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Modular Dvoretzky, Type-Cotype, Khinchin-Kahane and Grothendieck Inequality Problems

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10 February 2023

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10 February 2023

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Abstract
We recently formulated Modular Dvoretzky, Type-Cotype, Khinchin-Kahane and Grothendieck Inequality problems in the Appendix of \textit{[arXiv:2302.03718v1]}. For the sake of wide accessibility we give a separate treatment of them.
Keywords: 
Subject: Computer Science and Mathematics  -   Analysis

1. Modular Problems

Our first kind of problems come from the Dvoretzky theorem [36,37,38,39,40,41,42,43,44,45,46,47,48,49,50]. Let X and Y be finite dimensional Banach spaces such that dim ( X ) = dim ( Y ) . Remember that the Banach-Mazur distance between X and Y is defined as
d B M ( X , Y ) : = inf { ∥ T ∥ ∥ T − 1 ∥ : T : X → Y is invertible linear operator } .
For n ∈ N , let ( R n , 〈 · , · 〉 ) be the standard Euclidean Hilbert space.
Theorem 1.
[28,51](John Theorem) If X is any n-dimensional real Banach space, then
d B M ( Y , ( R n , 〈 · , · 〉 ) ) ≤ n .
Theorem 2.
[28,52](Dvoretzky Theorem) There is a universal constant C > 0 satisfying the following property: If X is any n-dimensional real Banach space and 0 < ε < 1 3 , then for every natural number
k ≤ C log n ε 2 | log ε | ,
there exists a k-dimensional Banach subspace Y of X such that
d B M ( Y , ( R k , 〈 · , · 〉 ) ) < 1 + ε .
Let A be a unital C*-algebra with invariant basis number property (see [53] for a study on such C*-algebras) and E , F be finite rank Banach modules over A such that rank ( E ) = rank ( F ) . Modular Banach-Mazur distance between E and F is defined as
d M B M ( E , F ) : = inf { ∥ T ∥ ∥ T − 1 ∥ : T : E → F is invertible module homomorphism } .
Given a unital C*-algebra A and n ∈ N , by A n we mean the standard (left) module over A . We equip A n with the C*-valued inner product 〈 · , · 〉 : A n × A n → A defined by
〈 ( a j ) j = 1 n , ( b j ) j = 1 n 〉 : = ∑ j = 1 n a j b j * , ∀ ( a j ) j = 1 n , ( b j ) j = 1 n ∈ A n .
Hence norm on A n is given by
∥ ( a j ) j = 1 n ∥ : = ∑ j = 1 n a j a j * 1 2 , ∀ ( a j ) j = 1 n ∈ A n .
Then it is well-known that A n is a Hilbert C*-module. We denote this Hilbert C*-module by ( A n , 〈 · , · 〉 ) .
Problem 3.(Modular Dvoretzky Problem) Let A be the set of all unital C*-algebras with invariant basis number property. What is the best function Ψ : A × 0 , 1 3 × N → ( 0 , ∞ ) satisfying the following property: If E is any n-rank Banach module over a unital C*-algebra A with IBN property and 0 < ε < 1 3 , then for every natural number
k ≤ Ψ ( A , ε , n ) ,
there exists a k-rank Banach submodule F of E such that
d M B M ( F , ( A k , 〈 · , · 〉 ) ) < 1 + ε .
A particular case of Problem 3 is the following conjecture.
Conjecture 4.(Modular Dvoretzky Conjecture) Let A be a unital C*-algebra with IBN property. There is a universal constant C > 0 (which may depend upon A ) satisfying the following property: If E is any n-rank Banach module and 0 < ε < 1 3 , then for every natural number
k ≤ C log n ε 2 | log ε | ,
there exists a k-rank Banach submodule F of E such that
d M B M ( F , ( A k , 〈 · , · 〉 ) ) < 1 + ε .
Our second kind of problems come from the type-cotype theory of Banach spaces [8,12,13,18,19,28,54,55]. Let H be a Hilbert space, n ∈ N . Recall that for any n points x 1 , ⋯ , x n ∈ H , we have
1 2 n ∑ ε 1 , ⋯ , ε n ∈ { − 1 , 1 } ∑ j = 1 n ε j x j 2 = ∑ j = 1 n ∥ x j ∥ 2 .
It is Equality (1) which motivated the definition of type and cotype for Banach spaces.
Definition 4.
[28] Let 1 ≤ p ≤ 2 . A Banach space X is said to be of(Rademacher) type pif there exists T p ( X ) > 0 such that
1 2 n ∑ ε 1 , ⋯ , ε n ∈ { − 1 , 1 } ∑ j = 1 n ε j x j p 1 p ≤ T p ( X ) ∑ j = 1 n ∥ x j ∥ p 1 p , ∀ x 1 , ⋯ , x n ∈ X , ∀ n ∈ N .
Definition 4.
[28] Let 2 ≤ q < ∞ . A Banach space X is said to be of(Rademacher) cotype qif there exists C q ( X ) > 0 such that
∑ j = 1 n ∥ x j ∥ q 1 q ≤ C q ( X ) 1 2 n ∑ ε 1 , ⋯ , ε n ∈ { − 1 , 1 } ∑ j = 1 n ε j x j q 1 q , ∀ x 1 , ⋯ , x n ∈ X , ∀ n ∈ N .
Let E be a (left) Hilbert C*-module over a unital C*-algebra A , n ∈ N . We see that for any n points x 1 , ⋯ , x n ∈ E , we have
1 2 n ∑ ε 1 , ⋯ , ε n ∈ { − 1 , 1 } ∑ j = 1 n ε j x j , ∑ k = 1 n ε k x k = ∑ j = 1 n 〈 x j , x j 〉 .
Problem 5.(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 (2) for Hilbert C*-modules?
Problem 6.
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?
(i)
A Banach module M over a unital C*-algebra A has modular-type 2 and modular-cotype 2 if and only if M is isomorphic to a Hilbert C*-module over A .
(ii)
If M and N are Banach modules over a unital C*-algebra A of modular-type 2 and modular-cotype 2, respectively, then a bounded module morphism T : M → N factors through a Hilbert C*-module over A .
Problem 7.(Modular Khinchin-Kahane Inequalities Problems) Whether there is a Khinchin-Kahane inequalities for Banach modules over C*-algebras which reduce to Equality (2) for Hilbert C*-modules?
Our third kind of problems come from Grothendieck inequality [1,2,28,56,57,58,59,60,61,62,63].
Theorem 8.
[1,2,28,56,57,58](Grothendieck Inequality) There is a universal constant K G satisfying the following: For any Hilbert space H and any m , n ∈ N , if a scalar matrix [ a j , k ] 1 ≤ j ≤ m , 1 ≤ k ≤ n satisfy
∑ j = 1 m ∑ k = 1 n a j , k s j t k ≤ 1 , ∀ s j , t k ∈ K , | s j | ≤ 1 , | t k | ≤ 1 ,
then
∑ j = 1 m ∑ k = 1 n a j , k 〈 u j , v k 〉 ≤ K G , ∀ u j , v k ∈ H , ∥ u j ∥ ≤ 1 , ∥ v k ∥ ≤ 1 .
Problem 8.(Modular Grothendieck Inequality Problem - 1) Let A be the set of all unital C*-algebras. Let E be a Hilbert C*-module over a unital C*-algebra A . Let A + be the set of all positive elemnts in A . What is the best function Ψ : A × N × N → A + satisfying the following property: If [ a j , k ] 1 ≤ j ≤ m , 1 ≤ k ≤ n ∈ M m × n ( A ) satisfy
∑ j = 1 m ∑ k = 1 n a j , k s j t k , ∑ p = 1 m ∑ q = 1 n a p , q s p t q ≤ 1 , ∀ s j , t k ∈ A ,                             s j s j * = s j * s j = 1 , ∀ 1 ≤ j ≤ m , t k t k * = t k * t k = 1 , ∀ 1 ≤ k ≤ n ,
then
∑ j = 1 m ∑ k = 1 n a j , k 〈 u j , v k 〉 , ∑ p = 1 m ∑ q = 1 n a p , q 〈 u p , v q 〉 ≤ Ψ ( A , m , n ) , ∀ u j , v k ∈ E ,                             〈 u j , u j 〉 = 1 , ∀ 1 ≤ j ≤ m , 〈 v k , v k 〉 = 1 , ∀ 1 ≤ k ≤ n .
In particular, whether Ψ depends on m and n?
Problem 8.(Modular Grothendieck Inequality Problem - 2) Let A be the set of all unital C*-algebras. Let E be a Hilbert C*-module over a unital C*-algebra A . Let A + be the set of all positive elemnts in A . What is the best function Ψ : A × N × N → A + satisfying the following property: If [ a j , k ] 1 ≤ j ≤ m , 1 ≤ k ≤ n ∈ M m × n ( A ) satisfy
∑ j = 1 m ∑ k = 1 n a j , k s j t k , ∑ p = 1 m ∑ q = 1 n a p , q s p t q ≤ 1 , ∀ s j , t k ∈ A , ∥ s j ∥ ≤ 1 , ∀ 1 ≤ j ≤ m , ∥ t k ∥ ≤ 1 , ∀ 1 ≤ k ≤ n ,
then
〈 ∑ j = 1 m ∑ k = 1 n a j , k 〈 u j , v k 〉 , ∑ p = 1 m ∑ q = 1 n a p , q 〈 u p , v q 〉 〉 ≤ Ψ ( A , m , n ) , ∀ u j , v k ∈ E ,                             ∥ u j ∥ ≤ 1 , ∀ 1 ≤ j ≤ m , ∥ v k ∥ ≤ 1 , ∀ 1 ≤ k ≤ n .
In particular, whether Ψ depends on m and n?
We believe strongly that Ψ depends on A .
Remark 8.
Modular Bourgain-Tzafriri restricted invertibility conjectureandModular Johnson-Lindenstrauss flattening conjecturehave been stated in [64,65].

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