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Absolutely Summing Morphisms between Hilbert C*-Modules and Modular Pietsch Factorization Problem

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06 February 2023

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08 February 2023

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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: 
Subject: Computer Science and Mathematics  -   Analysis

MSC:  47B10; 47L20; 46L08; 46L05; 42C15

1. Introduction

In his Resume, A. Grothendieck studied 1-absolutely and 2-absolutely summing operators between Banach spaces [1] (also see [2]). In 1967, for each 1≤ p < ∞ , A. Pietsch introduced the notion of p-absolutely summing operators which became an area around the end of 20 century [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]. In 1979, Tomczak-Jaegermann studied p-summing operators by fixing by fixing number of points [21]. In 1970, Kwapien defined the notion of 0-summing operators [22]. In 2003, Farmer and Johnson introduced the notion of Lipschitz p-summing operators between metric spaces [23] (also see [24,25,26,27]).
Definition 1. 
[7,28] Let X and Y be Banach spaces, X * be the dual of X and 1 ≤ p < ∞ . A bounded linear operator T : X → Y is said to be p-absolutely summing if there is a real constant C > 0 satisfying following: for every n ∈ N and for all x 1 , ⋯ , x n ∈ X ,
∑ j = 1 n ∥ T x j ∥ p 1 p ≤ C sup f ∈ X * , ∥ f ∥ ≤ 1 ∑ j = 1 n | f ( x j ) | p 1 p .
In this case, the p-absolutely summing norm of T is defined as
π p ( T ) : = inf { C : C satisfies Inequality ( 1 ) } .
The set of all p-absolutely summing operators from X to Y is denoted by Π p ( X , Y ) .
Following are most important results in the theory of p-absolutely summing operators.
Theorem 1. 
[7,11] Let 1 ≤ p < ∞ and X , Y be Banach spaces. Then ( Π p ( X , Y ) , π p ( · ) ) is an operator ideal.
Theorem 2. 
[7,28] Let H and K be Hilbert spaces. Then a bounded linear operator T : H → K is 2-absolutely summing if and only if it is Hilbert-Schmidt. Moreover, ∥ T ∥ H S = π 2 ( T ) .
Theorem 3. 
[7,28] (Pietsch Factorization Theorem) Let X and Y be Banach spaces. A bounded linear operator T : X → Y is p-absolutely summing if and only if there is a real constant C > 0 and a regular Borel probability measure on B X * { f : f ∈ X * , ∥ f ∥ ≤ 1 } in weak*-topology such that
∥ T x ∥ ≤ C ∫ B X * | f ( x ) | p d μ B X * ( f ) 1 p , ∀ x ∈ X .
Moreover, π p ( T ) = inf { C : C satisfies Inequality ( 2 ) } .
In this paper, we define the notion of p-absolutely summing morphisms between Hilbert C*-modules over commutative C*-algebras (Definition 2). We derive in Theorem 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 as follows. For the theory of Hilbert C*-modules we refer [29,30,31].
Definition 2. 
Let 1 ≤ p < ∞ . Let M and N be Hilbert C*-modules over a commutative C*-algebra A . An adjointable morphism T : M → N is said to be modular p-absolutely summing if there is a real constant C > 0 satisfying following: for every n ∈ N and for all x 1 , ⋯ , x n ∈ M ,
∑ j = 1 n 〈 T x j , T x j 〉 p 2 1 p ≤ C sup x ∈ M , ∥ x ∥ ≤ 1 ∑ j = 1 n ( 〈 x , x j 〉 〈 x j , x 〉 ) p 2 1 p .
In this case, the p-absolutely summing norm of T is defined as
π p ( T ) inf { C : C satisfies Inequality ( 3 ) } .
The set of all p-absolutely summing morphisms from M to N is denoted by Π p ( M , N ) .
In 2021, Stern and van Suijlekom introduced the notion of modular Schatten class morphisms [32].
Definition 3. 
[32] Let 1 ≤ p < ∞ . Let A be a C*-algebra and A ^ be its Gelfand spectrum. Let M and N be Hilbert C*-modules over A . Let T : M → N be an adjointable morphism. We say that T is in the modular p-Schatten class if the function
Tr | T | p : A ^ ∋ χ ↦ Tr | χ * T | p ∈ R ∪ { ∞ }
lies in A . The modular p-Schatten norm of T is defined as
∥ T ∥ p ∥ Tr | T | p ∥ A 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 ∥ H S .
Using the theory of modular frames for Hilbert C*-modules (see [33]) Stern and van Suijlekom were able to derive following result.
Theorem 4. 
[32] Let M and N be Hilbert C*-modules over A . Let T : M → N be an adjointable morphism. Then T is modular Hilbert-Schmidt if and only if for every modular Parseval frame { τ n } n for M , the series ∑ n = 1 ∞ 〈 | T | p τ n , τ n 〉 converges in norm in A and
Tr | T | p = ∑ n = 1 ∞ 〈 | T | p τ n , τ n 〉 .
We now derive modular version of Theorem 2 with the following notion.
Definition 4. 
[34] A C*-algebra A is said to be monotone closed if every bounded increasing net in A has the least upper bound in A .
Theorem 5. 
Let M and N be Hilbert C*-modules over a commutative C*-algebra A . Assume A is monotone closed. Let T : M → N be an adjointable morphism. Then T ∈ Π 2 ( M , N ) if and only if T is modular Hilbert-Schmidt. Moreover, ∥ T ∥ H S = π 2 ( T ) .
Proof. 
( ⇒ ) Let T ∈ Π 2 ( M , N ) . Let { τ n } n = 1 ∞ be a modular Parseval frame for M . Then
〈 x , x 〉 = ∑ n = 1 ∞ 〈 x , τ n 〉 〈 τ n , x 〉 , ∀ x ∈ M ,
where the series converges in the norm of A . To show T is modular Hilbert-Schmidt, using Theorem 4, it suffices to show that the series ∑ n = 1 ∞ 〈 T τ n , T τ n 〉 converges in norm in A . Note that the series ∑ n = 1 ∞ 〈 T τ n , T τ n 〉 is monotonically increasing. Since the C*-algebra is monotone closed, we are done if we show the sequence { ∑ j = 1 n 〈 T τ j , T τ j 〉 } n = 1 ∞ is bounded. Let n ∈ N . Since T is 2-summing, using Equation (4) we have
∑ j = 1 n 〈 T τ j , T τ j 〉 ≤ π 2 ( T ) 2 sup x ∈ M , ∥ x ∥ ≤ 1 ∑ j = 1 n 〈 x , τ j 〉 〈 τ j , x 〉 ≤ π 2 ( T ) 2 sup x ∈ M , ∥ x ∥ ≤ 1 ∑ j = 1 ∞ 〈 x , τ j 〉 〈 τ j , x 〉 = π 2 ( T ) 2 sup x ∈ M , ∥ x ∥ ≤ 1 ∥ x ∥ 2 = π 2 ( T ) 2 .
( ⇐ ) Let n ∈ N and x 1 , ⋯ , x n ∈ M . Let { ω n } n = 1 ∞ be an orthonormal basis for M . Define
S : M ∋ x ↦ ∑ j = 1 n 〈 x , ω j 〉 x j ∈ M .
Then
∥ S ∥ 2 = ∥ S * ∥ 2 = sup x ∈ M , ∥ x ∥ ≤ 1 ∥ S * x ∥ 2 = sup x ∈ M , ∥ x ∥ ≤ 1 ∑ n = 1 ∞ 〈 S * x , ω n 〉 ω n 2 = sup x ∈ M , ∥ x ∥ ≤ 1 ∑ n = 1 ∞ 〈 x , S ω n 〉 ω n 2 = sup x ∈ M , ∥ x ∥ ≤ 1 ∑ j = 1 n 〈 x , x j 〉 ω j 2 = sup x ∈ M , ∥ x ∥ ≤ 1 ∑ j = 1 n 〈 x , x j 〉 〈 x j , x 〉 .
Hence
∑ j = 1 n 〈 T x j , T x j 〉 = ∑ j = 1 n 〈 T S ω j , T S ω j 〉 = ∥ T S ∥ HS 2 ≤ ∥ T ∥ HS 2 ∥ S ∥ = ∥ T ∥ HS 2 sup x ∈ M , ∥ x ∥ ≤ 1 ∑ j = 1 n 〈 x , x j 〉 〈 x j , x 〉 .
 □
Note that we have not used monotonic closedness of C*-algebra in “if” part. In view of Theorem 3, we formulate following problem.
Question 6. 
Whether there exists a modular Pietsch factorization theorem?
We solve Question (6) partially in the following theorem. Integrals in the following theorem is in the Kasparov sense [35].
Theorem 7. 
Let M and N be Hilbert C*-modules over a commutative C*-algebra A . Let T : M → N be an adjointable morphism. Assume that there exists a Lie group G ⊆ B M { x : x ∈ M , ∥ x ∥ ≤ 1 } satisfying following.
(i) 
μ G ( G ) = 1 .
(ii) 
For each x ∈ M , the map G ∋ y ↦ 〈 x . y 〉 〈 y , x 〉 is continuous.
(iii) 
There exists a real C > 0 such that
〈 T x , T x 〉 p 2 ≤ C p ∫ G ( 〈 x . y 〉 〈 y , x 〉 ) p 2 d μ G ( y ) , ∀ x ∈ M .
Then T modular p-absolutely summing and π p ( T ) = C .
Proof. 
Let n ∈ N and x 1 , ⋯ , x n ∈ M . Then
∑ j = 1 n 〈 T x j , T x j 〉 p 2 ≤ C p ∑ j = 1 n ∫ G ( 〈 x j , y 〉 〈 y , x j 〉 ) p 2 d μ G ( y ) = C p ∫ G ∑ j = 1 n ( 〈 x j , y 〉 〈 y , x j 〉 ) p 2 d μ G ( y ) ≤ C p ∫ G ∑ j = 1 n ( 〈 x j , y 〉 〈 y , x j 〉 ) p 2 d μ G ( y ) ≤ C p ∫ G sup y ∈ M , ∥ y ∥ ≤ 1 ∑ j = 1 n ( 〈 x j , y 〉 〈 y , x j 〉 ) p 2 d μ G ( y ) = C p sup y ∈ M , ∥ y ∥ ≤ 1 ∑ j = 1 n ( 〈 x j , y 〉 〈 y , x j 〉 ) p 2 μ G ( G ) = C p sup y ∈ M , ∥ y ∥ ≤ 1 ∑ j = 1 n ( 〈 x j , y 〉 〈 y , x j 〉 ) p 2 .
 □

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 [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 ) . Remenber 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 8. 
[28,51] (John Theorem) If X is any n-dimensional real Banach space, then
d B M ( Y , ( R n , 〈 · , · 〉 ) ) ≤ n .
Theorem 9. 
[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 1. 
(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 1 is the following conjecture.
Conjecture 10. 
(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 (5) which motivated the definition of type and cotype for Banach spaces.
Definition 5. 
[28] Let 1 ≤ p ≤ 2 . A Banach space X is said to be of  (Rademacher) type p if 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 6. 
[28] Let 2 ≤ q < ∞ . A Banach space X is said to be of (Rademacher) cotype q if 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 2. 
(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. 
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 4. 
(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 [1,2,28,56,57,58,59,60,61,62,63].
Theorem 11. 
[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 5. 
(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 6. 
(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 1. 
Modular Bourgain-Tzafriri restricted invertibility conjecture  and Modular Johnson-Lindenstrauss flattening conjecture have been stated in [64,65].

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