# Lie Modules of Banach Space Nest Algebras

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (i)
- A net $\left\{{T}_{\alpha}\right\}$ in $\mathcal{B}\left(X\right)$ converges to the operator T in the strong operator topology if $\left\{{T}_{\alpha}x\right\}$ converges to $Tx$, for all $x\in X$.
- (ii)
- A net $\left\{{T}_{\alpha}\right\}$ in $\mathcal{B}\left(X\right)$ converges to the operator T in the weak operator topology if $\left\{f\left({T}_{\alpha}x\right)\right\}$ converges to $f\left(Tx\right)$, for all $x\in X$ and $f\in {X}^{*}$.

**Lemma 1.**

- (i)
- $N=\vee \{L\in \mathcal{N}:{L}_{-}<N\}$;
- (ii)
- $\mathrm{span}\{{M}^{\perp}:M\in \mathcal{N},{M}_{+}>N\}$ is dense in ${N}^{\perp}$ in the weak*-topology of ${X}^{*}$.

**Proof.**

**Lemma 2.**

- (i)
- The operator $f\otimes y$ lies in $\mathcal{A}$ if and only if there exists $N\in \mathcal{N}$ such that $y\in N$ and $f\in {N}_{-}^{\perp}$;
- (ii)
- If $f\otimes y$ lies in $\mathcal{U}$, then, for all $z\in {N}_{y}$, the rank one operator $f\otimes z$ lies in $\mathcal{U}$;
- (iii)
- If $f\otimes y$ lies in $\mathcal{U}$, then, for all $g\in {\widehat{N}}_{f}^{\perp}$, the rank one operator $g\otimes y$ lies in ${\overline{\mathcal{U}}}^{w}$;
- (iv)
- If $\mathcal{U}$ is weakly closed and $f\otimes y$ lies in $\mathcal{U}$, then, for all $g\in {\widehat{N}}_{f}^{\perp}$, $z\in {N}_{y}$, the rank one operator $g\otimes z$ lies in $\mathcal{U}$.

**Proof.**

## 2. Bimodules

**Lemma 3.**

**Remark 1.**

**Lemma 4.**

**Proof.**

**Corollary 1.**

**Proof.**

**Theorem 1.**

**Proof.**

- Case (1)
- $\varphi \left(N\right)>{N}_{y}$.

- Case (2).
- $\varphi \left(N\right)={N}_{y}$.

_{j}) in span{${M}^{\perp}:M\in \mathcal{N},{M}_{+}{\hat{N}}_{f}$} that converges to g in the weak*-topology. Accordingly, let ${g}_{j}\in {M}_{j}^{\perp}$, where ${M}_{j}\in \mathcal{N}\mathrm{and}({M}_{j}{)}_{+}{\hat{N}}_{f}$.

**Proposition 1.**

**Proof.**

## 3. Lie Modules

**Lemma 5.**

- (i)
- If $P,Q\in \mathcal{A}$ are projections such that $PQ=0=QP$, then $PTQ\in \mathcal{L}$;
- (ii)
- If $\mathcal{L}$ is weakly closed and P is a projection such that $(I-P)\mathcal{L}P\ne \left\{0\right\}$, then:$$P\mathcal{L}(I-P)=PB\left(X\right)(I-P).$$

**Proof.**

**Remark 2.**

**Lemma 6.**

- (i)
- ${\mathcal{K}}_{V}$ is a weakly closed ideal of $\mathcal{A}$;
- (ii)
- $\mathcal{K}$ is a weakly closed $\mathcal{A}$-bimodule;
- (iii)
- $[\mathcal{K},\mathcal{A}]\subseteq \mathcal{L}$.

**Proof.**

**Remark 3.**

**Example 1.**

**Remark 4.**

**Theorem 2.**

**Proof.**

**Theorem 3.**

- (i)
- $\mathcal{J}$ is the largest weakly closed 𝒜-bimodule contained in $\mathcal{L}$ and is given by:$$\mathcal{J}=\{T\in \mathcal{B}(X):T(N)\subseteq \varphi (N),\mathit{for}\mathit{all}N\in \mathcal{N}\},$$
- (ii)
- $\mathcal{K}\text{}and\text{}{\mathcal{D}}_{\mathcal{K}}$ are as in (6) and (11), respectively, and $[\mathcal{K},\mathcal{A}]\subseteq \mathcal{L}.$

**Proof.**

**Example 2.**

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Capitão, P.; Oliveira, L.
Lie Modules of Banach Space Nest Algebras. *Mathematics* **2024**, *12*, 1251.
https://doi.org/10.3390/math12081251

**AMA Style**

Capitão P, Oliveira L.
Lie Modules of Banach Space Nest Algebras. *Mathematics*. 2024; 12(8):1251.
https://doi.org/10.3390/math12081251

**Chicago/Turabian Style**

Capitão, Pedro, and Lina Oliveira.
2024. "Lie Modules of Banach Space Nest Algebras" *Mathematics* 12, no. 8: 1251.
https://doi.org/10.3390/math12081251