Multiplicative weak convergence in Banach $f$-algebras

TL;DR

This paper explores the concept of multiplicative weak convergence within Banach f-algebras and their dual spaces, providing new insights into their structural properties and convergence behaviors.

Contribution

It introduces the notions of multiplicative weak convergence and multiplicative weak* convergence specific to Banach f-algebras, expanding the theoretical framework.

Findings

Established properties of multiplicative weak convergence

Analyzed the dual space's multiplicative weak* convergence

Provided foundational results for further research

Abstract

In this paper, we study multiplicative weak convergence in Banach f-algebra and multiplicative weak* convergence in its dual.

Equations

$$|x_n|u=|x_n|u\wedge |x_{n_0}|u=|x_n|u\wedge w\stackrel{w}{\to }0$$

$$|x_{\alpha }\vee y_{\beta }-x\vee y|u\le |x_{\alpha }\vee y_{\beta }-x_{\alpha }\vee y|u+|x_{\alpha }\vee y-x\vee y|u$$

$$\le |y_{\beta }-y|u+|x_{\alpha }-x|u\stackrel{w}{\to }0$$

$$|x_{\alpha }{y}_{\beta }-xy|u=|x_{\alpha }{y}_{\beta }-x_{\alpha }y+x_{\alpha }y-xy|u$$

$$\le ||x_{\alpha }-x+x||y_{\beta }-y|u+|x_{\alpha }-x||y|u$$

$$\le |x_{\alpha }-x||y_{\beta }-y|u+|y_{\beta }-y||x|u+|x_{\alpha }-x||y|u$$

$$|x_n|u\le |x_n|(u-u\wedge ne)+|x_n|(u\wedge ne)\le \frac{1}{n}{u}^2|x_n|+n|x_n|(u\wedge e)$$

$$f(|x_{\alpha }-x|-u|x_{\alpha }-x|)\le ϵ.$$

$$V_{u,ϵ,f}=\{x\in E:f(|x|u)<ϵ\},$$

Full text

1. Introduction

A net $(x_{\alpha})$ in a Riesz space $E$ is called order convergent to $x\in E$ if $|x_{\alpha}-x|\leq y_{\beta}$ for $y_{\beta}\downarrow 0\in E_{+}$. A net $(x_{\alpha})$ in a Riesz space $E$ is called unbounded order convergent to $x\in E$ if $|x_{\alpha}-x|\wedge u\xrightarrow{o}0$ for all $u\in E_{+}$[4,5]. A net $(x_{\alpha})$ in a Banach lattice $E$ is called unbounded norm convergent to $x\in E$ if $|x_{\alpha}-x|\wedge u\xrightarrow{\|\|}0$ for all $u\in E_{+}$[6,7]. A net $(x_{\alpha})$ in a Banach lattice $E$ is called unbounded absolute weak convergent to $x\in E$ if $|x_{\alpha}-x|\wedge u\xrightarrow{w}0$ for all $u\in E_{+}$[8,9].

In [1],a vector lattice $E$ under an associative multiplication is said to be a $Riesz$ $algebra$ whenever the multiplication makes $E$ an algebra (with the usual properties).A Riesz algebra $E$ is called commutative if $xy=yx$ for all $x,y\in E$. A Riesz algebra $E$ is called f-algebra if $E$ has additionally property that $x\wedge y=0$ implies $(xz)\wedge y=(zx)\wedge y=0$ and $xy\in E_{+}$ for every $x,y\in E_{+}$.A vector lattice $E$ is called Archimedean whenever $\frac{1}{n}x\downarrow 0$ holds in $E$ for each $x\in E_{+}$.Every Archimedean $f$-algebra is commutative.In this article, unless otherwise, all vector lattices are assumed to be real and Archimedean, and so $f$-algebras are commutative. An $f$-algebra $E$ which is at the same time a Banach lattice is called a Banach $f$-algebra whenever $\|xy\|\leq\|x\||y\|$ holds for all $x,y\in E$.

A net $(x_{\alpha})$ in an $f$-algebra $E$ is called multiplicative order convergent to $x\in E$ if $|x_{\alpha}-x|u\xrightarrow{o}0$ for all $u\in E_{+}$ in[11]. A net $(x_{\alpha})$ in a Banach $f$-algebra $E$ is called multiplicative norm convergent to $x\in E$ if $|x_{\alpha}-x|u\rightarrow 0$ for all $u\in E_{+}$ in[12].

Definition 1.1**.**

A net $(x_{\alpha})$ in a Banach $f$-algebra $E$ is called multiplicative weak convergent to $x\in E$ if $|x_{\alpha}-x|u\xrightarrow{w}0$ for all $u\in E_{+}$ $(x_{\alpha}\xrightarrow{mw}x)$.

A net $(x^{{}^{\prime}}_{\alpha})$ in a Banach $f$-algebra $E^{{}^{\prime}}$ is called multiplicative weak star convergent to $x^{{}^{\prime}}\in E$ if $|x^{{}^{\prime}}_{\alpha}-x^{{}^{\prime}}|u\xrightarrow{w^{*}}0$ for all $u^{{}^{\prime}}\in E^{{}^{\prime}}_{+}$ $(x^{{}^{\prime}}_{\alpha}\xrightarrow{mw^{*}}x^{{}^{\prime}})$.

Remark 1.2**.**

For a net $(x\alpha)$ in a Banach $f$-algebra $E$,$(x_{\alpha}\xrightarrow{mw}x)$ implies $(x_{\alpha}y\xrightarrow{mw}xy)$for all $y\in E$ because of $|x_{\alpha}y-xy|u=|x_{\alpha}-x||y|u$ for all $u\in E_{+}$. The converse holds true in Banach $f$-algebras with the multiplication unit. Indeed, assume$(x_{\alpha}y\xrightarrow{mw}xy)$for each $y\in E$. Fix $u\in E_{+}$, so, $|x_{\alpha}-x|u=|x_{\alpha}e-xe|u\xrightarrow{w}0$. Similarily, the $mw^{*}$-convergence has those properties.

Remark 1.3**.**

In Banach $f$-algebras, the weak convergence does not implies the $mw$-convergence, unless it has Schur property. The converse holds true in Banach $f$-algebras with the multiplication unit. Assume$(x_{\alpha}y\xrightarrow{mw}xy)$for each $y\in E$, then $|x_{\alpha}-x|=|x_{\alpha}-x|e\xrightarrow{w}0$, so $x_{\alpha}\xrightarrow{w}x$.

Remark 1.4**.**

In order continuous Banach $f$-algebras, order convergence and $mo$-convergence imply the $mn$-convergence.

2. Results

Lemma 2.1**.**

Let $(x_{\alpha})$and $(y_{\alpha})$ be two nets in a Banach $f$-algebra $E$. Then the following holds true:

$(1)$* $x_{\alpha}\xrightarrow{mw}x$ iff $x_{\alpha}-x\xrightarrow{mw}0$ iff $|x_{\alpha}-x|\xrightarrow{mw}0$;*

$(2)$* if $x_{\alpha}\xrightarrow{mw}x$ then $y_{\beta}\xrightarrow{mw}x$ for each subnet $(y_{\beta})$ of $(x_{\alpha})$;*

$(3)$* suppose $x_{\alpha}\xrightarrow{mw}x$ and $y_{\beta}\xrightarrow{mw}y$, then $ax_{\alpha}+by_{\beta}\xrightarrow{mw}ax+by$ for any $a,b\in R$;*

$(4)$* if $x_{\alpha}\xrightarrow{mw}x$ and $x_{\alpha}\xrightarrow{mw}y$, then $x=y$;*

$(5)$* if $x_{\alpha}\xrightarrow{mw}x$ then $|x_{\alpha}|\xrightarrow{mw}|x|$.*

The $mw^{*}$-convergence also has those properties.

Proof.

We only need $|x-y|\leq|x-x_{\alpha}|+|x_{\alpha}-y|$ and \big{|}|x_{\alpha}|-|x|\big{|}\leq|x_{\alpha}-x|. ∎

Lemma 2.2**.**

For a Banach $f$-algebra $E$ is order continuous, the $mo$-convergence implies $mw$-convergence.

Proof.

According to [12,remark 1.2], we have the conclusion. ∎

Lemma 2.3**.**

Every disjoint decreasing sequence in a Banach $f$-algebra $mw$-converges to zero.

Proof.

Suppose $(x_{n})$ is a disjoint and decreasing sequence in an Banach $f$-algebra $E$. So, $|x_{n}|u$ is also a disjoint sequence in $E$ for all $u\in E_{+}$ by[2,definition 2.53 and 3,definition 3.1.8].Fix $u\in E_{+}$, by[8.lemma 2], we have $|x_{n}|u\xrightarrow{uaw}0$ in $E$. So, $|xn|u\wedge w\xrightarrow{w}0$ in $E$ for all $w\in E_{+}$. Thus, in particular for fixed $n_{0}$, taking $w$ as $|x_{n_{0}}|u$. Then, for all $n\geq n_{0}$, we get

[TABLE]

Since $|x_{n}|u\leq|x_{n_{0}}|u$, therefore, $x_{n}\xrightarrow{mw}0$ in $E$. ∎

Lemma 2.4**.**

Let $E$ be an Banach $f$-algebra, $B$ be a projection band of $E$ and $P_{B}$ be the corresponding band projection. Then $x_{\alpha}\xrightarrow{mw}x$ in $E$ implies $P_{B}(x_{\alpha})\xrightarrow{mw}P_{B}(x)$ in both $E$ and $B$.

Proof.

The proof is similarly to [11,proposition 2.7]. ∎

Lemma 2.5**.**

Let $(x_{\alpha})$ be a net in a Banach $f$-algebra $E$ with order continuous norm. Then we have that

$(1)$* $0\leq x_{\alpha}\xrightarrow{mw}x$ implies $x\in E_{+}$;*

$(2)$* if $(x_{\alpha})$ is monotone and $x_{\alpha}\xrightarrow{mw}x$ then $x_{\alpha}\xrightarrow{w}x$.*

Proof.

$(1)$: Assume $(x_{\alpha})$ consists of non-zero elements and $mw$-converges to $x\in E$. Then, by Lemma 2.1(5), we have $x_{\alpha}=x^{+}_{\alpha}\xrightarrow{mw}x^{+}\geq 0$. Therefore, we have $x\in E_{+}$.

$(2)$: For a fixed $\alpha$ ,we have $x_{\beta}-x_{\alpha}\in E_{+}$ for $\beta\geq\alpha$. By (1), we have $x_{\beta}-x_{\alpha}\xrightarrow{mw}x-x_{\alpha}\in E_{+}$, so $x_{\alpha}\uparrow x$, since $E$ is order continuous, therefore, $x_{\alpha}\xrightarrow{w}x$. ∎

The lattice operations in Banach lattice $f$-algebras are $mw$-continuous in the following sense.

Proposition 2.6**.**

Let $(x_{\alpha})_{\alpha\in A}$ and $(y_{\beta})_{\beta\in B}$ be two nets in a Banach $f$-algebra $E$. If $x_{\alpha}\xrightarrow{mw}x$ and $y_{\beta}\xrightarrow{mw}y$ then $(x_{\alpha}\vee y_{\beta})_{(\alpha,\beta)\in A\times B}\xrightarrow{mw}x\vee y$.($\wedge$ and $||$ are similarily)

Proof.

Assume $x_{\alpha}\xrightarrow{mw}x$ and $y_{\beta}\xrightarrow{mw}y$. Since we have

[TABLE]

[TABLE]

for every $u\in E_{+}$.That is, $(x_{\alpha}\vee y_{\beta})_{(\alpha,\beta)\in A\times B}\xrightarrow{mw}x\vee y$. ∎

The multiplication in $f$-algebra is $mw$-continuous in the following sense.

Proposition 2.7**.**

Let $E$ be a Banach lattice $f$-algebra, and $(x_{\alpha})_{\alpha\in A}$ and $(y_{\beta})_{\beta\in A}$ be two nets in $E$. If $x_{\alpha}\xrightarrow{mw}x$ and $y_{\beta}\xrightarrow{mw}y$ for some $x,y\in E$ ,$(x_{\alpha})$ or $(y_{\beta})$ is monotone, then we have $x_{\alpha}y_{\beta}\xrightarrow{mw}xy$.

Proof.

Assume $x_{\alpha}\xrightarrow{mw}x$ and $y_{\beta}\xrightarrow{mw}y$, then we have $|x_{\alpha}-x|u\xrightarrow{w}0$ and $|y_{\beta}-x|u\xrightarrow{w}0$ for every $u\in E_{+}$.Since

[TABLE]

[TABLE]

[TABLE]

The second and the third terms are weak converges to zero, we show first term is weak converges to zero. Assume $(x_{\alpha})$ is increasing, then $|x_{\alpha}-x|\leq 2|x|$ and $2|y_{\beta}-y||x|u$ is weak converges to zero, so we have the conclusion. ∎

Theorem 2.8**.**

Let $E$ be a order continuous Banach $f$-algebra with a multiplicative unit $e$ and $(x_{n})\downarrow$ be a sequence in $E$. Then $x_{n}\xrightarrow{mw}0$ iff $|x_{n}|(u\wedge n)\xrightarrow{w}0$ for all $u\in E_{+}$.

Proof.

For the forward implication, assume $x_{n}\xrightarrow{mw}0$, then $|x_{n}|u\xrightarrow{w}0$ for all $u\in E_{+}$. Since $|x_{n}|(u\wedge e)\leq|x_{n}|u\xrightarrow{w}0$,therefore, $|x_{n}|(u\wedge n)\xrightarrow{w}0$.

For the reverse implication, by applying [2,theorem 2.57],we have

[TABLE]

Since $(x_{n})\downarrow$ and $E$ is a order continuous Banach $f$-algebra, we have the first term converges weakly to zero, and it is similarily to the proof of [6,lemma 2.11], the second term weak convergent to zero, so $x_{n}\xrightarrow{mw}0$. ∎

It is similarily to [12,proposition 2.4], we have the following result.

Theorem 2.9**.**

Let $0\leq(x_{\alpha})_{\alpha\in A}\downarrow$ be a net in a Banach $f$-algebra $E$ with a quasi-interior point $e$. Then $x_{\alpha}\xrightarrow{mw}0$ iff $x_{\alpha}e\xrightarrow{w}0$.

Corollary 2.10**.**

Let $0\leq(x_{\alpha})_{\alpha\in A}\downarrow$ be a net in an order continuous Banach $f$-algebra $E$ with a weak order unit $e$. Then $x_{\alpha}\xrightarrow{mw}0$ iff $x_{\alpha}e\xrightarrow{w}0$.

Corollary 2.11**.**

Let $0\leq(x_{\alpha})_{\alpha\in A}\downarrow$ be a net in a separable Banach $f$-algebra $E$. Then $x_{\alpha}\xrightarrow{mw}0$ iff $x_{\alpha}e\xrightarrow{w}0$.

Corollary 2.12**.**

Let $0\leq(x_{\alpha})_{\alpha\in A}\downarrow$ be a net in a Banach $f$-algebra $E$ with a quasi-interior point $e$. Then $x_{\alpha}\xrightarrow{mw}0$ iff $x_{\alpha}(e+u)\xrightarrow{w}0$ for all $u\in E_{+}$.

Definition 2.13**.**

A subset $A$ of $E$ is called a $f$-weak-almost order bounded if for any $\epsilon>0$, there is $u\in E_{+},f\in E_{+}^{{}^{\prime}}$ such that $f(|x|-u|x|)\leq\epsilon$.

Next, we have the following work, it is similarily to [10,proposition 2.8].

Theorem 2.14**.**

Let $E$ be a Banach $f$-algebra. If $(x_{\alpha})$ is $f$-weak-almost order bounded and $mw$-converges to $x$, then $x_{\alpha}\xrightarrow{|\sigma|(E,E^{{}^{\prime}})}x$.

Proof.

If $(x_{\alpha})$ is $f$-weak-almost order bounded net. Then the net $(|x_{\alpha}-x|)$ is also. For any $\epsilon>0$, there exists $u>0,f\in E^{{}^{\prime}}_{+}$ such that

[TABLE]

Since $x_{\alpha}\xrightarrow{mw}x$, we have $|x_{\alpha}-x|u\xrightarrow{w}0$. Therefore, we have $x_{\alpha}\xrightarrow{|\sigma|(E,E^{{}^{\prime}})}x$. ∎

Definition 2.15**.**

Let $(x_{\alpha})$ be a net in a Banach lattice $f$-algebra $E$. Then

$(1)$ $(x_{\alpha})$ is said to be $mw$-Cauchy if the net $(x_{\alpha}-x_{\alpha^{{}^{\prime}}})_{(\alpha,\alpha^{{}^{\prime}})\in A\times A}$ $mw$-converges to 0.

$(2)$ $E$ is called $(\sigma)$-$mw$-complete if every $mw$-Cauchy net(sequence) in $E$ is $mw$-convergent.

$(3)$ $E$ is called $mw$-continuous if $x_{\alpha}\xrightarrow{o}0$ implies that $x_{\alpha}\xrightarrow{mw}0$.

Lemma 2.16**.**

A Banach $f$-algebra is $mw$-continuous iff $x_{\alpha}\downarrow 0$ implies $x_{\alpha}\xrightarrow{mw}0$.

Proof.

Let $(x_{\alpha})\xrightarrow{o}0$, we have there exists $y_{\beta}\downarrow 0$ and $|x_{\alpha}|\leq y_{\beta}$. Since $y_{\beta}\downarrow 0$, so $y_{\beta}\xrightarrow{mw}0$, we have $x_{\alpha}\xrightarrow{mw}0$. ∎

Theorem 2.17**.**

Let $E$ be an $mw$-complete Banach $f$-algebra. Then the following statements are quivalent:

$(1)$* $E$ is $mw$-continuous;*

$(2)$* if $0\leq x_{\alpha}\uparrow\leq x$ holds in $E$ then $(x_{\alpha})$ is an $mw$-Cauchy net;*

$(3)$* $x_{\alpha}\downarrow 0$ implies $x_{\alpha}\xrightarrow{mw}0$ in $E$.*

Proof.

$(1)\Rightarrow(2)$: By the proof of [2,lemma 1.37], we have $(y_{\beta}-x_{\alpha})_{\alpha,\beta}\downarrow 0$; according to lemma 2.16, we have $(y_{\beta}-x_{\alpha})_{\alpha,\beta}\xrightarrow{mw}0$, so $(x_{\alpha})$ is an $mw$-Cauchy net.

$(2)\Rightarrow(3)$: Fix arbitrary $\alpha_{0}$, we have $0\leq(x_{\alpha_{0}}-x_{\alpha})_{(\alpha\geq\alpha_{0})}\uparrow\leq x_{\alpha_{0}}$. So $(x_{\alpha_{0}}-x_{\alpha})_{(\alpha\geq\alpha_{0})}$ is $mw$-Cauchy net ,so $(x_{\alpha^{{}^{\prime}}}-x_{\alpha})\xrightarrow{mw}0$. Since $E$ is $mw$-complete, so $x_{\alpha}\xrightarrow{mw}x$. Because of lemma 2.5, we have $x=0$, therefore, $x_{\alpha}\xrightarrow{mw}0$.

$(3)\Rightarrow(1)$: By lemma 2.16. ∎

Corollary 2.18**.**

$(\sigma)$-$mw$-complete also has those properties of the last theorem.

Corollary 2.19**.**

Every $mw$-continuous and $mw$-complete Banach $f$-algebra is Dedekind complete.

Proof.

Suppose $E$ is $mw$-continuous and $mw$-complete and $0\leq x_{\alpha}\uparrow\leq y$ in $E$. By theorem 2.17, $(x_{\alpha})$ is $mw$-Cauchy, so $x_{\alpha}\xrightarrow{mw}x$ and by the proof of lemma 2.5, we have $x_{\alpha}\uparrow x$, so $E$ is Dedekind complete. ∎

It was observed in [7,8,10], we now turn our attention to a topology on Banach $f$-algebras. The sets of the form

[TABLE]

where $u\in E_{+},\epsilon>0,f\in E_{+}^{{}^{\prime}}$ form a base of zero neighborhoods for a Hausdorff topology, and the convergence in this topology is exactly the $mw$-convergen.

Similarily to [7,8,10], we these conclusions.

Lemma 2.20**.**

$V_{u,\epsilon,f}$* is either contained in $[-u,u]$ or contains a non-trivial ideal.*

Lemma 2.21**.**

If $V_{u,\epsilon,f}$ is contained in $[-u,u]$, then $u$ is a strong unit.

Similarily to [10]

Theorem 2.22**.**

Let $E$ be a Banach lattice.$E^{{}^{\prime}}$ has a strong order unit when one of the following conditions is valid:

(1) $mw$-topology agrees with norm topology.

(2) $mw$-topology agrees with weak topology.

Problem 2.23**.**

How to describe the compactness of $mw$-topology.

When the $mw$-topology is metrizability.

Acknowledgement. xxxxxx