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\fancyhead[LO]{\textbf{MODULAR  DVORETZKY, TYPE-COTYPE, KHINCHIN-KAHANE AND GROTHENDIECK INEQUALITY  PROBLEMS}}
\fancyhead[RE]{\textbf{K. MAHESH KRISHNA}}


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{\bf{Modular Dvoretzky, Type-Cotype, Khinchin-Kahane and Grothendieck Inequality Problems}}\\
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\textbf{K. MAHESH KRISHNA}\\
School of Mathematics and Natural Sciences\\
Chanakya University Global Campus\\
NH-648, Haraluru Village\\
Devanahalli Taluk, 	Bengaluru  North District\\
Karnataka State, 562 110, India\\
Email: kmaheshak@gmail.com\\

Date: \today
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\textbf{Abstract}:  We ask for noncommutative versions of following four: (1) Dvoretzky Theorem, (2) Type-Cotype Definition, (3)  Khinchin-Kahane Inequalities, (4) Grothendieck Inequality.

\textbf{Keywords}: Dvoretzky Theorem, Type-Cotype, Khinchin-Kahane Inequality, Grothendieck Inequality, C*-algebra, Hilbert C*-module.

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

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 \section{Modular Dvoretzky Problem}
  Let $\mathcal{X}$ and $\mathcal{Y}$ be finite dimensional Banach spaces such that $\dim (\mathcal{X})=\dim(\mathcal{Y})$. Remember that the \textbf{Banach-Mazur distance} between $\mathcal{X}$ and $\mathcal{Y}$ is defined as 
  \begin{align*}
  	d_{BM}(\mathcal{X}, \mathcal{Y})\coloneqq \inf\{\|T\|\|T^{-1}\|: T: \mathcal{X} \to \mathcal{Y} \text{ is invertible linear operator}\}.
  \end{align*}
  For $n \in \mathbb{N}$, let $(\mathbb{R}^n, \langle \cdot, \cdot \rangle)$ be the standard Euclidean Hilbert space.
  \begin{theorem}\cite{ALBIACKALTON, JOHN} \textbf{(John Theorem)}
  		If  $\mathcal{X}$ is any $n$-dimensional real Banach space, then 
  		\begin{align*}
  			d_{BM}(\mathcal{X}, (\mathbb{R}^n, \langle \cdot, \cdot \rangle))\leq \sqrt{n}.	
  	\end{align*}
  \end{theorem}
   \begin{theorem}\cite{DVORETZKY, ALBIACKALTON} \textbf{(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 	
  		\begin{align*}
  			k\leq C \frac{\varepsilon^2}{|\log \varepsilon|}\log n,
  		\end{align*}
  		there exists  a $k$-dimensional Banach subspace $\mathcal{Y}$ of $\mathcal{X}$ such that 
  		\begin{align*}
  			d_{BM}(\mathcal{Y}, (\mathbb{R}^k, \langle \cdot, \cdot \rangle))<1+\varepsilon.
  	\end{align*}
  \end{theorem}
We refer \cite{MILMANSCHECHTMAN, MILMAM, MILMAN1992, FIGIEL2, SZANKOWSKI, MATOUSEK, MATSAKPLICHKO, PISIERBOOK, GORDONGUEDONMEYER, FIGIEL, FIGIEL3, FIGIELLINDENSTRAUSSMILMAN, SCHECHTMAN, GORDON2, PISIERBOOK2} for more information on Dvoretzky theorem. 
  Let $\mathcal{A}$  be a unital C*-algebra with invariant basis number property (see \cite{GIPSON} for a study on such C*-algebras)  and    $\mathcal{E}$, $\mathcal{F}$ be finite rank  Banach modules over $\mathcal{A}$  such that $\operatorname{rank} (\mathcal{E})=\operatorname{rank}(\mathcal{F})$. \textbf{Modular Banach-Mazur distance} between $\mathcal{E}$ and $\mathcal{F}$ is defined as 
  \begin{align*}
  	d_{MBM}(\mathcal{E}, \mathcal{F})\coloneqq \inf\{\|T\|\|T^{-1}\|: T: \mathcal{E} \to \mathcal{F} \text{ is invertible module homomorphism}\}.
  \end{align*}
  Given a unital C*-algebra $\mathcal{A}$ and $n \in \mathbb{N}$, by $\mathcal{A}^n$ we mean the standard (right) 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 
  \begin{align*}
  	\langle (a_j)_{j=1}^n, (b_j)_{j=1}^n \rangle \coloneqq \sum_{j=1}^na_j^*b_j, \quad \forall (a_j)_{j=1}^n, (b_j)_{j=1}^n \in 	\mathcal{A}^n.
  \end{align*}
  Hence the norm on $\mathcal{A}^n$ is given by
  \begin{align*}
  	\|(a_j)_{j=1}^n\|\coloneqq \left\|\sum_{j=1}^na_j^*a_j\right\|^\frac{1}{2} , \quad \forall (a_j)_{j=1}^n \in \mathcal{A}^n.	
  \end{align*}
  Then it is well-known that $\mathcal{A}^n$ is a Hilbert C*-module. We denote this Hilbert C*-module by $(\mathcal{A}^n, \langle \cdot, \cdot \rangle)$. 
  \begin{problem}
  	\textbf{(Modular John Problem)} 
  		Let $\mathscr{A}$ be the set of all unital C*-algebras with invariant basis number property. What is the best function $\Phi: \mathscr{A} \times \mathbb{N} \to (0, \infty)$  satisfying the following property: If $\mathcal{E}$ is any $n$-rank Banach module  over a unital C*-algebra $\mathcal{A}$ with IBN property, then 
  	\begin{align*}
  		d_{MBM}(\mathcal{E}, (\mathcal{A}^n, \langle \cdot, \cdot \rangle))\leq \Phi(\mathcal{A}, n).
  	\end{align*}
   \end{problem}
  
  
  
  \begin{problem}\label{MODULARDVORETZKYPROBLEM} \textbf{(Modular Dvoretzky Problem)}
  		Let $\mathscr{A}$ be the set of all unital C*-algebras with invariant basis number property. What is the best function $\Psi: \mathscr{A} \times \mathbb{R} \times \mathbb{N} \to (0, \infty)$  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 $, then for every natural number 	
  		\begin{align*}
  			k\leq \Psi(\mathcal{A}, \varepsilon, n),
  		\end{align*}
  		there exists  a $k$-rank Banach submodule  $\mathcal{F}$ of $\mathcal{E}$ such that 
  		\begin{align*}
  			d_{MBM}(\mathcal{F}, (\mathcal{A}^k, \langle \cdot, \cdot \rangle))<1+\varepsilon.
  	\end{align*}
  \end{problem}


 \section{Modular Type-Cotype and Khinchin-Kahane Problems}

  Let $\mathcal{H}$ be a Hilbert space, $n\in \mathbb{N}$. Recall that for any $n$ points $h_1, \dots , h_n \in \mathcal{H}$, we have 
  \begin{align}\label{H2}
  	\frac{1}{2^n}\sum_{\varepsilon_1, \dots,  \varepsilon_n \in \{-1, 1\}}	\left\|\sum_{j=1}^{n}\varepsilon_j h_j\right\|^2=\sum_{j=1}^{n}\|h_j\|^2.
  \end{align}
  It is Equality (\ref{H2}) which motivated the definition of type and cotype for Banach spaces. 
  \begin{definition}\cite{ALBIACKALTON}
  	Let $1\leq p \leq 2$. A Banach space $\mathcal{X}$ is said to be of \textbf{(Rademacher) type $p$} if there exists $T_p(\mathcal{X})>0$ such that 
  	\begin{align*}
  		\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}.
  	\end{align*}
  \end{definition}
  \begin{definition}\cite{ALBIACKALTON}
  	Let $2\leq q  <\infty$. A Banach space $\mathcal{X}$ is said to be of  \textbf{(Rademacher) cotype $q$} if there exists $C_q(\mathcal{X})>0$ such that 
  	\begin{align*}
  		\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}.
  	\end{align*}	
  \end{definition}
We refer \cite{LATALAOLESZKIEWICZ, SZAREK, HANDBOOK1, HANDBOOK2, ALBIACKALTON, LIQUEFFELEC1, LIQUEFFELEC2, DIESTELJARCHOWTONGE} for more information on type-cotype for Banach spaces. 
 Let $\mathcal{E}$ be a  (right) 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 
  \begin{align}\label{HMODULE}
  	\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. 
  \end{align}
  \begin{problem} 
  	\textbf{(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 (\ref{HMODULE}) for Hilbert C*-modules?
  \end{problem}
 \begin{problem}
  	Whether there is a notion of type and cotype for Banach modules over C*-algebras  such that Kwapien theorems hold?, In other words, whether following statements hold?
  		\begin{enumerate}[\upshape(i)]
  			\item 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 $\mathcal{A}$.
  			\item 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 $\mathcal{A}$.
  	\end{enumerate}
  \end{problem}
\begin{problem}
\textbf{(Modular Khinchin-Kahane Inequalities Problems)} Whether there is a Khinchin-Kahane	inequalities for Banach modules over C*-algebras  which reduce to Equality (\ref{HMODULE}) for Hilbert C*-modules?
\end{problem}

 \section{Modular Grothendieck Inequality Problem}
In 1968, Lindenstrauss and Pelczynski derived the following form of Grothendieck inequality \cite{LP}.
\begin{theorem}\cite{BLEI, FRIEDLANDLIMZHANG, RIETZ, GROTHENDIECK, ALBIACKALTON,  DIESTELFOURIESWART, LP} \textbf{(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 $[a_{j,k}]_{1\leq j \leq m, 1\leq k \leq n}$ satisfies
  	\begin{align*}
  		\left|\sum_{j=1}^m \sum_{k=1}^na_{j,k}s_jt_k\right|\leq \left(\max_{1\leq j \leq m}|s_j|\right) \left(\max_{1\leq k \leq n}|t_k|\right), \quad  \forall s_j , t_k \in \mathbb{K},  
  	\end{align*}
  then 
  \begin{align*}
  \left| \sum_{j=1}^m \sum_{k=1}^na_{j,k}\langle u_j, v_k\rangle\right|	\leq K_G\left(\max_{1\leq j \leq m}\|u_j\|\right) \left(\max_{1\leq k \leq n}\|v_k\|\right), \quad  \forall u_j , v_k \in \mathcal{H}. 
  \end{align*}
  	 \end{theorem}
   We refer \cite{BLEI, GROTHENDIECK, ALBIACKALTON, DIESTELFOURIESWART, FRIEDLANDLIMZHANG, RIETZ, HAAGERUP, JAMESON2, KAIJSER, BLEI2014, PISIERGROTHENDIECK} for more information on Grothendieck inequality.  
   \begin{problem}
 	\textbf{(Modular Grothendieck Inequality Problem - 1)} Let $\mathcal{A}$ be a  unital C*-algebra. Whether there is a universal real constant $R_\mathcal{A}>0$ (which can depend upon $\mathcal{A}$) satisfying the following: Let $\mathcal{E}$ be a Hilbert C*-module over $\mathcal{A}$.  If $[a_{j,k}]_{1\leq j \leq m, 1\leq k \leq n}\in \mathbb{M}_{m\times n}(\mathcal{A})$ satisfies
 		\begin{align*}
 		\left(\sum_{j=1}^m \sum_{k=1}^ns_j^*a_{j,k}t_k\right)^*\left( \sum_{p=1}^m \sum_{q=1}^ns_p^*a_{p,q}t_q\right) \leq 1, \quad  \forall s_j , t_k \in \mathcal{A}, s_j^*s_j=1, \forall 1\leq j\leq m, t_k^*t_k=1, \forall 1\leq k\leq n,
 	\end{align*}
 	then 
 	\begin{align*}
 		\left( \sum_{j=1}^m \sum_{k=1}^n\langle u_j, a_{j,k}v_k\rangle\right)^* \left(\sum_{p=1}^m \sum_{q=1}^n\langle u_p, a_{p,q}v_q\rangle\right)	\leq R_\mathcal{A}, \quad  \forall u_j , v_k \in \mathcal{E}, \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.
 	\end{align*}
 \end{problem}
   \begin{problem}
  	\textbf{(Modular Grothendieck Inequality Problem - 2)} Let $\mathcal{A}$ be a  unital C*-algebra. Whether there is a universal real constant $R_\mathcal{A}>0$ (which can depend upon $\mathcal{A}$) satisfying the following: Let $\mathcal{E}$ be a Hilbert C*-module over $\mathcal{A}$.  If $[a_{j,k}]_{1\leq j \leq m, 1\leq k \leq n}\in \mathbb{M}_{m\times n}(\mathcal{A})$ satisfies
  	\begin{align*}
  		\left\|\sum_{j=1}^m \sum_{k=1}^ns^*_ja_{j,k}t_k\right\|  \leq  \left(\max_{1\leq j \leq m}\|s_j\|\right) \left(\max_{1\leq k \leq n}\|t_k\|\right), \quad  \forall s_j , t_k \in \mathcal{A},
  	\end{align*}
  	then 
  	\begin{align*}
  		\left\| \sum_{j=1}^m \sum_{k=1}^n\langle u_j, a_{j,k}v_k\rangle\right\|\leq R_\mathcal{A}\left(\max_{1\leq j \leq m}\|u_j\|\right) \left(\max_{1\leq k \leq n}\|v_k\|\right), \quad  \forall u_j , v_k \in \mathcal{E}.
  	\end{align*}
  \end{problem}
  

  
  
  
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	\bibitem{SCHECHTMAN}
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	\newblock In {\em Geometric aspects of functional analysis (1987--88)}, volume
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	\bibitem{SZANKOWSKI}
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	\bibitem{SZAREK}
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\end{thebibliography}

 


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