flows as well as maps. Numerical difficulties are almost absent in
the simulation of maps. Almost all routes to chaos observed in flows are
observed in maps as well [1]. Most chaos control schemes are applicable in the
flows as well as maps. They appear naturally in scientific contexts where time
is discrete. They have found applications in convecting fluids, lasers, heart
cells, chemical oscillators, etc [2]. Circle map has found applications in
several systems described by a damped driven pendulum. Examples include
Josephson junction in microwave field [3], charge density waves, lasers [4, 5]
cardiac arrhythmia [6] and even air-bubble formation [7]. Logistic maps have
found applications in chemical physics and population dynamics [8, 9] These
systems have been extended to a spatially extended version popularly known as
coupled map lattice. Coupled map lattices have found applications in diverse
fields such as austenite-martensite structural transformation, convection and
crystal growth [10, 11, 12].

Thus it can be useful to investigate coupled fractional maps to understand the
dynamics of spatiotemporal systems in presence of memory. Studies in
fractional maps are unfrequent compared to fractional differential equations.
Simulation of the fractional differential equation is computationally
cumbersome compared to ordinary differential equation. It also needs domain
expertise in numerical analysis. Simulating high dimensional system of
fractional differential equations will need extensive computational resources.
Though simulation of fractional maps is more time-consuming than integer-order
maps, the computational resources required are far less than that for
fractional differential equations.

Systems with power-law memory occur in several physical situations ranging
from electromagnetic waves in dielectric media to adaptation in biological
systems [13, 14]. In this work, we study coupled fractional maps and
investigate a very basic problem of existence and stability of fixed-point
solution and state conditions for a synchronized fixed-point solution. In
certain important cases, such as coupled map lattice in finite dimension,
explicit bounds can be derived for stability.

##  2 Preliminaries

In this section, we present some basic definitions and results. Let
$h>0,\;a\in\mathbb{R}$, $(h\mathbb{N})_{a}=\\{a,a+h,a+2h,\ldots\\}$ and
$\mathbb{N}_{a}=\\{a,a+1,a+2,\ldots\\}$.

###### Definition 2.1

(see [15, 16, 17]). For a function $x:(h\mathbb{N})_{a}\rightarrow\mathbb{R}$,
the forward h-difference operator if defined as

$(\Delta_{h}x)(t)=\frac{x(t+h)-x(t)}{h},$

where t $\in(h\mathbb{N})_{a}$.

Throughout this paper, we take $a=0$ and $h=1$.

###### Definition 2.2

[17] For a function $x:\mathbb{N_{\circ}}\rightarrow\mathbb{R}$ the fractional
sum of order $\alpha>0$ is given by

$(\Delta^{-\alpha}x)(t)=\frac{1}{\Gamma(\alpha)}\sum_{s=0}^{n}\frac{\Gamma(\alpha+n-s)}{\Gamma(n-s+1)}x(s),$
(1)

where, $t=\alpha+n,\;n\in\mathbb{N_{\circ}}$.

###### Definition 2.3

[17, 18] Let $\mu>0$ and $m-1<\mu<m$, where $m\in\mathbb{N}$,
$m=\lceil\mu\rceil$. The $\mu$th fractional Caputo like difference is defined
as

$\Delta^{\mu}x(t)=\Delta^{-(m-\mu)}\left(\Delta^{m}x(t)\right),$ (2)

where $t\in\mathbb{N}_{m-\mu}$ and

$\Delta^{m}x(t)=\sum_{k=0}^{m}\binom{m}{k}(-1)^{m-k}x(t+k).$ (3)

###### Definition 2.4

[17] The Z-transform of a sequence $\\{y(n)\\}_{n=0}^{\infty}$ is a complex
function given by $Y(z)=Z[y](z)=\sum_{k=0}^{\infty}y(k)z^{-k}$ where
$z\in\mathbb{C}$ is a complex number for which the series converges
absolutely.

###### Definition 2.5

[17] Let $\tilde{\phi}_{\alpha}(n)$ be a family of binomial functions defined
on $\mathbb{Z}$, parametrized by $\alpha$ defined by

$\displaystyle\tilde{\phi}_{\alpha}(n)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(n+\alpha-1)}{\Gamma(\alpha)\Gamma(n)}$ (8)
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{c}n+\alpha-1\\\ n\\\
\end{array}\right)=(-1)^{n}\left(\begin{array}[]{c}-\alpha\\\
n\end{array}\right).$

Then

$Z(\tilde{\phi}_{\alpha}(t))=\frac{1}{(1-z^{-1})^{\alpha}},\quad|z|>1.$

###### Definition 2.6

[17] The convolution $\phi*x$ of the functions $\phi$ and $x$ defined on
$\mathbb{N}$ is defined as

$\left(\phi*x\right)(n)=\sum_{s=0}^{n}\phi(n-s)x(s)=\sum_{s=0}^{n}\phi(s)x(n-s).$

Then the Z-transform of this convolution is

$Z\left(\phi*x\right)(n)=\left(Z\left(\phi\right)(n)\right)\left(Z\left(x\right)(n)\right).$
(9)

###### Lemma 2.1

[18] The discrete function $x(t)$ is solution of an initial value problem

$\displaystyle\Delta^{\alpha}x(t)$ $\displaystyle=$ $\displaystyle
f(x(t+\alpha-1)),\quad t\in N_{1-\alpha},\quad 0<\alpha<1,$ $\displaystyle
x(0)$ $\displaystyle=$ $\displaystyle x_{0}$ (10)

if and only if $x(t)$ is solution of following fractional discrete dynamical
system

$\displaystyle x(t)$ $\displaystyle=$ $\displaystyle
x_{0}+\sum_{s=1-\alpha}^{t-\alpha}\frac{\Gamma(t-s)}{\Gamma(\alpha)\Gamma(t-s-\alpha+1)}f\left(x(s+\alpha-1)\right)$
(11) $\displaystyle=$ $\displaystyle
x_{0}+\sum_{j=0}^{t-1}\frac{\Gamma(t-j+\alpha-1)}{\Gamma(\alpha)\Gamma(t-j)}f\left(x(j)\right).$

##  3 Fractional order coupled map lattices: Linear systems

Consider the linear coupled map lattice of fractional order $\alpha\in(0,1)$

$x_{t+1}(k)=x_{0}(k)+\sum_{j=0}^{t}\sum_{m=1}^{N}\frac{\Gamma(t-j+\alpha)}{\Gamma(\alpha)\Gamma(t-j+1)}\left(A_{km}x_{j}(m)-x_{j}(k)\right),$
(12)

where $x_{t}(k)$ is the variable at time $t$ associated with the $k$-th
lattice point, $k=1,2,\cdots,N$, $x_{t}(0)=x_{t}(N)$ and $x_{t}(N+1)=x_{t}(1)$
and $A=(A_{km})$ is $N\times N$ connectivity matrix.
If we write $X_{t}=\left(x_{t}(1),x_{t}(2),\cdots,x_{t}(N)\right)$, a column
vector in $\mathbb{R}^{N}$ then the system (12) is equivalent to

$\displaystyle X_{t+1}$ $\displaystyle=$ $\displaystyle
X_{0}+\sum_{j=0}^{t}\frac{\Gamma(t-j+\alpha)}{\Gamma(\alpha)\Gamma(t-j+1)}\left(A-I\right)X_{j},$
(13) $\displaystyle=$ $\displaystyle
X_{0}+\left(A-I\right)\left(\tilde{\phi}_{\alpha}(t)*X_{t}\right),$

where $I$ is $N\times N$ identity matrix. Applying Z-transform and using the
properties given in Section 2, we get

$z\bar{X}(z)-zX_{0}=\frac{1}{1-z^{-1}}X_{0}+\frac{1}{(1-z^{-1})^{\alpha}}\bar{X}(z)\left(A-I\right),\quad|z|>1,$

where $\bar{X}(z)$ is the Z-transform of $X_{t}$. Therefore, the
characteristic equation of system (13) is given as

$det\left(z(1-z^{-1})^{\alpha}I-(A-I)\right)=0.$ (14)

Motivated from [19, 20], we propose the following stability theorem.

###### Theorem 3.1

The zero solution of the system (12) or (13) is asymptotically stable if and
only if all the roots of the characteristic equation (14) satisfy $|z|<1$.

###  3.1 Stable Region

At the boundary of stable region, the root $z$ of characteristic equation (14)
should satisfy $|z|=1$. Therefore, we obtain the parametric boundary curve
$\beta(t)$ of stable region by substituting $z=e^{\iota t}$, $0\leq t\leq
2\pi$ in the (14) as

$\beta(t)=\left(2^{\alpha}\left(\sin(t/2)\right)^{\alpha}\cos\left(\alpha\frac{\pi}{2}+t(1-\alpha/2)\right)+1,2^{\alpha}\left(\sin(t/2)\right)^{\alpha}\sin\left(\alpha\frac{\pi}{2}+t(1-\alpha/2)\right)\right).$
(15)

The boundary curves $\beta(t)$ for different values of $\alpha\in(0,1]$ are
sketched in Figure 1.

Figure 1: The stability region of fractional order map

If the eigenvalues of $A$ are complex, we need to consider if the given
eigenvalue is in the stable region defined by the cardioid given above and the
solution is stable only if all eigenvalues lie in the stable region.

We have following result [19, 21, 22, 20].

###### Theorem 3.2

If all the eigenvalues of matrix $A$ lie inside the region bounded by the
curve $\beta(t)$, $0\leq t\leq 2\pi$ defined by (15) then the system (12) is
asymptotically stable.

Thus the stability of synchronized fixed point $x_{t}(k)=0$ as
$t\rightarrow\infty$ $\forall i$ depends only on eigenvalues of connectivity
matrix $A$. Let us consider a particular case of coupled map lattice on one-
dimensional lattice with translationally invariant coupling and periodic
boundary conditions. The matrix $A$ such that $A_{ii}=a_{1}$,
$A_{i,i+1}=a_{2}$ and $A_{i,i-1}=a_{0}$ is given by.

$A=\begin{pmatrix}a_{1}&a_{2}&0&0&\cdots&0&0&a_{0}\\\
a_{0}&a_{1}&a_{2}&0&\cdots&0&0&0\\\ 0&a_{0}&a_{1}&a_{2}&\cdots&0&0&0\\\
\vdots&\vdots&\vdots&\vdots&&\vdots&\vdots&\vdots\\\
0&0&0&0&\cdots&a_{0}&a_{1}&a_{2}\\\ a_{2}&0&0&0&\cdots&0&a_{0}&a_{1}\\\
\end{pmatrix}$

For the special case $N=2$, we define

$A=\begin{pmatrix}a_{1}&a_{0}+a_{2}\\\ a_{0}+a_{2}&a_{1}\\\ \end{pmatrix}$

and for $N=1$

$A=\begin{pmatrix}a_{1}+a_{0}+a_{2}\end{pmatrix}$

using periodic boundary conditions.

$A$ is a circulant matrix with eigenvalues
$\lambda_{l}=a_{1}+a_{2}\omega^{l}+a_{0}\omega^{-l}$ where
$\omega=\exp(\iota{\frac{2\pi}{N}})$ is primitive $N$th root of unity [23].
For symmetric case, where $a_{2}=a_{0}$, we get
$\lambda_{l}=a_{1}+2a_{2}\cos(\theta_{l})$, where $\theta_{l}={\frac{2\pi
l}{N}}$ for $0\leq l\leq N-1$. We note that $\lambda_{l}=\lambda_{N-l}$ in
this case. For case $a_{2}=-a_{0}$, we obtain $\lambda_{l}=a_{1}+\iota
2a_{2}\sin(\theta_{l})$ where $\theta_{l}={\frac{2\pi l}{N}}$. If $N=4K$,
$\lambda_{K}=a_{1}+\iota 2a_{2}$ and $\lambda_{3K}=a1-\iota 2a2$. These are
limiting cases in this system. Coupled map lattice in one dimension is a
widely explored system and we will study the above cases in further detail in
this section.

First we consider the bounds on real part of the eigenvalue.
Note: The stable region of the real eigenvalue $\lambda$ is
$1-2^{\alpha}<\lambda<1$.

###### Example 3.1

Consider $N=3$ and $\alpha=0.4$. The parameter values $a_{0}=0.2$,
$a_{1}=-0.5$ and $a_{2}=0.1$ produce the eigenvalues $-0.2$ and $-0.65\pm
0.0866025\iota$. Since the eigenvalues $-0.65\pm 0.0866025\iota$ are outside
the stable region, we get the unstable solutions as shown in Fig. 2.

Figure 2: Unstable solution for $N=3$, $\alpha=0.4$, $a_{0}=0.2$, $a_{1}=-0.5$
and $a_{2}=0.1$

On the other hand, if we set $N=3$, $\alpha=0.8$, $a_{0}=0.2$, $a_{1}=-0.3$
and $a_{2}=0.1$ then all the eigenvalues viz. $0$ and $-0.45\pm 0.0866\iota$
lie inside the stable region and we get the stable solutions (cf. Fig. 3).

Figure 3: Stable solution for $N=3$, $\alpha=0.8$, $a_{0}=0.2$, $a_{1}=-0.3$
and $a_{2}=0.1$

In the following two subsections, we consider two important particular cases
and discuss the stability.

####  3.1.1 Symmetric Case

In this section, we assume that the coefficient matrix $A$ in (13) is
symmetric i.e. $a_{0}=a_{2}$.

###### Theorem 3.3

The stable region of the system (13) with $a_{0}=a_{2}$ is bounded by the
quadrilateral with vertices $Q_{1}=(0,1)$,
$Q_{2}=\left(-2^{\alpha-2},2^{\alpha-1}+1-2^{\alpha}\right)$,
$Q_{3}=(0,1-2^{\alpha})$ and $Q_{4}=\left(2^{\alpha-2},1-2^{\alpha-1}\right)$
for even values of lattice points $N$ and $Q_{1}=(0,1)$,
$Q_{2}^{\prime}=\left(-\frac{2^{\alpha-1}}{1+\cos(\pi/N)},\frac{2^{\alpha}}{1+\cos(\pi/N)}+1-2^{\alpha}\right)$,
$Q_{3}=(0,1-2^{\alpha})$ and
$Q_{4}^{\prime}=\left(\frac{2^{\alpha-1}}{1+\cos(\pi/N)},-\frac{2^{\alpha}}{1+\cos(\pi/N)}+1\right)$
for odd $N$ in the $a_{2}a_{1}$-plane.

Proof: The eigenvalues of $A$ in the symmetric case are

$\lambda_{j}=a_{1}+2a_{2}\cos\left(\frac{2\pi j}{N}\right),\quad
j=0,1,\cdots,N-1.$ (16)

Note that $\cos\left(\frac{2\pi
j}{N}\right)=\cos\left(\frac{2\pi(N-j)}{N}\right)$. Therefore, to obtain the
distinct values we take $j=0,1,\cdots,[N/2]$, where $[r]$ is an integer-part
of the real number $r$. Since, all these eigenvalues are real, the stable
region in the $a_{2}a_{1}$-plane is an intersection of the regions

$1-2^{\alpha}<a_{1}+2a_{2}\cos\left(\frac{2\pi j}{N}\right)<1,\quad
j=0,1,\cdots,[N/2].$ (17)

The boundaries of these regions are straight lines defined by following two
sets

$\displaystyle S1_{j}:\quad a_{1}$ $\displaystyle=$
$\displaystyle-2\cos\left(\frac{2\pi
j}{N}\right)a_{2}+(1-2^{\alpha}),\quad\text{and}$ (18) $\displaystyle
S2_{j}:\quad a_{1}$ $\displaystyle=$ $\displaystyle-2\cos\left(\frac{2\pi
j}{N}\right)a_{2}+1,$ (19)

where $j=0,1,\cdots,[N/2]$.
Note that

$[N/2]=\begin{cases}N/2,\quad\text{if $N$ is even}\\\ (N-1)/2,\quad\text{if
$N$ is odd.}\end{cases}$ (20)

The stable region of (13) will be bounded by the straight lines $S1_{j}$ and
$S2_{j}$ which are close to origin in the $a_{2}a_{1}$-plane, as shown in
Figure 4.

Figure 4: Stable region of symmetric case

The lines in the set $S1_{j}$ intersect each other at $(0,1-2^{\alpha})$
whereas those in the set $S2_{j}$ have intersection at $(0,1)$ in the
$a_{2}a_{1}$-plane. Further, the innermost line $a_{1}=-2a_{2}+(1-2^{\alpha})$
in the set (18) with $j=0$ intersects the innermost line
$a_{1}=-2\cos\left(\frac{2\pi[N/2]}{N}\right)a_{2}+1$ in the set (19) with
$j=[N/2]$ in the $a_{2}a_{1}$-plane at the point
$\left(-2^{\alpha-2},2^{\alpha-1}+1-2^{\alpha}\right)$ when $N$ is even and at
$\left(-\frac{2^{\alpha-1}}{1+\cos(\pi/N)},\frac{2^{\alpha}}{1+\cos(\pi/N)}+1-2^{\alpha}\right)$
when $N$ is odd. Secondly, the intersection between the innermost lines
$a_{1}=-2\cos\left(\frac{2\pi[N/2]}{N}\right)a_{2}+(1-2^{\alpha})$ in the set
(18) with $j=[N/2]$ and $a_{1}=-2a_{2}+1$ in the set (19) with $j=0$ is the
point $\left(2^{\alpha-2},1-2^{\alpha-1}\right)$ when $N$ is even and
$\left(\frac{2^{\alpha-1}}{1+\cos(\pi/N)},-\frac{2^{\alpha}}{1+\cos(\pi/N)}+1\right)$
when $N$ is odd.
Thus, the stable region which is an intersection of all the regions (17) is
bounded by the quadrilateral with vertices described in the statement of this
theorem. This proves the result.

We note that stable region does not change for even $N$. Two extreme values
for $a_{1}+2a_{2}\cos(\theta_{l})$ are given by $\lambda_{0}=a_{1}+2a_{2}$ and
$\lambda_{N/2}=a_{1}-2a_{2}$. For odd $N$, one of the limits
$\lambda_{0}=a_{1}+2a_{2}$ is still realized. Other limit is slightly bigger
by a leading to a slightly higher stability range and the it is approached as
$1/N^{2}$ for large $N$. Thus, in the thermodynamic limit
$N\rightarrow\infty$, stability region for $N\rightarrow\infty$ coincides with
stability region for $N=2$. Thus the stability of extreme eigenvalues in the
thermodynamic limit determine the stability region.

###### Example 3.2

Consider the symmetric system (13) with even number $N=8$ of lattice points
and $\alpha=0.2$. The stable region using Theorem 3.3 is sketched in Figure 5.

Figure 5: Stable region of symmetric system (13) with $N=8$ and $\alpha=0.2$

We verified that the solutions starting in a neighborhood of origin converge
to origin if we take $(a_{2},a_{1})$ in the stable region. Figure 6 shows the
converging trajectories for the parameter values $a_{1}=0.1$, and
$a_{2}=-0.05$. The unstable solution is sketched in Figure 7 with
$a_{1}=-0.02$, and $a_{2}=0.1$ which are outside the stable region.

Figure 6: Stable solution of symmetric system (13) with $N=8$, $\alpha=0.2$,
$a_{1}=0.1$, and $a_{2}=-0.05$ Figure 7: Unstable solution of symmetric system
(13) with $N=8$, $\alpha=0.2$, $a_{1}=-0.02$, and $a_{2}=0.1$

###### Example 3.3

Let us consider the symmetric system (13) with odd number $N=9$ of lattice
points. The stable region in this case with $\alpha=0.5$ is shown in Figure 8.
The stable solution for the parameter values $a_{1}=0.6$, and $a_{2}=-0.1$is
shown in Figure 9 whereas the unstable solution for $a_{1}=0.2$, and
$a_{2}=0.6$ is in Figure 10.

Figure 8: Stable region of symmetric system (13) with $N=9$ and $\alpha=0.5$
Figure 9: Stable solution of symmetric system (13) with $N=9$, $\alpha=0.5$,
$a_{1}=0.6$, and $a_{2}=-0.1$ Figure 10: Unstable solution of symmetric system
(13) with $N=9$, $\alpha=0.5$, $a_{1}=0.2$, and $a_{2}=0.6$

####  3.1.2 Asymmetric Case

Now, we consider the asymmetric system (13) with $a_{0}=-a_{2}$. We define the
cardioids

$\gamma_{j}=\left(Re\left[e^{\iota t}\left(1-e^{-\iota
t}\right)^{\alpha}\right]+1,\frac{1}{2\sin(2\pi j/N)}Im\left[e^{\iota
t}\left(1-e^{-\iota t}\right)^{\alpha}\right]\right),\quad j=1,2,\cdots,[N/2]$
(21)

in the $a_{1}a_{2}$-plane, provided $\sin(2\pi j/N)\neq 0$.
The stability result in this case is discussed below. Note that $\lceil
r\rceil$ is the ceiling function of real number $r$.

###### Theorem 3.4

Consider the system (13) with $a_{0}=-a_{2}$. We have following stability
results:

  * 1.

If $N=1$ or $N=2$ then the stable region is $1-2^{\alpha}<a_{1}<1$.

  * 2.

If $N\geq 3$ is an odd number then the stable region is bounded by the line
$a_{1}=1$ and the cardioid $\gamma_{\lceil\frac{N-1}{4}\rceil}$ in the
$a_{1}a_{2}$-plane.

  * 3.

If $N\geq 4$ is an even number then the stable region is bounded by the line
$a_{1}=1$ and the cardioid $\gamma_{\left[\frac{N}{4}\right]}$ in the
$a_{1}a_{2}$-plane.

Proof: Suppose that $a_{0}=-a_{2}$ in (13) . Therefore, the eigenvalues of the
coefficient matrix $A$ are

$\lambda_{j}=a_{1}+\iota 2a_{2}\sin\left(\frac{2\pi j}{N}\right),\quad
j=0,1,\cdots,N-1.$ (22)

It is observed that for $j=[N/2]+1,[N/2]+2,\cdots,N-1$, the values
$\lambda_{j}$ are complex conjugates of those for $j=0,1,\cdots,[N/2]$.
Therefore, the stable region corresponding to $\lambda_{j}$ is given by the
cardioid $\gamma_{j}$ defined in (21).
Since, $\lambda_{0}=a_{1}\in\mathbb{R}$, one of the stability conditions is

$1-2^{\alpha}<a_{1}<1.$ (23)

Further, if $N=1$ or $N=2$ then $a_{1}$ is the only eigenvalue of matrix $A$.
Therefore, the stability condition is given by (23).
For $N\geq 3$, the stable region is an intersection of the cardioids
$\gamma_{j}$ and the region (23). It is observed that, this region is bounded
by the “innermost” cardioid and the line $a_{1}=1$ in the $a_{1}a_{2}$-plane,
as shown in the Figure 11.

Figure 11: Stable region of asymmetric system with $a_{0}=-a_{2}$

Now, we have to find the $j$ for which the cardioid $\gamma_{j}$ is innermost.
The innermost cardioid is generated by $\gamma_{j}$ for which the value
$\sin(2\pi j/N)$ is maximum. Further, the value $\sin(2\pi j/N)$ is maximum
for the number $2\pi j/N$ which is closest to $\pi/2$ i.e. if the value
$|\frac{2\pi j}{N}-\frac{\pi}{2}|=\frac{\pi}{2N}|4j-N|$ is minimum.
Thus, our problem is reduced to find minimum of the set

$S=\left\\{|4j-N|:j=1,2,\cdots,[N/2]\right\\}.$ (24)

If $N$ is even number, then the minimum of $S$ occurs at $j=[N/4]$. On the
other hand, if $N$ is an odd number, then the minimum of $S$ occurs at
$j=\lceil\frac{N-1}{4}\rceil$.
The result is proved.

We note that if the number of maps is multiple of 4, say $N=4K$,
$\lambda_{K}=a_{1}+\iota 2a_{2}$ and $\lambda_{N-K}=a_{1}-\iota 2a_{2}$. Also,
$\lambda_{0}=a_{1}$ for any $N$. The cardioid $\gamma_{K}$ defined above
reduces to cardioid for given value of $\alpha$ for $j=0$ where real part is
given by $a_{1}$ and imaginary part is $2a_{2}$. For $j=0$, $\gamma_{j}$ is
strip between $1-2^{\alpha}\leq a_{0}\leq 1$ with no condition on $a_{2}$. The
stability region is given by intersection of this strip with the cardioid
$\gamma_{K}$ for given $\alpha$ for $N=4K$. If $N$ is not an exact multiple of
4, the stability region is slightly bigger and as expected it shrinks to
stability region for $N=4$ in the thermodynamic limit.

###### Example 3.4

We consider the system (13) with $a_{0}=-a_{2}$, $N=6$ and $\alpha=0.3$. Here,
$N$ is even and $[N/4]=1$. According to Theorem 3.4, the stable region is
bounded by the cardioid $\gamma_{1}$ and the line $a_{1}=1$ in the
$a_{1}a_{2}$-plane, as shown in Figure 12.

Figure 12: Stable region of system (13) with $a_{0}=-a_{2}$, $N=6$ and
$\alpha=0.3$

The point $a_{1}=-0.3,a_{2}=0.5$ is outside the stable region and therefore we
get unstable solution (cf. Figure 13). On the other hand, we get the stable
solution (cf. Figure 14) for the parameter values $a_{1}=-0.1,a_{2}=-0.22$.

Figure 13: Unstable solution of (13) with $a_{1}=-0.3,a_{2}=0.5$,
$a_{0}=-a_{2}$, $N=6$ and $\alpha=0.3$ Figure 14: Stable solution of system
(13) with $a_{1}=-0.1,a_{2}=-0.22$, $a_{0}=-a_{2}$, $N=6$ and $\alpha=0.3$

We also verified the Theorem 3.4 for odd values of $N$ but not presented the
example for brevity.

####  3.1.3 Thermodynamic limit

The coupled map lattice (13) in the thermodynamic limit
$N\longrightarrow\infty$ is an interesting system studied in the literature
[24, 25, 26]. This limit may gives rise to some important phenomena such as
rescaling of the Lyapunov spectrum [27]. The physical interpretation of this
limit [28, 29] is that a coupled map lattice with a very large number of
lattice points may be identified as a chain of relatively small-sized
independently evolving subsystems.

As $N\longrightarrow\infty$, $\cos(\pi/N)\longrightarrow 1$. Therefore, the
stable region of the symmetric system (13) in the $a_{1}a_{2}$-plane according
to Theorem 3.3 is bounded by the quadrilateral $Q_{1}Q_{2}Q_{3}Q_{4}$ in the
thermodynamic limit.

As $N\longrightarrow\infty$, the interval $[0,2\pi]$ will contain an
infinitely many values of the form $2\pi j/N$, $j=1,2,\cdots,N-1$. Therefore,
the maximum of $\sin(2\pi j/N)$ will approach to $1$ as
$N\longrightarrow\infty$. Therefore, the stable region in the
$a_{1}a_{2}$-plane of the asymmetric system (13) with $a_{0}=-a_{2}$ according
to Theorem 3.4 is bounded by the line $a_{1}=1$ and the cardioid

$\gamma_{\infty}=\left(Re\left[e^{\iota t}\left(1-e^{-\iota
t}\right)^{\alpha}\right]+1,\frac{1}{2}Im\left[e^{\iota t}\left(1-e^{-\iota
t}\right)^{\alpha}\right]\right)$

in the thermodynamic limit.

Though we have studied 1-dimensional case in detail, the formulation is very
generic and can be extended to any case where the eigenvalues of the
connectivity matrix can be computed analytically. Consider a 2-dimensional
case with $NM$ maps with couplings $A_{(i,j),(i\pm 1,j)}=a_{0}$
$A_{(i,j),(i,j\pm 1)}=a_{2}$ $A_{(i,j),(i,j)}=a_{1}$. This is a block-
circulant matrix with circulant blocks and the eigenvalues are given by
$\lambda_{k1,k2}=a_{1}+2a_{0}\cos(\theta_{k1})+2a_{2}\cos(\theta_{k2})$ where
$\theta_{k1}={\frac{2\pi k_{1}}{N}}$ and $\theta_{k2}={\frac{2\pi k_{2}}{M}}$.
The indices $k_{1}$ and $k_{2}$ run from 0 to $N-1$ and 0 to $M-1$
respectively [30]. The bounds are given by $a_{1}+2a_{0}+2a_{2}$ and
$a_{1}-2a_{0}-2a_{2}$ in the thermodynamic limit (assuming all off-diagonal
couplings positive) and the stability region is given by quadrilateral where
both these bounds are in the range $[-2^{\alpha}+1,1]$. Thus the formulation
allows us to analytically find the stability of a coupled map lattice with any
connectivity matrix if the eigenvalues can be determined analytically. If we
couple each site to $B$ nearest neighbors instead of just one neighbor or to
$k$ randomly chosen sites [31, 32] the eigenvalues of the connectivity matrix
can be found analytically. The stability conditions for synchronized fixed
point can be studied even in such cases. If the eigenvalues can be determined
only numerically, we can still use the stability conditions to explore the
stability of the system by systematically increasing the system size.

##  4 Fractional order coupled map lattices: Nonlinear systems

Consider the nonlinear coupled map lattice of fractional order
$\alpha\in(0,1]$

$x_{t+1}(k)=x_{0}(k)+\sum_{j=0}^{t}\frac{\Gamma(t-j+\alpha)}{\Gamma(\alpha)\Gamma(t-j+1)}\left(f_{0}\left(x_{j}(k-1)\right)+f_{1}\left(x_{j}(k)\right)-x_{j}(k)+f_{2}\left(x_{j}(k+1)\right)\right),$
(25)

where $k=1,2,\cdots,N$, $x_{t}(0)=x_{t}(N)$, $x_{t}(N+1)=x_{t}(1)$ and the
functions $f_{k}:\mathbb{R}\longrightarrow\mathbb{R}$, $k=0,1,2$ are
continuously differentiable functions.
If we define $X_{t}$ as in Section 3 and
$F:\mathbb{R}^{N}\longrightarrow\mathbb{R}^{N}$ as

$F(X_{t})=\begin{bmatrix}f_{0}\left(x_{j}(N)\right)+f_{1}\left(x_{j}(1)\right)+f_{2}\left(x_{j}(2)\right)\\\
f_{0}\left(x_{j}(1)\right)+f_{1}\left(x_{j}(2)\right)+f_{2}\left(x_{j}(3)\right)\\\
f_{0}\left(x_{j}(2)\right)+f_{1}\left(x_{j}(3)\right)+f_{2}\left(x_{j}(4)\right)\\\
\vdots\\\
f_{0}\left(x_{j}(N-2)\right)+f_{1}\left(x_{j}(N-1)\right)+f_{2}\left(x_{j}(N)\right)\\\
f_{0}\left(x_{j}(N-1)\right)+f_{1}\left(x_{j}(N)\right)+f_{2}\left(x_{j}(1)\right)\end{bmatrix}$

then the system (25) is equivalent to

$X_{t+1}=X_{0}+\sum_{j=0}^{t}\frac{\Gamma(t-j+\alpha)}{\Gamma(\alpha)\Gamma(t-j+1)}\left[F\left(X_{j}\right)-X_{j}\right].$
(26)

A point $X_{*}=\left(x_{*}(1),x_{*}(2),\cdots,x_{*}(N)\right)$ is called an
equilibrium point of (26) if it is a fixed point of function $F$ [33].
Therefore, such a point must satisfy

$f_{0}\left(x_{*}(j-1)\right)+f_{1}\left(x_{*}(j)\right)+f_{2}\left(x_{*}(j+1)\right)=x_{*}(j),\quad
j=1,2,\cdots,N.$ (27)

For simplicity, we assume that the equilibrium point is homogeneous, i.e.
$X_{*}=\left(x_{*},x_{*},\cdots,x_{*}\right)$ so that the conditions (27) get
reduced to a single condition

$f_{0}\left(x_{*}\right)+f_{1}\left(x_{*}\right)+f_{2}\left(x_{*}\right)=x_{*}.$
(28)

If we identify $a_{0}=f_{0}^{\prime}\left(x_{*}\right)$,
$a_{1}=f_{1}^{\prime}\left(x_{*}\right)$ and
$a_{2}=f_{2}^{\prime}\left(x_{*}\right)$ then the linearization of (26) at
homogeneous equilibrium point $X_{*}$ is given by the equation (13).
Furthermore, if we assume the condition (28) then all the stability results
viz. Theorems 3.1, 3.2, 3.3 and 3.4 can be used to analyze the stability of
$X_{*}$. We illustrate these results in the following examples.

###### Example 4.1

Consider $f_{1}(x)=\mu x(1-x)$, the logistic map [34] and
$f_{2}(x)=f_{0}(x)=4x^{3}-\delta x$.

Here, the origin $X_{*}=(0,0,\cdots,0)$ is an equilibrium point of (26).
Further, $a_{1}=f_{1}^{\prime}(0)=\mu$ and
$a_{0}=a_{2}=f_{2}^{\prime}(0)=-\delta$. Therefore, for $\alpha=0.6$ and $N=4$
the stable region in $\delta\mu$-plane is sketched in Figure 15. The stable
orbits for $\mu=0.05$, $\delta=-0.1$ are plotted in Figure 16.

Figure 15: Stable region of origin of (26) in Ex. 4.1 with $N=4$ and
$\alpha=0.6$ Figure 16: Stable orbits of system (26) in Ex. 4.1 with
$\mu=0.05$, $\delta=-0.1$, $N=4$ and $\alpha=0.6$

###### Example 4.2

We consider $f_{1}(x)=\mu x(1-x)$ and the circle map [35]
$f_{2}(x)=x+\delta\sin(x)$. We also set $f_{0}(x)=-f_{2}(x)$ so that the
system (26) is asymmetric.

Again, we have origin as equilibrium $X_{*}$ and $a_{1}=\mu$,
$a_{2}=-a_{0}=1+\delta$. We take $\alpha=0.8$ and $N=7$. The stable region
shown in Figure 17 is bounded by the line $\mu=1$ and the cardioid
$\gamma_{2}$. The parameter values $\mu=0.6,\delta=-0.8$ in the stable region
give rise to stable orbits as shown in Figure 18. If we take $\mu=1.1$ and
$\delta=-1.2$ in the unstable region, then the trajectories repelled by origin
are attracted by another homogeneous equilibrium point with $x_{*}=1-1/\mu$
for the sufficiently small positive initial conditions (cf. Figure 19). Note
that the trajectories will be unbounded if we take negative initial
conditions, in this case.

Figure 17: Stable region of origin of (26) in Ex. 4.2 with $N=7$ and
$\alpha=0.8$ Figure 18: Stable orbits of system (26) in Ex. 4.2 with
$\mu=0.6,\delta=-0.8$, $N=7$ and $\alpha=0.8$ Figure 19: Orbits of system (26)
in Ex. 4.2 diverging from origin for $\mu=1.1$, $\delta=-1.2$, $N=4$ and
$\alpha=0.6$

###### Example 4.3

Consider $f(x)=\mu x(1-x)$ and define $f_{1}(x)=(1-\epsilon)f(x)$ and
$f_{0}(x)=f_{2}(x)=\frac{\epsilon}{2}f(x)$.

If $x_{*}$ is a fixed point of $f$ then $f(x_{*})=x_{*}$ and hence the
condition (28) is satisfied. Therefore, for this choice of functions the
system (26) will have two homogeneous equilibrium points viz.
$X_{1*}=(0,0,\cdots,0)$ and $X_{2*}=(q,q,\cdots,q)$, where
$q=\frac{\mu-1}{\mu}$.
Stability of $X_{1*}$:
Here, $a_{1}=f_{1}^{\prime}(0)=\mu(1-\epsilon)$ and
$a_{2}=f_{2}^{\prime}(0)=\epsilon\mu/2$. Therefore,
$\epsilon=\frac{2a_{2}}{a_{1}+2a_{2}}$ and $\mu=a_{1}+2a_{2}$. The stable
region of $X_{1*}$ in the $\epsilon\mu$-plane can now be obtained using the
Theorem 3.3 by substituting the values of $a_{1}$ and $a_{2}$ in the
expressions of $\epsilon$ and $\mu$ for various values of $N$ and $\alpha$.
Stability of $X_{2*}$:
In this case, $a_{1}=f_{1}^{\prime}(q)=(1-\epsilon)(2-\mu)$ and
$a_{2}=f_{2}^{\prime}(q)=\epsilon(2-\mu)/2$. On simplifying, we get
$\epsilon=\frac{2a_{2}}{a_{1}+2a_{2}}$ and $\mu=2-a_{1}-2a_{2}$. The stable
region of $X_{2*}$ can now be traced in $\epsilon\mu$-plane by utilizing
Theorem 3.3.
The asymmetric case $f_{0}(x)=-f_{2}(x)$ can also be done in a similar way.

##  5 Discussion

As mentioned above, if the eigenvalues of underlying connectivity matrix can
be found analytically, the stability of the synchronized state becomes very
simple even for coupled fractional maps with an altered stability condition.
One possible extension is stability analysis of spatially periodic fixed
point. If an unsynchronized but spatially periodic fixed point is realized in
fractional coupled maps (which is possible only in nonlinear systems), the
Jacobian can be block diagonalized. These blocks have a dimension of
periodicity in space [30]. This simplifies the stability analysis
considerably.

Transition to a frozen or absorbing state is an extensively studied transition
in nonequilibrium statistical physics which includes systems such as coupled
oscillators. (Such transition is not possible in equilibrium systems because
detailed balance cannot be violated.) The above work allows us to study such
dynamical systems in presence of memory. The thermodynamic and asymptotic
limit is important because phase can be defined only for the state of an
infinite system after infinite time. The above analysis gives an analytic
estimate for critical point for such system and also gives important
information about the nature of instability. Of course, such systems can have
a very different nature. For coupled fractional maps, a power-law decay is
obtained throughout the absorbing phase and not just the critical point [36].
Thus nature of transition can be very different. In integer order maps, the
bifurcation depends on whether the eigenvalue crosses the unit circle at 1,
-1, or complex value [2]. It also depends on which eigenmodes become unstable
[37]. Similar studies can be carried out in fractional systems.

## Acknowledgment

S. Bhalekar acknowledges the University of Hyderabad for Institute of
Eminence-Professional Development Fund (IoE-PDF) by MHRD (F11/9/2019-U3(A)).
P. M. Gade thanks DST-SERB for financial assistance (Ref. CRG/2020/003993).

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# Notes on real interpolation of operator $L_{p}$-spaces

Marius Junge  Department of Mathematics, University of Illinois, Urbana, IL
61801, USA<EMAIL_ADDRESS>and  Quanhua Xu  Institute for Advanced Study
in Mathematics, Harbin Institute of Technology, Harbin 150001, China; and
Laboratoire de Mathématiques, Université de Bourgogne Franche-Comté, 25030
Besançon Cedex, France<EMAIL_ADDRESS>
###### Abstract.

Let $\mathcal{M}$ be a semifinite von Neumann algebra. We equip the associated
noncommutative $L_{p}$-spaces with their natural operator space structure
introduced by Pisier via complex interpolation. On the other hand, for
$1<p<\infty$ let

$L_{p,p}(\mathcal{M})=\big{(}L_{\infty}(\mathcal{M}),\,L_{1}(\mathcal{M})\big{)}_{\frac{1}{p},\,p}$

be equipped with the operator space structure via real interpolation as
defined by the second named author (J. Funct. Anal. 139 (1996), 500–539). We
show that $L_{p,p}(\mathcal{M})=L_{p}(\mathcal{M})$ completely isomorphically
if and only if $\mathcal{M}$ is finite dimensional. This solves in the
negative the three problems left open in the quoted work of the second author.

We also show that for $1<p<\infty$ and $1\leq q\leq\infty$ with $p\neq q$

$\big{(}L_{\infty}(\mathcal{M};\ell_{q}),\,L_{1}(\mathcal{M};\ell_{q})\big{)}_{\frac{1}{p},\,p}=L_{p}(\mathcal{M};\ell_{q})$

with equivalent norms, i.e., at the Banach space level if and only if
$\mathcal{M}$ is isomorphic, as a Banach space, to a commutative von Neumann
algebra.

Our third result concerns the following inequality:

$\big{\|}\big{(}\sum_{i}x_{i}^{q}\big{)}^{\frac{1}{q}}\big{\|}_{L_{p}(\mathcal{M})}\leq\big{\|}\big{(}\sum_{i}x_{i}^{r}\big{)}^{\frac{1}{r}}\big{\|}_{L_{p}(\mathcal{M})}$

for any finite sequence $(x_{i})\subset L_{p}^{+}(\mathcal{M})$, where
$0<r<q<\infty$ and $0<p\leq\infty$. If $\mathcal{M}$ is not isomorphic, as a
Banach space, to a commutative von Meumann algebra, then this inequality holds
if and only if $p\geq r$.

2000 Mathematics Subject Classification: Primary: Secondary:

Key words: Operator spaces, $L_{p}$-spaces, real interpolation, column
Hilbertian spaces

##  1\. Introduction

Interpolation of $L_{p}$-spaces is a classical subject. Our reference for
interpolation theory is [1]. Let $(\Omega,\mu)$ be a measure space. Let $1\leq
p_{0},p_{1},p\leq\infty$ and $0<\theta<1$ with
$\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}$. The following well
known interpolation equalities hold

(1)
$\big{(}L_{p_{0}}(\Omega),\,L_{p_{1}}(\Omega)\big{)}_{\theta}=L_{p}(\Omega)\;\text{
with equal norms},$ (2)
$\big{(}L_{p_{0}}(\Omega),\,L_{p_{1}}(\Omega)\big{)}_{\theta,p}=L_{p}(\Omega)\;\text{
with equivalent norms}.$

Here $(\,\cdot,\,\cdot\,)_{\theta}$ and $(\,\cdot,\,\cdot\,)_{\theta,p}$
denote, respectively, the complex and real interpolation functors. It is also
well known that the above equalities hold for vector-valued $L_{p}$-spaces.
More precisely, under the same assumption on the parameters (assuming
additionally $p<\infty$), we have

(3)
$\big{(}L_{p_{0}}(\Omega;E_{0}),\,L_{p_{1}}(\Omega;E_{1})\big{)}_{\theta}=L_{p}\big{(}\Omega;(E_{0},\,E_{1})_{\theta}\big{)}\;\text{
with equal norms},$ (4)
$\big{(}L_{p_{0}}(\Omega;E_{0}),\,L_{p_{1}}(\Omega;E_{1})\big{)}_{\theta,p}=L_{p}\big{(}\Omega;(E_{0},\,E_{1})_{\theta,p}\big{)}\;\text{
with equivalent norms}$

for any compatible pair $(E_{0},\,E_{1})$ of Banach spaces.

The present note concerns interpolation theory in the category of operator
spaces. We refer to [2, 10] for operator space theory. The complex and real
interpolations for operator spaces are developed in [8] and [13],
respectively. Unless explicitly stated otherwise, all (commutative and
noncommutative) $L_{p}$-spaces in the sequel are equipped with their natural
operator space structure as defined in [9, 10]. Pisier proved that (1) and (3)
remain true in the category of operator spaces, that is, these equalities hold
completely isometrically, $(E_{0},\,E_{1})$ being, of course, assumed to be
operator spaces in the case of (3).

It is more natural to work with noncommutative $L_{p}$-spaces in the category
of operator spaces. Let ${\mathcal{M}}$ be a semifinite von Neumann algebra
equipped with a normal semifinite faithful trace $\tau$. Let
$L_{p}({\mathcal{M}})$ denote the associated noncommutative $L_{p}$-space (cf.
[11]). If ${\mathcal{M}}=B(\ell_{2})$ with the usual trace,
$L_{p}({\mathcal{M}})$ is the Schatten $p$-class $S_{p}$. If ${\mathcal{M}}$
is hyperfinite, Pisier [9] also introduced the vector-valued
$L_{p}({\mathcal{M}};E)$ for an operator space $E$, and showed that (3)
continues to hold in this more general setting:

(5)
$\big{(}L_{p_{0}}({\mathcal{M}};E_{0}),\,L_{p_{1}}({\mathcal{M}};E_{1})\big{)}_{\theta}=L_{p}\big{(}{\mathcal{M}};(E_{0},\,E_{1})_{\theta}\big{)}\;\text{
completely isometrically}$

for any hyperfinite ${\mathcal{M}}$ and any compatible pair $(E_{0},\,E_{1})$
of operator spaces.

However, real interpolation does not behave as smoothly as complex
interpolation in the category of operator spaces. The problem whether (5) can
be extended to real interpolation was left unsolved in [13], see Problems 6.1,
6.2 and 6.4 there. Let $1<p<\infty$. Using the Banach space equality

$L_{p}({\mathcal{M}})=\big{(}L_{\infty}({\mathcal{M}}),\,L_{1}({\mathcal{M}})\big{)}_{\frac{1}{p},p},$

we can equip $L_{p}({\mathcal{M}})$ with another operator space structure via
real interpolation as in [13], the resulting operator space is denoted by
$L_{p,p}({\mathcal{M}})$. Then Problem 6.1 of [13] asks whether
$L_{p,p}({\mathcal{M}})=L_{p}({\mathcal{M}})$ completely isomorphically for
$p\neq 2$ (the answer is affirmative for $p=2$). The following result resolves
this problem in the negative.

######  Theorem 1.

Let $1<p<\infty$ with $p\neq 2$. Then
$L_{p,p}({\mathcal{M}})=L_{p}({\mathcal{M}})$ completely isomorphically if and
only if ${\mathcal{M}}$ is finite dimensional.

Consequently, the answers to all three Problems 6.1, 6.2 and 6.4 of [13] are
negative. In particular, neither (2) nor (4) holds in the category of operator
spaces.

The next theorem provides an even worse answer to Problem 6.4. It shows that
(5) extends to real interpolation at the Banach space level only in the
commutative case. Recall that one can define $L_{p}({\mathcal{M}};\ell_{q})$
for any von Neumann algebra ${\mathcal{M}}$ (see section 3 for more
information). $L_{p}({\mathcal{M}};\ell_{q})$ coincides with Pisier’s space
when ${\mathcal{M}}$ is hyperfinite. Note that $L_{p}({\mathcal{M}};\ell_{q})$
is defined only as a Banach space if ${\mathcal{M}}$ is not hyperfinite.

######  Theorem 2.

Let $1<p<\infty$ and $1\leq q\leq\infty$ with $p\neq q$. Then

(6)
$\big{(}L_{\infty}({\mathcal{M}};\ell_{q}),\,L_{1}({\mathcal{M}};\ell_{q})\big{)}_{\frac{1}{p},p}=L_{p}({\mathcal{M}};\ell_{q})\;\text{
with equivalent norms}$

if and only if ${\mathcal{M}}$ is isomorphic, as a Banach space, to
$L_{\infty}(\Omega,\mu)$ for some measure space $(\Omega,\mu)$.

Our third theorem does not concern the real interpolation of the
$L_{p}({\mathcal{M}};\ell_{q})$ spaces but gives a result that is to be
compared with the norm of these spaces. In the commutative case, the norm of a
sequence $(x_{i})$ in $L_{p}(\Omega;\ell_{q})$ is given by

$\|(x_{i})\|_{L_{p}(\Omega;\ell_{q})}=\big{\|}\big{(}\sum_{i}|x_{i}|^{q}\big{)}^{\frac{1}{q}}\big{\|}_{L_{p}(\Omega)}\,.$

It is well known that this expression is no longer valid in the noncommutative
setting, which is one source of many difficulties in noncommutative analysis.
The following theorem shows that another classical property of the norm
$\|(x_{i})\|_{L_{p}(\Omega;\ell_{q})}$ does not extend to the noncommutative
case. The index $p$ is now allowed to go below $1$, $L_{p}^{+}({\mathcal{M}})$
denotes the positive cone of $L_{p}({\mathcal{M}})$.

######  Theorem 3.

Let $0<r<q<\infty$ and $0<p\leq\infty$.

  1. (i)

If $p\geq r$, then

(7)
$\big{\|}\big{(}\sum_{i}x_{i}^{q}\big{)}^{\frac{1}{q}}\big{\|}_{L_{p}({\mathcal{M}})}\leq\big{\|}\big{(}\sum_{i}x_{i}^{r}\big{)}^{\frac{1}{r}}\big{\|}_{L_{p}({\mathcal{M}})}$

for any finite sequence $(x_{i})\subset L_{p}^{+}({\mathcal{M}})$.

  2. (ii)

If $p<r$ and ${\mathcal{M}}$ is not isomorphic, as a Banach space, to
$L_{\infty}(\Omega,\mu)$ for some measure space $(\Omega,\mu)$, then there
exists no constant $C$ such that

(8)
$\big{\|}\big{(}\sum_{i}x_{i}^{q}\big{)}^{\frac{1}{q}}\big{\|}_{L_{p}({\mathcal{M}})}\leq
C\,\big{\|}\big{(}\sum_{i}x_{i}^{r}\big{)}^{\frac{1}{r}}\big{\|}_{L_{p}({\mathcal{M}})}$

for any finite sequence $(x_{i})\subset L_{p}^{+}({\mathcal{M}})$.

The previous theorems will be respectively proved in the next three sections.

##  2\. Proof of Theorem 1

We will need some preparations on column and row Hilbertian operator spaces.
Let $C_{p}$ (resp. $R_{p}$) denote the first column (resp. row) subspace of
$S_{p}$ consisting of matrices whose all entries but those in the first column
(resp. row) vanish. We have the following completely isometric
identifications:

(9) $(C_{p})^{*}\cong C_{p^{\prime}}\cong
R_{p}\quad\mbox{and}\quad(R_{p})^{*}\cong R_{p^{\prime}}\cong
C_{p}\,,\quad\forall\;1\leq p\leq\infty,$

where $p^{\prime}$ denotes the conjugate index of $p$. $C_{p}$ and $R_{p}$ can
be also defined via complex interpolation from $C=C_{\infty}$ and
$R=R_{\infty}$. We view $(C,R)$ as a compatible pair by identifying both of
them with $\ell_{2}$ (at the Banach space level), i.e., by identifying the
canonical bases $(e_{k,1})$ of $C_{p}$ and $(e_{1,k})$ of $R_{p}$ with
$(e_{k})$ of $\ell_{2}$. Then we have the following completely isometric
equalities

(10) $C_{p}=(C,\ R)_{\frac{1}{p}}=(C_{\infty},\
C_{1})_{\frac{1}{p}}\quad\mbox{and}\quad R_{p}=(R,\
C)_{\frac{1}{p}}=(R_{\infty},\ R_{1})_{\frac{1}{p}}.$

We refer to [8, 9] for more details.

Let ${\rm Rad}_{p}$ be the closed subspace spanned by the Rademacher sequence
$(\varepsilon_{n})$ in $L_{p}([0,\,1])$. Then the noncommutative Khintchine
inequality can be reformulated in terms of column and row spaces (see [6, 9]).
To this end, we introduce

$CR_{p}=C_{p}+R_{p}\;\text{for}\;p\leq 2\quad\text{and}\quad CR_{p}=C_{p}\cap
R_{p}\;\text{for}\;p>2.$

######  Lemma 4.

Let $1<p<\infty$. Then ${\rm Rad}_{p}=CR_{p}$ completely isomorphically.
Moreover, the orthogonal projection from $L_{2}([0,\,1])$ onto ${\rm Rad}_{2}$
extends to a completely bounded projection from $L_{p}([0,\,1])$ onto ${\rm
Rad}_{p}$. All relevant constants depend only on $p$.

If $E$ is an operator space, $C_{p}(E)$ (resp. $R_{p}(E)$) denotes the first
column (resp. row) subspace of the $E$-valued Schatten class $S_{p}(E)$. It is
clear that $C_{p}(E)$ and $R_{p}(E)$ are completely 1-complemented in
$S_{p}(E)$. Consequently, applying (5) to ${\mathcal{M}}=B(\ell_{2})$, we get

(11)
$\big{(}C_{p_{0}}(E_{0}),\,C_{p_{1}}(E_{1})\big{)}_{\theta}=C_{p}((E_{0},\,E_{1})_{\theta})$

for any compatible pair $(E_{0},\,E_{1})$ of operator spaces, where
$\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}$.

Note that $C_{p}(R_{p})=S_{p}$ and $C_{p}(C_{p})=S_{2}$ isometrically at the
Banach space level. Here we represent the elements in $C_{p}(R_{p})$ and
$C_{p}(C_{p})$ by infinite matrices in the canonical bases of $C_{p}$ and
$R_{p}$. The following elementary fact is known to experts and implicitly
contained in the proof of [14, Lemma 5.9]. We include its simple proof for
completeness.

######  Lemma 5.

Let $1\leq p,q\leq\infty$. Let $r$ be determined by
$\frac{1}{r}=\frac{1}{2p}+\frac{1}{2q^{\prime}}$. Then

$C_{p}(CR_{q})=S_{r}\;\text{ with equivalent norms}.$

###### Proof.

By (5), (11) and (9), we have the following isometric equalities

$\displaystyle C_{\infty}(C_{q})$
$\displaystyle=\big{(}C_{\infty}(C_{\infty}),\,C_{\infty}(C_{1})\big{)}_{\frac{1}{q}}$
$\displaystyle=\big{(}C_{\infty}(C_{\infty}),\,C_{\infty}(R_{\infty})\big{)}_{\frac{1}{q}}$
$\displaystyle=\big{(}S_{2},\,S_{\infty}\big{)}_{\frac{1}{q}}=S_{2q^{\prime}}\,.$

Similarly, $C_{1}(C_{q})=S_{(2q)^{\prime}}$ isometrically. Thus

$C_{p}(C_{q})=\big{(}C_{\infty}(C_{q}),\,C_{1}(C_{q})\big{)}_{\frac{1}{p}}=\big{(}S_{2q^{\prime}},\,S_{(2q)^{\prime}}\big{)}_{\frac{1}{p}}=S_{r}.$

Combining this with (9), we also have

$C_{p}(R_{q})=C_{p}(C_{p^{\prime}})=S_{t}\;\text{ isometrically},$

where $\frac{1}{t}=\frac{1}{2p}+\frac{1}{2q}$. Hence,

$C_{p}(CR_{q})=C_{p}(C_{q})+C_{p}(R_{q})=S_{r}+S_{t}=S_{r}$

for $q\leq 2$ and $C_{p}(CR_{q})=S_{r}\cap S_{t}=S_{r}$ for $q>2$ too. ∎

###### Proof of Theorem 1.

By the type decomposition of von Neumann algebras, if ${\mathcal{M}}$ is not
finite dimensional, then ${\mathcal{M}}$ contains an infinite dimensional
commutative $L_{\infty}(\Omega,\mu)$ as subalgebra which is moreover the image
of a trace preserving normal conditional expectation. Indeed, if the type I
summand of ${\mathcal{M}}$ is infinite dimensional, then ${\mathcal{M}}$
contains an infinite dimensional commutative $L_{\infty}(\Omega,\mu)$. On the
other hand, if the type II∞ summand of ${\mathcal{M}}$ exists, then
${\mathcal{M}}$ contains ${\mathcal{B}}(\ell_{2})$, so $\ell_{\infty}$ too.
Finally, if the type II1 summand of ${\mathcal{M}}$ exists, then
${\mathcal{M}}$ contains $L_{\infty}([0,\,1])$. See [12] for the type
decomposition of von Neumann algebras.

Note that if $L_{\infty}(\Omega,\mu)$ is infinite dimensional,
$L_{\infty}(\Omega,\mu)$ contains, as subalgebra, either $L_{\infty}([0,\,1])$
or $\ell_{\infty}$. On the other hand, if
$L_{p,p}({\mathcal{M}})=L_{p}({\mathcal{M}})$ held for
${\mathcal{M}}=\ell_{\infty}$, it would do so for
${\mathcal{M}}=\ell^{n}_{\infty}$ uniformly in $n\geq 1$. Then by a standard
approximation argument, we see that it would hold for
${\mathcal{M}}=L_{\infty}([0,\,1])$ too.

Thus we are reduced to showing the theorem for the special case
${\mathcal{M}}=L_{\infty}([0,\,1])$. Namely, we must show that
$L_{p,p}([0,\,1])\not=L_{p}([0,\,1])$ completely isomorphically. In the rest
of the proof, we will drop the interval $[0,\,1]$ from $L_{p}([0,\,1])$. By
[13, Theorem 5.4], we choose $1<p_{0}<p_{1}<\infty$ and $0<\theta<1$ such that
$\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}$ and

$L_{p,p}=\big{(}L_{p_{0}},\,L_{p_{1}}\big{)}_{\theta,p}\,.$

Then by [13, Proposition 6.3] and its proof, we have

$S_{p}(L_{p,p})=\big{(}S_{p}(L_{p_{0}}),\,S_{p}(L_{p_{1}})\big{)}_{\theta,p}\,.$

Using the complete complementation of $C_{p}(E)$ in $S_{p}(E)$, we deduce

$C_{p}(L_{p,p})=\big{(}C_{p}(L_{p_{0}}),\,C_{p}(L_{p_{1}})\big{)}_{\theta,p}\,.$

On the other hand, by the complete complementation of ${\rm Rad}_{p_{i}}$ in
$L_{p_{i}}$ for $i=0,1$, we get the following isomorphic embedding:

$\big{(}C_{p}({\rm Rad}_{p_{0}}),\,C_{p}({\rm
Rad}_{p_{1}})\big{)}_{\theta,p}\subset C_{p}(L_{p,p}).$

Now by Lemmas 4 and 5,

$\big{(}C_{p}({\rm Rad}_{p_{0}}),\,C_{p}({\rm
Rad}_{p_{1}})\big{)}_{\theta,p}=(S_{r_{0}},\,S_{r_{1}})_{\theta,p}=S_{2,p}\;\text{
with equivalent norms},$

where $\frac{1}{r_{i}}=\frac{1}{2p}+\frac{1}{2p_{i}^{\prime}}$ for $i=0,1$.
Thus the closed subspace spanned by the Rademacher functions in
$C_{p}(L_{p,p})$ is isomorphic to $S_{2,p}$. However, that spanned by the same
functions in $C_{p}(L_{p})$ is isomorphic to $S_{2}$. But $S_{2,p}=S_{2}$ only
if $p=2$. Thus the theorem is proved. ∎

##  3\. Proof of Theorem 2

We begin this proof by recalling the definition of the space
$L_{p}({\mathcal{M}};\ell_{q})$ that is introduced in [3] for $q=1$ and
$q=\infty$ (see also [4]), and in [5] for $1<q<\infty$. This definition is
inspired by Pisier’s description of the norm of
$L_{p}({\mathcal{M}};\ell_{q})$ in the hyperfinite case.

A sequence $(x_{i})$ in $L_{p}({\mathcal{M}})$ belongs to
$L_{p}({\mathcal{M}};\ell_{\infty})$ iff $(x_{i})$ admits a factorization
$x_{i}=ay_{i}b$ with $a,b\in L_{2p}({\mathcal{M}})$ and
$(y_{i})\in\ell_{\infty}(L_{\infty}({\mathcal{M}}))$. The norm of $(x_{i})$ is
then defined as

(12)
$\|(x_{i})\|_{L_{p}({\mathcal{M}};\ell_{\infty})}=\inf_{x_{i}=ay_{i}b}\,\|a\|_{L_{2p}({\mathcal{M}})}\,\|(y_{i})\|_{\ell_{\infty}(L_{\infty}({\mathcal{M}}))}\,\|b\|_{L_{2p}({\mathcal{M}})}\,.$

On the other hand, $L_{p}({\mathcal{M}};\ell_{1})$ is defined as the space of
all sequences $(x_{i})\subset L_{p}({\mathcal{M}})$ for which there exist
$a_{ij},b_{ij}\in L_{2p}({\mathcal{M}})$ such that

$x_{i}=\sum_{j}a_{ij}^{*}b_{ij}\,.$

$L_{p}({\mathcal{M}};\ell_{1})$ is equipped with the norm

$\|(x_{i})\|_{L_{p}({\mathcal{M}};\ell_{1})}=\inf_{x_{i}=\sum_{j}a_{ij}^{*}b_{ij}}\,\big{\|}\sum_{i,j}a_{ij}^{*}a_{ij}\big{\|}_{L_{p}({\mathcal{M}})}^{\frac{1}{2}}\,\big{\|}\sum_{i,j}b_{ij}^{*}b_{ij}\big{\|}_{L_{p}({\mathcal{M}})}^{\frac{1}{2}}\,.$

Now for $1<q<\infty$ we define $L_{p}({\mathcal{M}};\ell_{q})$ as a complex
interpolation space between $L_{p}({\mathcal{M}};\ell_{\infty})$ and
$L_{p}({\mathcal{M}};\ell_{1})$:

$L_{p}({\mathcal{M}};\ell_{q})=\big{(}L_{p}({\mathcal{M}};\ell_{\infty}),\;L_{p}({\mathcal{M}};\ell_{1})\big{)}_{\frac{1}{q}}\,.$

The following description of the norm of $L_{p}({\mathcal{M}};\ell_{q})$ is
proved in [9] for hyperfinite ${\mathcal{M}}$ and in [5] for a general
${\mathcal{M}}$.

######  Lemma 6.

Let $1\leq p,q\leq\infty$.

  1. (i)

If $p\leq q$,

$\|(x_{i})\|_{L_{p}({\mathcal{M}};\ell_{q})}=\inf_{x_{i}=ay_{i}b}\,\|a\|_{L_{2r}({\mathcal{M}})}\,\|(y_{i})\|_{\ell_{q}(L_{q}({\mathcal{M}}))}\,\|b\|_{L_{2r}({\mathcal{M}})}$

for any $(x_{i})\in L_{p}({\mathcal{M}};\ell_{q})$, where
$\frac{1}{r}=\frac{1}{p}-\frac{1}{q}$.

  2. (ii)

If $p\geq q$,

$\|(x_{i})\|_{L_{p}({\mathcal{M}};\ell_{q})}=\sup_{\|\alpha\|_{L_{2s}({\mathcal{M}})}\leq
1,\;\|\beta\|_{L_{2s}({\mathcal{M}})}\leq 1}\,\|(\alpha
x_{i}\beta)\|_{\ell_{q}(L_{q}({\mathcal{M}}))}$

for any $(x_{i})\in L_{p}({\mathcal{M}};\ell_{q})$, where
$\frac{1}{s}=\frac{1}{q}-\frac{1}{p}$.

We will again consider the column subspace $C_{p}(\ell_{q})$ of
$S_{p}(\ell_{q})$ for the proof of Theorem 2. As in the previous section, a
generic element $x\in C_{p}(\ell_{q})$ is viewed as an infinite matrix

$x=\sum_{i,j=1}^{\infty}x_{ij}e_{i,1}\otimes e_{j}.$

Let $D_{p,q}$ denote the diagonal subspace of $C_{p}(\ell_{q})$ consisting of
all $x$ with $x_{ij}=0$ for $i\neq j$.

######  Lemma 7.

Let $1\leq p,q\leq\infty$. Then $D_{p,q}$ is completely $1$-complemented in
$C_{p}(\ell_{q})$.

###### Proof.

The proof is very simple. It suffices to note that the canonical bases of
$C_{p}$ and $\ell_{q}$ are completely $1$-unconditional. A standard average
argument then yields the assertion. ∎

We will identify an element $x=(x_{i}e_{i,1})\in D_{p,q}$ with the sequence
$(x_{i})$.

######  Lemma 8.

Let $1\leq p,q\leq\infty$ and $r_{p,q}$ be determined by
$\frac{1}{r_{p,q}}=\frac{1}{2p}+\frac{1}{2q}$. Then $D_{p,q}=\ell_{r_{p,q}}$
with equal norms.

###### Proof.

First consider the cases $q=\infty$ and $q=1$. Let $x=(x_{i}e_{i,1})\in
D_{p,\infty}\subset S_{p}(\ell_{\infty})$. Let $a$ be the diagonal matrix with
the $x_{i}$’s as its diagonal entries. Then we have the following
factorization:

$x_{i}e_{i,1}=ae_{i,1}e_{11},\quad i\geq 1.$

Thus by the definition of the norm of $S_{p}(\ell_{\infty})$, we get

$\|x\|_{S_{p}(\ell_{\infty})}\leq\|a\|_{S_{2p}}\,\|(e_{i,1})\|_{\ell_{\infty}(B(\ell_{2}))}\,\|e_{11}\|_{S_{2p}}=\|x\|_{\ell_{2p}}\,.$

On the other hand, for $q=1$ we factorize $x$ as

$x_{i}e_{i,1}=[{\rm
sgn}(x_{i})|x_{i}|^{\alpha}e_{i,1}]\,[|x_{i}|^{1-\alpha}e_{1,1}]=a_{i}b_{i},$

where $\alpha=\frac{r_{p,1}}{2p}$. Therefore, by the definition of the norm of
$S_{p}(\ell_{1})$

$\displaystyle\|x\|_{S_{p}(\ell_{1})}$
$\displaystyle\leq\big{\|}\sum_{i}a_{i}a_{i}^{*}\big{\|}_{p}^{\frac{1}{2}}\big{\|}\sum_{i}b^{*}_{i}b_{i}\big{\|}_{p}^{\frac{1}{2}}$
$\displaystyle=\big{(}\sum_{i}|x_{i}|^{r_{p,1}}\big{)}^{\frac{1}{2p}}\big{(}\sum_{i}|x_{i}|^{r_{p,1}}\big{)}^{\frac{1}{2}}=\|x\|_{\ell_{r_{p,1}}}\,.$

Thus we have proved that for any $1\leq p\leq\infty$

$\ell_{r_{p,\infty}}\subset D_{p,\infty}\quad\text{ and
}\quad\ell_{r_{p,1}}\subset D_{p,1}\;\text{ contractively}.$

Dualizing these inclusions and using Lemma 7, we deduce the assertion for
$q=\infty$ and $q=1$. The case $1<q<\infty$ is then completed by complex
interpolation via (11) with the help of Lemma 7 again. ∎

######  Remark 9.

The previous lemma can be proved directly by Lemma 6 without passing to
complex interpolation.

###### Proof of Theorem 2.

If ${\mathcal{M}}$ is isomorphic, as Banach space, to some commutative
$L_{\infty}$, then ${\mathcal{M}}$ is a finite direct sum of algebras of the
form $L_{\infty}(\Omega,\mu)\otimes\mathbb{M}_{n}$, where $\mathbb{M}_{n}$ is
the $n\times n$ full matrix algebra. Then (6) goes back to (4).

Conversely, suppose that ${\mathcal{M}}$ is not isomorphic to a commutative
von Neumann algebra. Our first step is to reduce the non validity of (6) to
the special case where ${\mathcal{M}}=B(\ell_{\infty})$. To this end, we use
the type decomposition of ${\mathcal{M}}$. If the type I summand of
${\mathcal{M}}$ is infinite dimensional, then ${\mathcal{M}}$ contains
$\mathbb{M}_{n}$ for infinite many $n$’s. On the other hand, if the type II∞
summand of ${\mathcal{M}}$ exists, then ${\mathcal{M}}$ contains
${\mathcal{B}}(\ell_{2})$. Finally, it is well known that if the type II1
summand of ${\mathcal{M}}$ exists, then ${\mathcal{M}}$ contains the
hyperfinite II1 factor ${\mathcal{R}}$ (cf. e.g. [7]); ${\mathcal{R}}$ is the
von Neumann tensor of countable many copies of $(\mathbb{M}_{2},\,{\rm tr})$,
where ${\rm tr}$ is the normalized trace on $\mathbb{M}_{2}$; so
${\mathcal{M}}$ again contains $\mathbb{M}_{n}$ for infinite many $n$’s. Note
that in all the three cases, the $\mathbb{M}_{n}$’s contained in
${\mathcal{M}}$ are images of trace preserving normal conditional expectations
(up to a normalization in the type II1 case).

In summary, if ${\mathcal{M}}$ is not isomorphic to a commutative von Neumann
algebra, ${\mathcal{M}}$ contains $\mathbb{M}_{n}$ for infinite many $n$’s
which are images of trace preserving normal conditional expectations. This
shows that if (6) held for ${\mathcal{M}}$, then it would do so for
${\mathcal{M}}=\mathbb{M}_{n}$ for infinite many $n$’s; consequently, by
approximation, it would further hold for ${\mathcal{M}}=B(\ell_{2})$ too. This
finishes the announced reduction.

It remains to show that (6) fails for ${\mathcal{M}}=B(\ell_{2})$. Namely, we
must show

$\big{(}S_{\infty}(\ell_{q}),\,S_{1}(\ell_{q})\big{)}_{\frac{1}{p},p}\not=S_{p}(\ell_{q})\,.$

This is an easy consequence of the previous two lemmas. By Lemma 7,

$\big{(}D_{\infty,q},\,D_{1,q}\big{)}_{\frac{1}{p},p}\subset\big{(}S_{\infty}(\ell_{q}),\,S_{1}(\ell_{q})\big{)}_{\frac{1}{p},p},\quad\text{an
isometric embedding}.$

On the other hand, by Lemma 8,

$\big{(}D_{\infty,q},\,D_{1,q}\big{)}_{\frac{1}{p},p}=\big{(}\ell_{r_{\infty,q}},\,\ell_{r_{\infty,q}}\big{)}_{\frac{1}{p},p}=\ell_{r_{p,q},p}\quad\text{
with equivalent norms},$

where $\ell_{r,p}$ denotes the Lorentz sequence space. On the other hand, by
Lemma 8, the corresponding subspace of $S_{p}(\ell_{q})$ is equal to
$\ell_{r_{p,q}}$. However, $\ell_{r_{p,q},p}=\ell_{r_{p,q}}$ if and only if
$r_{p,q}=p$, i.e., $q=p$. The theorem is thus proved. ∎

##  4\. Proof of Theorem 3

In the sequel, $\|\,\|_{p}$ will denote the norm of $L_{p}({\mathcal{M}})$.
Fix a finite sequence $(x_{i})\subset L_{p}^{+}({\mathcal{M}})$. We claim that
the function
$q\mapsto\big{\|}\big{(}\sum_{i}x_{i}^{q}\big{)}^{\frac{1}{q}}\big{\|}_{p}^{q}$
is log-convex. Namely, for any $q_{0},q_{1}\in(0,\,\infty)$ and any
$\alpha\in(0,\,1)$

(13)
$\big{\|}\big{(}\sum_{i}x_{i}^{q}\big{)}^{\frac{1}{q}}\big{\|}_{p}^{q}\leq\big{\|}\big{(}\sum_{i}x_{i}^{q_{0}}\big{)}^{\frac{1}{q_{0}}}\big{\|}_{p}^{(1-\alpha)q_{0}}\,\big{\|}\big{(}\sum_{i}x_{i}^{q_{1}}\big{)}^{\frac{1}{q_{1}}}\big{\|}_{p}^{(1-\alpha)q_{1}}\,,$

where $q=(1-\alpha)q_{0}+\alpha q_{1}$. It suffices to show this inequality
for $\alpha=\frac{1}{2}$. Then it immediately follows from the Hölder
inequality for column spaces:

$\big{\|}\sum_{i}x_{i}^{q}\big{\|}_{\frac{p}{q}}=\big{\|}\sum_{i}x_{i}^{\frac{q_{0}}{2}}\,x_{i}^{\frac{q_{1}}{2}}\big{\|}_{\frac{p}{q}}\leq\big{\|}\big{(}\sum_{i}x_{i}^{q_{0}}\big{)}^{\frac{1}{q_{0}}}\big{\|}_{p}^{\frac{q_{0}}{2}}\,\big{\|}\big{(}\sum_{i}x_{i}^{q_{1}}\big{)}^{\frac{1}{q_{1}}}\big{\|}_{p}^{\frac{q_{1}}{2}}\,.$

Now let us show (7). Replacing $x_{i}$ by $x_{i}^{r}$ and dividing all indices
by $r$, we are reduced to the case where $r=1<q$ and $p\geq 1$. Thus it
suffices to show

(14)
$\big{\|}\big{(}\sum_{i}x_{i}^{q}\big{)}^{\frac{1}{q}}\big{\|}_{p}\leq\big{\|}\sum_{i}x_{i}\big{\|}_{p}\,.$

Set $x=\sum_{i}x_{i}$. We first consider the case where $q=2$. If $p\leq 2$,
then

$\displaystyle\big{\|}\big{(}\sum_{i}x_{i}^{2}\big{)}^{\frac{1}{2}}\big{\|}_{p}^{p}$
$\displaystyle=\tau\big{[}\big{(}\sum_{i}x_{i}^{2}\big{)}^{\frac{p}{2}}\big{]}\leq\sum_{i}\tau(x_{i}^{p})$
$\displaystyle\leq\sum_{i}\tau(x_{i}^{\frac{1}{2}}x^{p-1}x_{i}^{\frac{1}{2}})=\sum_{i}\tau(x^{p-1}x_{i})=\|x\|_{p}^{p}\,.$

Note that the same argument yields (14) in the case where $p\leq\min(q,2)$.
Assume $p>2$. Let $(\varepsilon_{n})$ be a Rademacher sequence, and let
$\mathbb{E}$ denote the corresponding expectation. Then by the triangle
inequality in $L_{\frac{p}{2}}({\mathcal{M}})$,

$\big{\|}\big{(}\sum_{i}x_{i}^{2}\big{)}^{\frac{1}{2}}\big{\|}_{p}\leq\big{[}\mathbb{E}\big{\|}\sum_{i}\varepsilon_{i}x_{i}\big{\|}_{p}^{2}\big{]}^{\frac{1}{2}}\leq\|x\|_{p}\,,$

where we have used the fact that $-x\leq\sum_{i}\varepsilon_{i}x_{i}\leq x$.
Thus inequality (14) is proved for $q=2$ and any $p\geq 1$.

Using (13) and the just proved cases, we deduce (14) for any $1<q\leq 2$ and
any $p\geq 1$.

Next, we show (14) for $q=p$. If $p\leq 2$, this was proved previously. The
case $p>2$ easily follows by an iteration argument. Indeed, if $2<p\leq 4$, by
the two cases already proved

$\displaystyle\big{\|}\big{(}\sum_{i}x_{i}^{p}\big{)}^{\frac{1}{p}}\big{\|}_{p}$
$\displaystyle=\big{\|}\big{(}\sum_{i}x_{i}^{p}\big{)}^{\frac{2}{p}}\big{\|}_{\frac{p}{2}}^{\frac{1}{2}}\leq\big{\|}\sum_{i}x_{i}^{2}\big{\|}_{\frac{p}{2}}^{\frac{1}{2}}$
$\displaystyle=\big{\|}\big{(}\sum_{n}x_{n}^{2}\big{)}^{\frac{1}{2}}\big{\|}_{p}\leq\|x\|_{p}.$

Repeating this argument, we obtain (14) for the case $q=p$.

We then deduce (14) for $1\leq p\leq q$ as follows

$\big{\|}\big{(}\sum_{i}x_{i}^{q}\big{)}^{\frac{1}{q}}\big{\|}_{p}\leq\big{\|}\big{(}\sum_{i}x_{i}^{p}\big{)}^{\frac{1}{p}}\big{\|}_{p}\leq\|x\|_{p}\,.$

Thus it remains to consider the case where $p\geq q\geq 2$. This is treated by
an iteration argument as above. Indeed, if $q\leq 4$, then

$\big{\|}\big{(}\sum_{i}x_{i}^{q}\big{)}^{\frac{1}{q}}\big{\|}_{p}=\big{\|}\big{(}\sum_{i}(x_{i}^{2})^{\frac{q}{2}}\big{)}^{\frac{2}{q}}\big{\|}_{\frac{p}{2}}^{\frac{1}{2}}\leq\big{\|}\big{(}\sum_{i}x_{i}^{2}\big{)}^{\frac{1}{2}}\big{\|}_{p}\leq\|x\|_{p}\,.$

Thus the proof of (14), so that of (7), is complete.

Now we turn to the proof of part (ii) of the theorem. As in the proof of
Theorem 2, it suffices to consider the case where ${\mathcal{M}}=B(\ell_{2})$.
Suppose that (8) holds with a constant $C$ and some indices $p,q,r$ such that
$0<p<r<q<\infty$. Then by dividing all indices by $r$, we are again reduced to
the case where $p<1=r<q$. Thus

(15) $\big{\|}\big{(}\sum_{i}x_{i}^{q}\big{)}^{\frac{1}{q}}\big{\|}_{p}\leq
C\,\big{\|}\sum_{i}x_{i}\big{\|}_{p}$

for all finite sequences $(x_{i})\subset L_{p}^{+}({\mathcal{M}})$. We claim
that $C$ must be equal to $1$. Indeed, given a positive integer $k$, applying
(15) to the family $(x_{i_{1}}\otimes\cdots\otimes x_{i_{k}})$ we get

$\big{\|}\big{(}\sum_{i_{1},...,i_{k}}x_{i_{1}}^{q}\otimes\cdots\otimes
x_{i_{k}}^{q}\big{)}^{\frac{1}{q}}\big{\|}_{p}\leq
C\,\big{\|}\sum_{i_{1},...,i_{k}}x_{i_{1}}\otimes\cdots\otimes
x_{i_{k}}\big{\|}_{p}\,.$

However,

$\big{\|}\sum_{i_{1},...,i_{k}}x_{i_{1}}\otimes\cdots\otimes
x_{i_{k}}\big{\|}_{p}=\big{\|}\big{(}\sum_{i}x_{i}\big{)}^{\otimes
k}\big{\|}_{p}=\big{\|}\sum_{i}x_{i}\big{\|}_{p}^{k}\,.$

Similarly,

$\big{\|}\big{(}\sum_{i_{1},...,i_{k}}x_{i_{1}}^{q}\otimes\cdots\otimes
x_{i_{k}}^{q}\big{)}^{\frac{1}{q}}\big{\|}_{p}=\big{\|}\big{(}\sum_{i}x_{i}^{q}\big{)}^{\frac{1}{q}}\big{\|}_{p}^{k}\,.$

It then follows that

$\big{\|}\big{(}\sum_{i}x_{i}^{q}\big{)}^{\frac{1}{q}}\big{\|}_{p}\leq
C^{\frac{1}{k}}\,\big{\|}\sum_{i}x_{i}\big{\|}_{p}\,;$

whence the claim by letting $k\to\infty$.

Now it is easy to construct counterexamples to (15) with $C=1$. Consider
${\mathcal{M}}=\mathbb{M}_{2}$ and the following matrices

$x=\begin{pmatrix}1&1\\\ 1&1\\\ \end{pmatrix}\quad\text{ and }\quad
y=\begin{pmatrix}0&0\\\ 0&t\\\ \end{pmatrix}$

with $t>0$ very small. Then

$x+y=\begin{pmatrix}1&1\\\ 1&1+t\\\ \end{pmatrix}\quad\text{and }\quad
x^{q}+y^{q}=2^{q-1}\begin{pmatrix}1&1\\\ 1&1+2^{1-q}\,t^{q}\\\ \end{pmatrix}.$

The two eigenvalues of $x+y$ are

$\frac{2+t\pm\sqrt{4+t^{2}}}{2}\,.$

Thus (recalling that $p<1$)

$\displaystyle\big{\|}x+y\big{\|}_{p}^{p}$
$\displaystyle=\frac{(2+t+\sqrt{4+t^{2}}\,)^{p}}{2^{p}}\,+\,\frac{(2+t-\sqrt{4+t^{2}}\,)^{p}}{2^{p}}$
$\displaystyle=2^{p}+2^{-p}\,t^{p}+{\rm o}(t^{p})\quad\mbox{as}\quad t\to 0.$

Similarly,

$\big{\|}(x^{q}+y^{q})^{\frac{1}{q}}\big{\|}_{p}^{p}=2^{p}+2^{-\frac{p}{q}}\,t^{p}+{\rm
o}(t^{p}).$

Hence, by (15) with $C=1$, we get

$2^{p}+2^{-\frac{p}{q}}\,t^{p}+{\rm o}(t^{p})\leq 2^{p}+2^{-p}\,t^{p}+{\rm
o}(t^{p}).$

It then follows that $2^{q}\leq 2$, which is a contradiction since $q>1$. Thus
Theorem 3 is completely proved.

Acknowledgements. The second named author is partially supported by the French
ANR project (No. ANR-19-CE40-0002) and the Natural Science Foundation of China
(No.12031004).

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