# This is the title

ABSOLUTELY SUMMING MORPHISMS BETWEEN HILBERT C*-MODULES AND MODULAR PIETSCH
FACTORIZATION PROBLEM
K. MAHESH KRISHNA
Post Doctoral Fellow

Statistics and Mathematics Unit

Indian Statistical Institute, Bangalore Centre

Karnataka 560 059, India

Email<EMAIL_ADDRESS>
Date:

Abstract: Motivated from the theory of Hilbert-Schmidt morphisms between
Hilbert C*-modules over commutative C*-algebras by Stern and van Suijlekom [J.
Funct. Anal., 2021], we introduce the notion of p-absolutely summing morphisms
between Hilbert C*-modules over commutative C*-algebras. We show that an
adjointable morphism between Hilbert C*-modules over monotone closed
commutative C*-algebra is 2-absolutely summing if and only if it is Hilbert-
Schmidt. We formulate version of Pietsch factorization problem for
p-absolutely summing morphisms and solve partially.

Keywords: Absolutely summing operator, Commutative C*-algebra, Hilbert
C*-module, Hilbert-Schmidt operator.

Mathematics Subject Classification (2020): 47B10, 47L20, 46L08, 46L05, 42C15.

##  1\. Introduction

In his Resume, A. Grothendieck studied 1-absolutely and 2-absolutely summing
operators between Banach spaces [22] (also see [8]). In 1967, for each $1\leq
p<\infty$, A. Pietsch introduced the notion of p-absolutely summing operators
which became an area around the end of 20 century [64, 7, 65, 40, 50, 9, 48,
30, 51, 28, 29, 52, 24, 26, 18, 39, 38, 41]. In 1979, Tomczak-Jaegermann
studied p-summing operators by fixing by fixing number of points [63]. In
1970, Kwapien defined the notion of 0-summing operators [36]. In 2003, Farmer
and Johnson introduced the notion of Lipschitz p-summing operators between
metric spaces [11] (also see [5, 49, 6, 4]).

######  Definition 1.1.

[1, 50] Let $\mathcal{X}$ and $\mathcal{Y}$ be Banach spaces,
$\mathcal{X}^{*}$ be the dual of $\mathcal{X}$ and $1\leq p<\infty$. A bounded
linear operator $T:\mathcal{X}\to\mathcal{Y}$ is said to be p-absolutely
summing if there is a real constant $C>0$ satisfying following: for every
$n\in\mathbb{N}$ and for all $x_{1},\dots,x_{n}\in\mathcal{X}$,

(1) $\displaystyle\left(\sum_{j=1}^{n}\|Tx_{j}\|^{p}\right)^{\frac{1}{p}}\leq
C\sup_{f\in\mathcal{X}^{*},\|f\|\leq
1}\left(\sum_{j=1}^{n}|f(x_{j})|^{p}\right)^{\frac{1}{p}}.$

In this case, the p-absolutely summing norm of $T$ is defined as

$\displaystyle\pi_{p}(T)\coloneqq\inf\\{C:C\text{ satisfies Inequality
}(\ref{ABSUMEQ})\\}.$

The set of all p-absolutely summing operators from $\mathcal{X}$ to
$\mathcal{Y}$ is denoted by $\Pi_{p}(\mathcal{X},\mathcal{Y})$.

Following are most important results in the theory of p-absolutely summing
operators.

######  Theorem 1.2.

[51, 50] Let $1\leq p<\infty$ and $\mathcal{X}$, $\mathcal{Y}$ be Banach
spaces. Then $(\Pi_{p}(\mathcal{X},\mathcal{Y}),\pi_{p}(\cdot))$ is an
operator ideal.

######  Theorem 1.3.

[1, 50] Let $\mathcal{H}$ and $\mathcal{K}$ be Hilbert spaces. Then a bounded
linear operator $T:\mathcal{H}\to\mathcal{K}$ is 2-absolutely summing if and
only if it is Hilbert-Schmidt. Moreover, $\|T\|_{\text{HS}}=\pi_{2}(T)$.

######  Theorem 1.4.

[1, 50] (Pietsch Factorization Theorem) Let $\mathcal{X}$ and $\mathcal{Y}$ be
Banach spaces. A bounded linear operator $T:\mathcal{X}\to\mathcal{Y}$ is
p-absolutely summing if and only if there is a real constant $C>0$ and a
regular Borel probability measure on
$B_{\mathcal{X}^{*}}\coloneqq\\{f:f\in\mathcal{X}^{*},\|f\|\leq 1\\}$ in
weak*-topology such that

(2) $\displaystyle\|Tx\|\leq
C\left(\int\limits_{B_{\mathcal{X}^{*}}}|f(x)|^{p}\,d\mu_{B_{\mathcal{X}^{*}}}(f)\right)^{\frac{1}{p}},\quad\forall
x\in\mathcal{X}.$

Moreover, $\pi_{p}(T)=\inf\\{C:C\text{ satisfies Inequality
}(\ref{FACINE})\\}$.

In this paper, we define the notion of p-absolutely summing morphisms between
Hilbert C*-modules over commutative C*-algebras (Definition 2.1). We derive in
Theorem 2.5 that an adjointable morphism between Hilbert C*-module over a
monote closed C*-algebra is 2-summing if and only if modular Hilbert-Schmidt.
We then formulate version of Pietsch factorization problem for p-absolutely
summing morphisms and solve partially.

##  2\. p-absolutely summing morphisms

We define modular version of Definition 1.1 as follows. For the theory of
Hilbert C*-modules we refer [47, 56, 32].

######  Definition 2.1.

Let $1\leq p<\infty$. Let $\mathcal{M}$ and $\mathcal{N}$ be Hilbert
C*-modules over a commutative C*-algebra $\mathcal{A}$. An adjointable
morphism $T:\mathcal{M}\to\mathcal{N}$ is said to be modular p-absolutely
summing if there is a real constant $C>0$ satisfying following: for every
$n\in\mathbb{N}$ and for all $x_{1},\dots,x_{n}\in\mathcal{M}$,

(3) $\displaystyle\left\|\sum_{j=1}^{n}\langle
Tx_{j},Tx_{j}\rangle^{\frac{p}{2}}\right\|^{\frac{1}{p}}\leq
C\sup_{x\in\mathcal{M},\|x\|\leq 1}\left\|\sum_{j=1}^{n}(\langle
x,x_{j}\rangle\langle x_{j},x\rangle)^{\frac{p}{2}}\right\|^{\frac{1}{p}}.$

In this case, the p-absolutely summing norm of $T$ is defined as

$\displaystyle\pi_{p}(T)\coloneqq\inf\\{C:C\text{ satisfies Inequality
}(\ref{MODABSUMEQ})\\}.$

The set of all p-absolutely summing morphisms from $\mathcal{M}$ to
$\mathcal{N}$ is denoted by $\Pi_{p}(\mathcal{M},\mathcal{N})$.

In 2021, Stern and van Suijlekom introduced the notion of modular Schatten
class morphisms [59].

######  Definition 2.2.

[59] Let $1\leq p<\infty$. Let $\mathcal{A}$ be a C*-algebra and
$\widehat{\mathcal{A}}$ be its Gelfand spectrum. Let $\mathcal{M}$ and
$\mathcal{N}$ be Hilbert C*-modules over $\mathcal{A}$. Let
$T:\mathcal{M}\to\mathcal{N}$ be an adjointable morphism. We say that $T$ is
in the modular p-Schatten class if the function

$\displaystyle\operatorname{Tr}|T|^{p}:\widehat{\mathcal{A}}\ni\chi\mapsto\operatorname{Tr}|\chi_{*}T|^{p}\in\mathbb{R}\cup\\{\infty\\}$

lies in $\mathcal{A}$. The modular p-Schatten norm of $T$ is defined as

$\displaystyle\|T\|_{p}\coloneqq\|\operatorname{Tr}|T|^{p}\|_{\mathcal{A}}^{\frac{1}{p}}.$

Modular 2-Schatten (resp. 1-Schatten) class morphism is called as modular
Hilbert-Schmidt (resp. modular trace class). We denote $\|T\|_{2}$ by
$\|T\|_{\text{HS}}$.

Using the theory of modular frames for Hilbert C*-modules (see [16]) Stern and
van Suijlekom were able to derive following result.

######  Theorem 2.3.

[59] Let $\mathcal{M}$ and $\mathcal{N}$ be Hilbert C*-modules over
$\mathcal{A}$. Let $T:\mathcal{M}\to\mathcal{N}$ be an adjointable morphism.
Then $T$ is modular Hilbert-Schmidt if and only if for every modular Parseval
frame $\\{\tau_{n}\\}_{n}$ for $\mathcal{M}$, the series
$\sum_{n=1}^{\infty}\langle|T|^{p}\tau_{n},\tau_{n}\rangle$ converges in norm
in $\mathcal{A}$ and

$\displaystyle\operatorname{Tr}|T|^{p}=\sum_{n=1}^{\infty}\langle|T|^{p}\tau_{n},\tau_{n}\rangle.$

We now derive modular version of Theorem 1.3 with the following notion.

######  Definition 2.4.

[62] A C*-algebra $\mathcal{A}$ is said to be monotone closed if every bounded
increasing net in $\mathcal{A}$ has the least upper bound in $\mathcal{A}$.

######  Theorem 2.5.

Let $\mathcal{M}$ and $\mathcal{N}$ be Hilbert C*-modules over a commutative
C*-algebra $\mathcal{A}$. Assume $\mathcal{A}$ is monotone closed. Let
$T:\mathcal{M}\to\mathcal{N}$ be an adjointable morphism. Then
$T\in\Pi_{2}(\mathcal{M},\mathcal{N})$ if and only if $T$ is modular Hilbert-
Schmidt. Moreover, $\|T\|_{\text{HS}}=\pi_{2}(T)$.

###### Proof.

$(\Rightarrow)$ Let $T\in\Pi_{2}(\mathcal{M},\mathcal{N})$. Let
$\\{\tau_{n}\\}_{n=1}^{\infty}$ be a modular Parseval frame for $\mathcal{M}$.
Then

(4) $\displaystyle\langle x,x\rangle=\sum_{n=1}^{\infty}\langle
x,\tau_{n}\rangle\langle\tau_{n},x\rangle,\quad\forall x\in\mathcal{M},$

where the series converges in the norm of $\mathcal{A}$. To show $T$ is
modular Hilbert-Schmidt, using Theorem 2.3, it suffices to show that the
series $\sum_{n=1}^{\infty}\langle T\tau_{n},T\tau_{n}\rangle$ converges in
norm in $\mathcal{A}$. Note that the series $\sum_{n=1}^{\infty}\langle
T\tau_{n},T\tau_{n}\rangle$ is monotonically increasing. Since the C*-algebra
is monotone closed, we are done if we show the sequence
$\\{\sum_{j=1}^{n}\langle T\tau_{j},T\tau_{j}\rangle\\}_{n=1}^{\infty}$ is
bounded. Let $n\in\mathbb{N}$. Since $T$ is 2-summing, using Equation (4) we
have

$\displaystyle\left\|\sum_{j=1}^{n}\langle T\tau_{j},T\tau_{j}\rangle\right\|$
$\displaystyle\leq\pi_{2}(T)^{2}\sup_{x\in\mathcal{M},\|x\|\leq
1}\left\|\sum_{j=1}^{n}\langle
x,\tau_{j}\rangle\langle\tau_{j},x\rangle\right\|\leq\pi_{2}(T)^{2}\sup_{x\in\mathcal{M},\|x\|\leq
1}\left\|\sum_{j=1}^{\infty}\langle
x,\tau_{j}\rangle\langle\tau_{j},x\rangle\right\|$
$\displaystyle=\pi_{2}(T)^{2}\sup_{x\in\mathcal{M},\|x\|\leq
1}\|x\|^{2}=\pi_{2}(T)^{2}.$

$(\Leftarrow)$ Let $n\in\mathbb{N}$ and $x_{1},\dots,x_{n}\in\mathcal{M}$. Let
$\\{\omega_{n}\\}_{n=1}^{\infty}$ be an orthonormal basis for $\mathcal{M}$.
Define

$\displaystyle S:\mathcal{M}\ni x\mapsto\sum_{j=1}^{n}\langle
x,\omega_{j}\rangle x_{j}\in\mathcal{M}.$

Then

$\displaystyle\|S\|^{2}$
$\displaystyle=\|S^{*}\|^{2}=\sup_{x\in\mathcal{M},\|x\|\leq
1}\|S^{*}x\|^{2}=\sup_{x\in\mathcal{M},\|x\|\leq
1}\left\|\sum_{n=1}^{\infty}\langle
S^{*}x,\omega_{n}\rangle\omega_{n}\right\|^{2}=\sup_{x\in\mathcal{M},\|x\|\leq
1}\left\|\sum_{n=1}^{\infty}\langle
x,S\omega_{n}\rangle\omega_{n}\right\|^{2}$
$\displaystyle=\sup_{x\in\mathcal{M},\|x\|\leq 1}\left\|\sum_{j=1}^{n}\langle
x,x_{j}\rangle\omega_{j}\right\|^{2}=\sup_{x\in\mathcal{M},\|x\|\leq
1}\left\|\sum_{j=1}^{n}\langle x,x_{j}\rangle\langle x_{j},x\rangle\right\|.$

Hence

$\displaystyle\left\|\sum_{j=1}^{n}\langle Tx_{j},Tx_{j}\rangle\right\|$
$\displaystyle=\left\|\sum_{j=1}^{n}\langle
TS\omega_{j},TS\omega_{j}\rangle\right\|=\|TS\|^{2}_{\operatorname{HS}}\leq\|T\|^{2}_{\operatorname{HS}}\|S\|=\|T\|^{2}_{\operatorname{HS}}\sup_{x\in\mathcal{M},\|x\|\leq
1}\left\|\sum_{j=1}^{n}\langle x,x_{j}\rangle\langle x_{j},x\rangle\right\|.$

∎

Note that we have not used monotonic closedness of C*-algebra in “if” part. In
view of Theorem 1.4, we formulate following problem.

######  Question 2.6.

Whether there exists a modular Pietsch factorization theorem?

We solve Question (2.6) partially in the following theorem. Integrals in the
following theorem is in the Kasparov sense [33].

######  Theorem 2.7.

Let $\mathcal{M}$ and $\mathcal{N}$ be Hilbert C*-modules over a commutative
C*-algebra $\mathcal{A}$. Let $T:\mathcal{M}\to\mathcal{N}$ be an adjointable
morphism. Assume that there exists a Lie group $G\subseteq
B_{\mathcal{M}}\coloneqq\\{x:x\in\mathcal{M},\|x\|\leq 1\\}$ satisfying
following.

  1. (i)

$\mu_{G}(G)=1$.

  2. (ii)

For each $x\in\mathcal{M}$, the map $G\ni y\mapsto\langle x.y\rangle\langle
y,x\rangle$ is continuous.

  3. (iii)

There exists a real $C>0$ such that

$\displaystyle\langle Tx,Tx\rangle^{\frac{p}{2}}\leq C^{p}\int_{G}(\langle
x.y\rangle\langle y,x\rangle)^{\frac{p}{2}}\,d\mu_{G}(y),\quad\forall
x\in\mathcal{M}.$

Then $T$ modular p-absolutely summing and $\pi_{p}(T)=C$.

###### Proof.

Let $n\in\mathbb{N}$ and $x_{1},\dots,x_{n}\in\mathcal{M}$. Then

$\displaystyle\left\|\sum_{j=1}^{n}\langle
Tx_{j},Tx_{j}\rangle^{\frac{p}{2}}\right\|\leq
C^{p}\left\|\sum_{j=1}^{n}\int_{G}(\langle x_{j},y\rangle\langle
y,x_{j}\rangle)^{\frac{p}{2}}\,d\mu_{G}(y)\right\|=C^{p}\left\|\int_{G}\sum_{j=1}^{n}(\langle
x_{j},y\rangle\langle y,x_{j}\rangle)^{\frac{p}{2}}\,d\mu_{G}(y)\right\|$
$\displaystyle\leq C^{p}\int_{G}\left\|\sum_{j=1}^{n}(\langle
x_{j},y\rangle\langle y,x_{j}\rangle)^{\frac{p}{2}}\right\|\,d\mu_{G}(y)\leq
C^{p}\int_{G}\sup_{y\in\mathcal{M},\|y\|\leq 1}\left\|\sum_{j=1}^{n}(\langle
x_{j},y\rangle\langle y,x_{j}\rangle)^{\frac{p}{2}}\right\|\,d\mu_{G}(y)$
$\displaystyle=C^{p}\sup_{y\in\mathcal{M},\|y\|\leq
1}\left\|\sum_{j=1}^{n}(\langle x_{j},y\rangle\langle
y,x_{j}\rangle)^{\frac{p}{2}}\right\|\mu_{G}(G)=C^{p}\sup_{y\in\mathcal{M},\|y\|\leq
1}\left\|\sum_{j=1}^{n}(\langle x_{j},y\rangle\langle
y,x_{j}\rangle)^{\frac{p}{2}}\right\|.$

∎

##  3\. Appendix

In this appendix we formulate some problems for Banach modules over
C*-algebras based on the results in Banach spaces which influenced a lot in
the modern development of Functional Analysis. Our first kind of problems come
from the Dvoretzky theorem [46, 45, 44, 14, 60, 42, 43, 54, 20, 13, 12, 15,
58, 21, 53]. Let $\mathcal{X}$ and $\mathcal{Y}$ be finite dimensional Banach
spaces such that $\dim(\mathcal{X})=\dim(\mathcal{Y})$. Remenber that the
Banach-Mazur distance between $\mathcal{X}$ and $\mathcal{Y}$ is defined as

$\displaystyle
d_{BM}(\mathcal{X},\mathcal{Y})\coloneqq\inf\\{\|T\|\|T^{-1}\|:T:\mathcal{X}\to\mathcal{Y}\text{
is invertible linear operator}\\}.$

For $n\in\mathbb{N}$, let $(\mathbb{R}^{n},\langle\cdot,\cdot\rangle)$ be the
standard Euclidean Hilbert space.

######  Theorem 3.1.

[1, 27] (John Theorem) If $\mathcal{X}$ is any $n$-dimensional real Banach
space, then

$\displaystyle
d_{BM}(\mathcal{Y},(\mathbb{R}^{n},\langle\cdot,\cdot\rangle))\leq\sqrt{n}.$

######  Theorem 3.2.

[10, 1] (Dvoretzky Theorem) There is a universal constant $C>0$ satisfying the
following property: If $\mathcal{X}$ is any $n$-dimensional real Banach space
and $0<\varepsilon<\frac{1}{3}$, then for every natural number

$\displaystyle k\leq C\log n\frac{\varepsilon^{2}}{|\log\varepsilon|},$

there exists a $k$-dimensional Banach subspace $\mathcal{Y}$ of $\mathcal{X}$
such that

$\displaystyle
d_{BM}(\mathcal{Y},(\mathbb{R}^{k},\langle\cdot,\cdot\rangle))<1+\varepsilon.$

Let $\mathcal{A}$ be a unital C*-algebra with invariant basis number property
(see [19] for a study on such C*-algebras) and $\mathcal{E}$, $\mathcal{F}$ be
finite rank Banach modules over $\mathcal{A}$ such that
$\operatorname{rank}(\mathcal{E})=\operatorname{rank}(\mathcal{F})$. Modular
Banach-Mazur distance between $\mathcal{E}$ and $\mathcal{F}$ is defined as

$\displaystyle
d_{MBM}(\mathcal{E},\mathcal{F})\coloneqq\inf\\{\|T\|\|T^{-1}\|:T:\mathcal{E}\to\mathcal{F}\text{
is invertible module homomorphism}\\}.$

Given a unital C*-algebra $\mathcal{A}$ and $n\in\mathbb{N}$, by
$\mathcal{A}^{n}$ we mean the standard (left) module over $\mathcal{A}$. We
equip $\mathcal{A}^{n}$ with the C*-valued inner product
$\langle\cdot,\cdot\rangle:\mathcal{A}^{n}\times\mathcal{A}^{n}\to\mathcal{A}$
defined by

$\displaystyle\langle(a_{j})_{j=1}^{n},(b_{j})_{j=1}^{n}\rangle\coloneqq\sum_{j=1}^{n}a_{j}b_{j}^{*},\quad\forall(a_{j})_{j=1}^{n},(b_{j})_{j=1}^{n}\in\mathcal{A}^{n}.$

Hence norm on $\mathcal{A}^{n}$ is given by

$\displaystyle\|(a_{j})_{j=1}^{n}\|\coloneqq\left\|\sum_{j=1}^{n}a_{j}a_{j}^{*}\right\|^{\frac{1}{2}},\quad\forall(a_{j})_{j=1}^{n}\in\mathcal{A}^{n}.$

Then it is well-known that $\mathcal{A}^{n}$ is a Hilbert C*-module. We denote
this Hilbert C*-module by $(\mathcal{A}^{n},\langle\cdot,\cdot\rangle)$.

######  Problem 3.3.

(Modular Dvoretzky Problem) Let $\mathscr{A}$ be the set of all unital
C*-algebras with invariant basis number property. What is the best function
$\Psi:\mathscr{A}\times\left(0,\frac{1}{3}\right)\times\mathbb{N}\to(0,\infty)$
satisfying the following property: If $\mathcal{E}$ is any $n$-rank Banach
module over a unital C*-algebra $\mathcal{A}$ with IBN property and
$0<\varepsilon<\frac{1}{3}$, then for every natural number

$\displaystyle k\leq\Psi(\mathcal{A},\varepsilon,n),$

there exists a $k$-rank Banach submodule $\mathcal{F}$ of $\mathcal{E}$ such
that

$\displaystyle
d_{MBM}(\mathcal{F},(\mathcal{A}^{k},\langle\cdot,\cdot\rangle))<1+\varepsilon.$

A particular case of Problem 3.3 is the following conjecture.

######  Conjecture 3.4.

(Modular Dvoretzky Conjecture) Let $\mathcal{A}$ be a unital C*-algebra with
IBN property. There is a universal constant $C>0$ (which may depend upon
$\mathcal{A}$) satisfying the following property: If $\mathcal{E}$ is any
$n$-rank Banach module and $0<\varepsilon<\frac{1}{3}$, then for every natural
number

$\displaystyle k\leq C\log n\frac{\varepsilon^{2}}{|\log\varepsilon|},$

there exists a $k$-rank Banach submodule $\mathcal{F}$ of $\mathcal{E}$ such
that

$\displaystyle
d_{MBM}(\mathcal{F},(\mathcal{A}^{k},\langle\cdot,\cdot\rangle))<1+\varepsilon.$

Our second kind of problems come from the type-cotype theory of Banach spaces
[37, 61, 28, 29, 1, 38, 39, 9]. Let $\mathcal{H}$ be a Hilbert space,
$n\in\mathbb{N}$. Recall that for any $n$ points
$x_{1},\dots,x_{n}\in\mathcal{H}$, we have

(5)
$\displaystyle\frac{1}{2^{n}}\sum_{\varepsilon_{1},\dots,\varepsilon_{n}\in\\{-1,1\\}}\left\|\sum_{j=1}^{n}\varepsilon_{j}x_{j}\right\|^{2}=\sum_{j=1}^{n}\|x_{j}\|^{2}.$

It is Equality (5) which motivated the definition of type and cotype for
Banach spaces.

######  Definition 3.5.

[1] Let $1\leq p\leq 2$. A Banach space $\mathcal{X}$ is said to be of
(Rademacher) type $p$ if there exists $T_{p}(\mathcal{X})>0$ such that

$\displaystyle\left(\frac{1}{2^{n}}\sum_{\varepsilon_{1},\dots,\varepsilon_{n}\in\\{-1,1\\}}\left\|\sum_{j=1}^{n}\varepsilon_{j}x_{j}\right\|^{p}\right)^{\frac{1}{p}}\leq
T_{p}(\mathcal{X})\left(\sum_{j=1}^{n}\|x_{j}\|^{p}\right)^{\frac{1}{p}},\quad\forall
x_{1},\dots,x_{n}\in\mathcal{X},~{}\forall n\in\mathbb{N}.$

######  Definition 3.6.

[1] Let $2\leq q<\infty$. A Banach space $\mathcal{X}$ is said to be of
(Rademacher) cotype $q$ if there exists $C_{q}(\mathcal{X})>0$ such that

$\displaystyle\left(\sum_{j=1}^{n}\|x_{j}\|^{q}\right)^{\frac{1}{q}}\leq
C_{q}(\mathcal{X})\left(\frac{1}{2^{n}}\sum_{\varepsilon_{1},\dots,\varepsilon_{n}\in\\{-1,1\\}}\left\|\sum_{j=1}^{n}\varepsilon_{j}x_{j}\right\|^{q}\right)^{\frac{1}{q}},\quad\forall
x_{1},\dots,x_{n}\in\mathcal{X},~{}\forall n\in\mathbb{N}.$

Let $\mathcal{E}$ be a (left) Hilbert C*-module over a unital C*-algebra
$\mathcal{A}$, $n\in\mathbb{N}$. We see that for any $n$ points
$x_{1},\dots,x_{n}\in\mathcal{E}$, we have

(6)
$\displaystyle\frac{1}{2^{n}}\sum_{\varepsilon_{1},\dots,\varepsilon_{n}\in\\{-1,1\\}}\left\langle\sum_{j=1}^{n}\varepsilon_{j}x_{j},\sum_{k=1}^{n}\varepsilon_{k}x_{k}\right\rangle=\sum_{j=1}^{n}\langle
x_{j},x_{j}\rangle.$

######  Problem 3.7.

(Modular Type-Cotype Problems) Whether there is a way to define type (we call
modular-type) and cotype (we call modular-cotype) for Banach modules over
C*-algebras which reduces to Equality (6) for Hilbert C*-modules?

######  Problem 3.8.

Whether there is a notion of type and cotype for Banach modules over
C*-algebras such that Kwapien theorems holds?, In other words, whether
following statements hold?

  1. (i)

A Banach module $\mathcal{M}$ over a unital C*-algebra $\mathcal{A}$ has
modular-type 2 and modular-cotype 2 if and only if $\mathcal{M}$ is isomorphic
to a Hilbert C*-module over $\mathcal{A}$.

  2. (ii)

If $\mathcal{M}$ and $\mathcal{N}$ are Banach modules over a unital C*-algebra
$\mathcal{A}$ of modular-type 2 and modular-cotype 2, respectively, then a
bounded module morphism $T:\mathcal{M}\to\mathcal{N}$ factors through a
Hilbert C*-module over $\mathcal{A}$.

######  Problem 3.9.

(Modular Khinchin-Kahane Inequalities Problems) Whether there is a Khinchin-
Kahane inequalities for Banach modules over C*-algebras which reduce to
Equality (6) for Hilbert C*-modules?

Our third kind of problems come from Grothendieck inequality [3, 22, 1, 8, 17,
57, 23, 25, 31, 2, 55].

######  Theorem 3.10.

[3, 17, 57, 22, 1, 8] (Grothendieck Inequality) There is a universal constant
$K_{G}$ satisfying the following: For any Hilbert space $\mathcal{H}$ and any
$m,n\in\mathbb{N}$, if a scalar matrix $[a_{j,k}]_{1\leq j\leq m,1\leq k\leq
n}$ satisfy

$\displaystyle\left|\sum_{j=1}^{m}\sum_{k=1}^{n}a_{j,k}s_{j}t_{k}\right|\leq
1,\quad\forall s_{j},t_{k}\in\mathbb{K},|s_{j}|\leq 1,|t_{k}|\leq 1,$

then

$\displaystyle\left|\sum_{j=1}^{m}\sum_{k=1}^{n}a_{j,k}\langle
u_{j},v_{k}\rangle\right|\leq K_{G},\quad\forall
u_{j},v_{k}\in\mathcal{H},\|u_{j}\|\leq 1,\|v_{k}\|\leq 1.$

######  Problem 3.11.

(Modular Grothendieck Inequality Problem - 1) Let $\mathscr{A}$ be the set of
all unital C*-algebras. Let $\mathcal{E}$ be a Hilbert C*-module over a unital
C*-algebra $\mathcal{A}$. Let $\mathcal{A}^{+}$ be the set of all positive
elemnts in $\mathcal{A}$. What is the best function
$\Psi:\mathscr{A}\times\mathbb{N}\times\mathbb{N}\to\mathcal{A}^{+}$
satisfying the following property: If $[a_{j,k}]_{1\leq j\leq m,1\leq k\leq
n}\in\mathbb{M}_{m\times n}(\mathcal{A})$ satisfy

$\displaystyle\left\langle\sum_{j=1}^{m}\sum_{k=1}^{n}a_{j,k}s_{j}t_{k},\sum_{p=1}^{m}\sum_{q=1}^{n}a_{p,q}s_{p}t_{q}\right\rangle\leq$
$\displaystyle 1,\quad\forall s_{j},t_{k}\in\mathcal{A},$ $\displaystyle
s_{j}s_{j}^{*}=s_{j}^{*}s_{j}=1,\forall 1\leq j\leq
m,t_{k}t_{k}^{*}=t_{k}^{*}t_{k}=1,\forall 1\leq k\leq n,$

then

$\displaystyle\left\langle\sum_{j=1}^{m}\sum_{k=1}^{n}a_{j,k}\langle
u_{j},v_{k}\rangle,\sum_{p=1}^{m}\sum_{q=1}^{n}a_{p,q}\langle
u_{p},v_{q}\rangle\right\rangle\leq$
$\displaystyle\Psi(\mathcal{A},m,n),\quad\forall u_{j},v_{k}\in\mathcal{E},$
$\displaystyle\langle u_{j},u_{j}\rangle=1,\forall 1\leq j\leq m,\langle
v_{k},v_{k}\rangle=1,\forall 1\leq k\leq n.$

In particular, whether $\Psi$ depends on $m$ and $n$?

######  Problem 3.12.

(Modular Grothendieck Inequality Problem - 2) Let $\mathscr{A}$ be the set of
all unital C*-algebras. Let $\mathcal{E}$ be a Hilbert C*-module over a unital
C*-algebra $\mathcal{A}$. Let $\mathcal{A}^{+}$ be the set of all positive
elemnts in $\mathcal{A}$. What is the best function
$\Psi:\mathscr{A}\times\mathbb{N}\times\mathbb{N}\to\mathcal{A}^{+}$
satisfying the following property: If $[a_{j,k}]_{1\leq j\leq m,1\leq k\leq
n}\in\mathbb{M}_{m\times n}(\mathcal{A})$ satisfy

$\displaystyle\left\langle\sum_{j=1}^{m}\sum_{k=1}^{n}a_{j,k}s_{j}t_{k},\sum_{p=1}^{m}\sum_{q=1}^{n}a_{p,q}s_{p}t_{q}\right\rangle\leq
1,\quad\forall s_{j},t_{k}\in\mathcal{A},\|s_{j}\|\leq 1,\forall 1\leq j\leq
m,\|t_{k}\|\leq 1,\forall 1\leq k\leq n,$

then

$\displaystyle\left\langle\sum_{j=1}^{m}\sum_{k=1}^{n}a_{j,k}\langle
u_{j},v_{k}\rangle,\sum_{p=1}^{m}\sum_{q=1}^{n}a_{p,q}\langle
u_{p},v_{q}\rangle\right\rangle$
$\displaystyle\leq\Psi(\mathcal{A},m,n),\quad\forall
u_{j},v_{k}\in\mathcal{E},$ $\displaystyle\|u_{j}\|\leq 1,\forall 1\leq j\leq
m,\|v_{k}\|\leq 1,\forall 1\leq k\leq n.$

In particular, whether $\Psi$ depends on $m$ and $n$?

We believe strongly that $\Psi$ depends on $\mathcal{A}$.

######  Remark 3.13.

Modular Bourgain-Tzafriri restricted invertibility conjecture and Modular
Johnson-Lindenstrauss flattening conjecture have been stated in [34, 35].

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