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

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$<\infty$, 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 $\mathcal{X}$ and $\mathcal{Y}$ be Banach spaces, ${\mathcal{X}}^{*}$ be the dual of $\mathcal{X}$ and $1\le p<\infty$. 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 $x_1,\cdots ,x_n\in \mathcal{X}$,

$$\begin{array}{c}\hfill {\left(\sum _{j=1}^n{\parallel T{x}_j\parallel }^p\right)}^{\frac{1}{p}}\le C\underset{f\in {\mathcal{X}}^{*},\parallel f\parallel \le 1}{sup}{\left(\sum _{j=1}^n{\left|f\left(x_j\right)\right|}^p\right)}^{\frac{1}{p}}.\end{array}$$

(1)

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

$$\begin{array}{c}\hfill {\pi }_p\left(T\right):=inf\{C:C\mathit{satisfies}\mathit{Inequality}\left(1\right)\}.\end{array}$$

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. 

[7,11] Let $1\le p<\infty$ and $\mathcal{X}$, $\mathcal{Y}$ be Banach spaces. Then $({\Pi }_p(\mathcal{X},\mathcal{Y}),{\pi }_p(·))$ is an operator ideal.

Theorem 2. 

[7,28] 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, ${\parallel T\parallel }_{HS}={\pi }_2\left(T\right)$.

Theorem 3. 

[7,28] (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 $B_{{\mathcal{X}}^{*}}\{f:f\in {\mathcal{X}}^{*},\parallel f\parallel \le 1\}$ in weak*-topology such that

$$\begin{array}{c}\hfill \parallel Tx\parallel \le C{\left(\underset{B_{{\mathcal{X}}^{*}}}{\int }{\left|f\left(x\right)\right|}^{p}d{\mu }_{B_{{\mathcal{X}}^{*}}}\left(f\right)\right)}^{\frac{1}{p}},\forall x\in \mathcal{X}.\end{array}$$

(2)

Moreover, ${\pi }_p\left(T\right)=inf\{C:C\mathit{satisfies}\mathit{Inequality}\left(2\right)\}$.

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 $1\le p<\infty$. 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 $x_1,\cdots ,x_n\in \mathcal{M}$,

$$\begin{array}{c}\hfill {∥\sum _{j=1}^n{\langle T{x}_j,T{x}_j\rangle }^{\frac{p}{2}}∥}^{\frac{1}{p}}\le C\underset{x\in \mathcal{M},\parallel x\parallel \le 1}{sup}{∥\sum _{j=1}^n{\left(\langle x,x_j\rangle \langle x_j,x\rangle \right)}^{\frac{p}{2}}∥}^{\frac{1}{p}}.\end{array}$$

(3)

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

$$\begin{array}{c}\hfill {\pi }_p\left(T\right)inf\{C:C\mathit{satisfies}\mathit{Inequality}\left(3\right)\}.\end{array}$$

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 [32].

Definition 3. 

[32] Let $1\le p<\infty$. 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

$$\begin{array}{c}\hfill {Tr\left|T\right|}^p:\widehat{\mathcal{A}}\ni \chi \mapsto Tr{\left|{\chi }_{*}T\right|}^p\in \mathbb{R}\cup \{\infty \}\end{array}$$

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

$$\begin{array}{c}\hfill {\parallel T\parallel }_p{\parallel Tr\left|T\right|}^p{\parallel }_{\mathcal{A}}^{\frac{1}{p}}.\end{array}$$

Modular 2-Schatten (resp. 1-Schatten) class morphism is called as modular Hilbert-Schmidt (resp. modular trace class). We denote ${\parallel T\parallel }_2$ by ${\parallel T\parallel }_{HS}$.

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 $\mathcal{M}$ 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 ${\left\{{\tau }_n\right\}}_n$ for $\mathcal{M}$, the series ${\sum }_{n=1}^{\infty }{\langle \left|T\right|}^p{\tau }_n,{\tau }_n\rangle$ converges in norm in $\mathcal{A}$ and

$$\begin{array}{c}\hfill {Tr\left|T\right|}^p=\sum _{n=1}^{\infty }{\langle \left|T\right|}^p{\tau }_n,{\tau }_n\rangle .\end{array}$$

We now derive modular version of Theorem 2 with the following notion.

Definition 4. 

[34] 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 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, ${\parallel T\parallel }_{HS}={\pi }_2\left(T\right)$.

Proof. 

$(\Rightarrow )$ Let $T\in {\Pi }_2(\mathcal{M},\mathcal{N})$. Let ${\left\{{\tau }_n\right\}}_{n=1}^{\infty }$ be a modular Parseval frame for $\mathcal{M}$. Then

$$\begin{array}{c}\hfill \langle x,x\rangle =\sum _{n=1}^{\infty }\langle x,{\tau }_n\rangle \langle {\tau }_n,x\rangle ,\forall x\in \mathcal{M},\end{array}$$

(4)

where the series converges in the norm of $\mathcal{A}$. To show T is modular Hilbert-Schmidt, using Theorem 4, 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 ${\left\{{\sum }_{j=1}^n\langle T{\tau }_j,T{\tau }_j\rangle \right\}}_{n=1}^{\infty }$ is bounded. Let $n\in \mathbb{N}$. Since T is 2-summing, using Equation (4) we have

$$\begin{array}{cc}\hfill ∥\sum _{j=1}^n\langle T{\tau }_j,T{\tau }_j\rangle ∥& \le {\pi }_2{\left(T\right)}^2\underset{x\in \mathcal{M},\parallel x\parallel \le 1}{sup}∥\sum _{j=1}^n\langle x,{\tau }_j\rangle \langle {\tau }_j,x\rangle ∥\le {\pi }_2{\left(T\right)}^2\underset{x\in \mathcal{M},\parallel x\parallel \le 1}{sup}∥\sum _{j=1}^{\infty }\langle x,{\tau }_j\rangle \langle {\tau }_j,x\rangle ∥\hfill \\ \hfill & ={\pi }_2{\left(T\right)}^2\underset{x\in \mathcal{M},\parallel x\parallel \le 1}{sup}{\parallel x\parallel }^2={\pi }_2{\left(T\right)}^2.\hfill \end{array}$$

$(\Leftarrow )$ Let $n\in \mathbb{N}$ and $x_1,\cdots ,x_n\in \mathcal{M}$. Let ${\left\{{\omega }_n\right\}}_{n=1}^{\infty }$ be an orthonormal basis for $\mathcal{M}$. Define

$$\begin{array}{c}\hfill S:\mathcal{M}\ni x\mapsto \sum _{j=1}^n\langle x,{\omega }_j\rangle x_j\in \mathcal{M}.\end{array}$$

Then

$$\begin{array}{cc}\hfill {\parallel S\parallel }^2& =\parallel S^{*}{\parallel }^2=\underset{x\in \mathcal{M},\parallel x\parallel \le 1}{sup}{\parallel S^{*}x\parallel }^2=\underset{x\in \mathcal{M},\parallel x\parallel \le 1}{sup}{∥\sum _{n=1}^{\infty }\langle S^{*}x,{\omega }_n\rangle {\omega }_n∥}^2=\underset{x\in \mathcal{M},\parallel x\parallel \le 1}{sup}{∥\sum _{n=1}^{\infty }\langle x,S{\omega }_n\rangle {\omega }_n∥}^2\hfill \\ \hfill & =\underset{x\in \mathcal{M},\parallel x\parallel \le 1}{sup}{∥\sum _{j=1}^n\langle x,x_j\rangle {\omega }_j∥}^2=\underset{x\in \mathcal{M},\parallel x\parallel \le 1}{sup}∥\sum _{j=1}^n\langle x,x_j\rangle \langle x_j,x\rangle ∥.\hfill \end{array}$$

Hence

$$\begin{array}{cc}\hfill ∥\sum _{j=1}^n\langle T{x}_j,T{x}_j\rangle ∥& =∥\sum _{j=1}^n\langle TS{\omega }_j,TS{\omega }_j\rangle ∥={\parallel TS\parallel }_{HS}^2\le {\parallel T\parallel }_{HS}^2\parallel S\parallel ={\parallel T\parallel }_{HS}^2\underset{x\in \mathcal{M},\parallel x\parallel \le 1}{sup}∥\sum _{j=1}^n\langle x,x_j\rangle \langle x_j,x\rangle ∥.\hfill \end{array}$$

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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 $\mathcal{M}$ 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 $G\subseteq B_{\mathcal{M}}\{x:x\in \mathcal{M},\parallel x\parallel \le 1\}$ satisfying following.

(i) 

${\mu }_G\left(G\right)=1$.

(ii) 

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

(iii) 

There exists a real $C>0$ such that

$$\begin{array}{c}\hfill {\langle Tx,Tx\rangle }^{\frac{p}{2}}\le C^p{\int }_G{\left(\langle x.y\rangle \langle y,x\rangle \right)}^{\frac{p}{2}}d{\mu }_G\left(y\right),\forall x\in \mathcal{M}.\end{array}$$

Then T modular p-absolutely summing and ${\pi }_p\left(T\right)=C$.

Proof. 

Let $n\in \mathbb{N}$ and $x_1,\cdots ,x_n\in \mathcal{M}$. Then

$$\begin{array}{cc}\hfill & ∥\sum _{j=1}^n{\langle T{x}_j,T{x}_j\rangle }^{\frac{p}{2}}∥\le C^p∥\sum _{j=1}^n{\int }_G{\left(\langle x_j,y\rangle \langle y,x_j\rangle \right)}^{\frac{p}{2}}d{\mu }_G\left(y\right)∥=C^p∥{\int }_G\sum _{j=1}^n{\left(\langle x_j,y\rangle \langle y,x_j\rangle \right)}^{\frac{p}{2}}d{\mu }_G\left(y\right)∥\hfill \\ \hfill & \le C^p{\int }_G∥\sum _{j=1}^n{\left(\langle x_j,y\rangle \langle y,x_j\rangle \right)}^{\frac{p}{2}}∥d{\mu }_G\left(y\right)\le C^p{\int }_G\underset{y\in \mathcal{M},\parallel y\parallel \le 1}{sup}∥\sum _{j=1}^n{\left(\langle x_j,y\rangle \langle y,x_j\rangle \right)}^{\frac{p}{2}}∥d{\mu }_G\left(y\right)\hfill \\ \hfill & =C^p\underset{y\in \mathcal{M},\parallel y\parallel \le 1}{sup}∥\sum _{j=1}^n{\left(\langle x_j,y\rangle \langle y,x_j\rangle \right)}^{\frac{p}{2}}∥{\mu }_G\left(G\right)=C^p\underset{y\in \mathcal{M},\parallel y\parallel \le 1}{sup}∥\sum _{j=1}^n{\left(\langle x_j,y\rangle \langle y,x_j\rangle \right)}^{\frac{p}{2}}∥.\hfill \end{array}$$

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