C*-Metric Bourgain-Figiel-Milman, Enflo Type and Mendel-Naor Cotype Problems

1. C*-Metric Bourgain-Figiel-Milman Problem

Let $\mathcal{X}$ and $\mathcal{Y}$ be finite dimensional Banach spaces such that $dim\left(\mathcal{X}\right)=dim\left(\mathcal{Y}\right)$. The Banach-Mazur distance between $\mathcal{X}$ and $\mathcal{Y}$ is defined as

$$\begin{array}{c}\hfill d_{BM}(\mathcal{X},\mathcal{Y})inf\{\parallel T\parallel \parallel T^{-1}\parallel :T:\mathcal{X}\to \mathcal{Y}\mathrm{is}\mathrm{invertible}\mathrm{linear}\mathrm{operator}\}.\end{array}$$

For $n\in \mathbb{N}$, let $({\mathbb{R}}^n,\langle ·,·\rangle )$ be the standard Euclidean Hilbert space. In 1960, Dvoretzky proved the following surprising result [1].

Theorem 1.1. 

[1,2] (Dvoretzky Theorem) There is a universal constant $C>0$ satisfying the following property: If $\mathcal{X}$ is any n-dimensional real Banach space and $0<\epsilon <{\displaystyle \frac{1}{3}}$, then for every natural number

$$\begin{array}{c}\hfill k\le Clogn{\displaystyle \frac{{\epsilon }^2}{|log\epsilon |}},\end{array}$$

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

$$\begin{array}{c}\hfill d_{BM}(\mathcal{Y},({\mathbb{R}}^k,\langle ·,·\rangle ))<1+\epsilon .\end{array}$$

We refer [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17] for more information on Dvoretzky theorem. Having Theorem 1.1, we ask about the metric space version of that. Let $\mathcal{M}$ and $\mathcal{N}$ be finite metric spaces with $o\left(\mathcal{M}\right)=o\left(\mathcal{N}\right)$. Given a map $f:\mathcal{M}\to \mathcal{N}$, define

$$\begin{array}{c}\hfill {\parallel f\parallel }_{Lip}sup\left\{{\displaystyle \frac{d\left(f\right(x),f(y\left)\right)}{d(x,y)}}:x,y\in \mathcal{M},x\ne y\right\}.\end{array}$$

The Bourgain-Figiel-Milman distance between $\mathcal{M}$ and $\mathcal{N}$ is defined as

$$\begin{array}{c}\hfill d_{BFM}{(\mathcal{M},\mathcal{N})inf\{\parallel f\parallel }_{Lip}\parallel f^{-1}{\parallel }_{Lip}:f:\mathcal{M}\to \mathcal{N}\mathrm{is}\mathrm{injective}\}.\end{array}$$

In 1986, Bourgain, Figiel and Milman proved the following nonlinear Dvoretzky theorem [18].

Theorem 1.2. 

[18] (Bourgain-Figiel-Milman Theorem) For every $\epsilon >0$, there is a universal constant $C\left(\epsilon \right)>0$ satisfying the following: Every finite metric space $\mathcal{M}$ contains a subset $\mathcal{N}$ satisfying following conditions.

(i) 

There is an injective map $f:\mathcal{N}\to {\ell }^2(\mathbb{N},\mathbb{R})$ such that

$$d_{BFM}(\mathcal{N},f\left(\mathcal{N}\right))\le 1+\epsilon .$$

(ii) 

$$o\left(\mathcal{N}\right)\ge C\left(\epsilon \right)log\left(o\right(\mathcal{M}\left)\right).$$

Further, following holds: If $\epsilon <1$, then we can take

$$\begin{array}{c}\hfill C\left(\epsilon \right)={\displaystyle \frac{c_1\epsilon }{log\left(\frac{c_2}{\epsilon }\right)}}\mathrm{for}\mathrm{some}{c}_1>0,c_2>0.\end{array}$$

We will formulate problem based on Theorem 1.2 to C*-algebra valued metric spaces. These are generalizations of metric spaces introduced in 2014 by Ma, Jiang and Sun [19].

Definition 1.3. 

[19] Let $\mathcal{A}$ be a unital C*-algebra and ${\mathcal{A}}^{+}\{a{a}^{*}:a\in \mathcal{A}\}$ be the set of all positive elements in $\mathcal{A}$. Let $\mathcal{M}$ be a set. A map $d^{*}:\mathcal{M}\times \mathcal{M}\to {\mathcal{A}}^{+}$ is said to be a C*-valued metric if it satisfies following conditions.

(i) 

If $x,y\in \mathcal{M}$ are such that $d^{*}(x,y)=0$, then $x=y$.

(ii) 

$d^{*}(x,y)=d^{*}(y,x)$ for all $x,y\in \mathcal{M}$.

(iii) 

$d^{*}(x,y)\le d^{*}(x,z)+d^{*}(z,y)$ for all $x,y,z\in \mathcal{M}$.

In this case, $(\mathcal{M},d^{*})$ is called as a C*-metric space.

Given a unital C*-algebra $\mathcal{A}$, we have the standard Hilbert C*-module over $\mathcal{A}$:

$$\begin{array}{c}\hfill {\ell }^2(\mathbb{N},\mathcal{A})\left\{{\left\{a_n\right\}}_{n=1}^{\infty }:a_n\in \mathcal{A},\forall n\in \mathbb{N},\sum _{n=1}^{\infty }{a}_n^{*}{a}_n\in \mathcal{A}\right\}\end{array}$$

equipped with the inner product

$$\begin{array}{c}\hfill \langle {\left\{a_n\right\}}_{n=1}^{\infty },{\left\{b_n\right\}}_{n=1}^{\infty }\rangle \sum _{n=1}^{\infty }{a}_n^{*}{b}_n,\forall {\left\{a_n\right\}}_{n=1}^{\infty },{\left\{b_n\right\}}_{n=1}^{\infty }\in {\ell }^2(\mathbb{N},\mathcal{A}).\end{array}$$

Hence the norm on ${\ell }^2(\mathbb{N},\mathcal{A})$ is given by

$$\begin{array}{c}\hfill \parallel {\left\{a_n\right\}}_{n=1}^{\infty }\parallel {∥\sum _{n=1}^{\infty }{a}_n^{*}{a}_n∥}^{{\displaystyle \frac{1}{2}}},\forall {\left\{a_n\right\}}_{n=1}^{\infty }\in {\ell }^2(\mathbb{N},\mathcal{A}).\end{array}$$

Let $\mathsf{\Omega }$ be a compact Hausdorff space. Let $\mathcal{A}C\left(\mathsf{\Omega }\right)$ be the C*-algebra of set of all continuous functions on $\mathsf{\Omega }$ with standard involution and sup norm. Define

$$\begin{array}{c}\hfill d^{*}({\left\{f_n\right\}}_{n=1}^{\infty },{\left\{g_n\right\}}_{n=1}^{\infty }){\left(\sum _{n=1}^{\infty }{|f_n-g_n|}^2\right)}^{{\displaystyle \frac{1}{2}}}\in \mathcal{A},\forall {\left\{a_n\right\}}_{n=1}^{\infty },{\left\{b_n\right\}}_{n=1}^{\infty }\in {\ell }^2(\mathbb{N},\mathcal{A}).\end{array}$$

Then $({\ell }^2(\mathbb{N},\mathcal{A}),d^{*})$ is a C*-metric space. We now ask for C*-metric version of Bourgain-Figiel-Milman theorem.

Problem 1.4. 

(C*-metric Bourgain-Figiel-Milman Problem) Let $\mathcal{A}$ be a unital C*-algebra. For every $\epsilon >0$, whether there is a universal constant $C(\epsilon ,\mathcal{A})>0$ satisfying the following: Every finite C*-metric space $\mathcal{M}$ contains a subset $\mathcal{N}$ satisfying following conditions.

(i) 

There is an injective map $f:\mathcal{N}\to {\ell }^2(\mathbb{N},\mathcal{A})$ such that

$$\begin{array}{c}\hfill d_{BFM}(\mathcal{N},f\left(\mathcal{N}\right))\le 1+\epsilon .\end{array}$$

(ii) 

$$\begin{array}{c}\hfill o\left(\mathcal{N}\right)\ge C(\epsilon ,\mathcal{A})log\left(o\right(\mathcal{M}\left)\right).\end{array}$$

2. C*-Metric Enflo Type and Mendel-Naor Cotype 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

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2^n}}\sum _{{\epsilon }_1,\dots ,{\epsilon }_n\in \{-1,1\}}{∥\sum _{j=1}^n{\epsilon }_j{h}_j∥}^2=\sum _{j=1}^n{\parallel h_j\parallel }^2.\end{array}$$

(1)

It is Equality (1) which motivated the definition of Type and Cotype for Banach spaces.

Definition 2.1. 

[2] Let $1\le p\le 2$. A Banach space $\mathcal{X}$ is said to be of (Rademacher) Type p if there exists $T_p\left(\mathcal{X}\right)>0$ such that

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{1}{2^n}}\sum _{{\epsilon }_1,\dots ,{\epsilon }_n\in \{-1,1\}}{∥\sum _{j=1}^n{\epsilon }_j{x}_j∥}^p\right)}^{{\displaystyle \frac{1}{p}}}\le T_p\left(\mathcal{X}\right){\left(\sum _{j=1}^n{\parallel x_j\parallel }^p\right)}^{{\displaystyle \frac{1}{p}}},\forall x_1,\dots ,x_n\in \mathcal{X},\forall n\in \mathbb{N}.\end{array}$$

Definition 2.2. 

[2] Let $2\le q<\infty$. A Banach space $\mathcal{X}$ is said to be of(Rademacher) Cotype qif there exists $C_q\left(\mathcal{X}\right)>0$ such that

$$\begin{array}{c}\hfill {\left(\sum _{j=1}^n{\parallel x_j\parallel }^q\right)}^{{\displaystyle \frac{1}{q}}}\le C_q\left(\mathcal{X}\right){\left({\displaystyle \frac{1}{2^n}}\sum _{{\epsilon }_1,\dots ,{\epsilon }_n\in \{-1,1\}}{∥\sum _{j=1}^n{\epsilon }_j{x}_j∥}^q\right)}^{{\displaystyle \frac{1}{q}}},\forall x_1,\dots ,x_n\in \mathcal{X},\forall n\in \mathbb{N}.\end{array}$$

There is a vast literature on Type-Cotype of Banach spaces, see [2,20,21,22,23,24,25,26,27,28].

In 1970, Enflo introduced the notion of Type for metric spaces [29,30].

Definition 2.3. 

[29,30] Let $p>0$. A metric space $(\mathcal{M},d)$ is said to be of Enflo Type p if there exists $E_p\left(\mathcal{M}\right)>0$ satisfying following conditions: For every $n\in \mathbb{N}$ and for every $f:{\{-1,1\}}^n\to \mathcal{M}$ it holds

$$\begin{array}{cc}\hfill & {\left({\displaystyle \frac{1}{2^n}}\sum _{{\epsilon }_1,\dots ,{\epsilon }_n\in \{-1,1\}}d{(f({\epsilon }_1,\dots ,{\epsilon }_n),d(-{\epsilon }_1,\dots ,-{\epsilon }_n))}^p\right)}^{{\displaystyle \frac{1}{p}}}\le \hfill \\ \hfill & E_p\left(\mathcal{M}\right){\left(\sum _{j=1}^n{\displaystyle \frac{1}{2^n}}\sum _{{\epsilon }_1,\dots ,{\epsilon }_n\in \{-1,1\}}d{(f({\epsilon }_1,\dots ,{\epsilon }_{j-1},{\epsilon }_j,{\epsilon }_{j+1},\dots ,{\epsilon }_n),d({\epsilon }_1,\dots ,{\epsilon }_{j-1},-{\epsilon }_j,{\epsilon }_{j+1},\dots ,{\epsilon }_n))}^p\right)}^{{\displaystyle \frac{1}{p}}}.\hfill \end{array}$$

In 2008, Mendel and Naor introduced the notion of Cotype for metric spaces [30].

Definition 2.4. 

[30] Let $q>0$. A metric space $(\mathcal{M},d)$ is said to be of Mendel-Naor Cotype q if there exists ${\Gamma }_q\left(\mathcal{M}\right)>0$ satisfying following conditions: For every $n\in \mathbb{N}$, there exists an even integer $m\in \mathbb{N}$ such that for every $f:{\mathbb{Z}}_m^n\to \mathcal{M}$ it holds

$$\begin{array}{cc}\hfill & \sum _{j=1}^n{\displaystyle \frac{1}{m^n}}\sum _{x\in {\mathbb{Z}}_m^n}d{\left(f\left(x+{\displaystyle \frac{m}{2}}{e}_j,f\left(x\right)\right)\right)}^q\hfill \\ \hfill & \le {\Gamma }_q{\left(\mathcal{M}\right)}^q{m}^q{\displaystyle \frac{1}{m^n}}\sum _{x\in {\mathbb{Z}}_m^n}{\displaystyle \frac{1}{3^n}}\sum _{\epsilon \in {\{-1,0,1\}}^n}d{(f(x+\epsilon ),f\left(x\right))}^q,\epsilon =({\epsilon }_1,\dots ,{\epsilon }_n),\hfill \end{array}$$

where ${\left\{e_j\right\}}_{j=1}^n$ is the standard basis for ${\mathbb{R}}^n$.

We now ask for C*-metric versions of Enflo Type and Mendel-Naor Cotype.

Problem 2.5. 

(C*-metric Enflo Type Problem)Whether there is a way to define C*-metric Enflo Type?

Problem 2.6. 

(C*-metric Mendel-Naor Cotype Problem)Whether there is a way to define C*-metric Mendel-Naor Cotype?

Note that there are also notions of Type for metric spaces by Bourgain, Milman and Wolfson [31] and by Ball [32]. These lead to following problems.

Problem 2.7. 

(C*-metric Bourgain-Milman-Wolfson Type Problem)Whether there is a way to define C*-metric Bourgain-Milman-Wolfson Type?

Problem 2.8. 

(C*-metric Markov/Ball Type Problem)Whether there is a way to define C*-metric Markov/Ball Type?

Remark 2.9. 

In 2023, we formulated Modular Dvoretzky and Type-Cotype problems which are still open [33].