Pointwise amenability for dual Banach algebras

TL;DR

This paper introduces and studies two new notions of pointwise amenability tailored for dual Banach algebras, focusing on their properties in specific sequence and semigroup algebras with respect to the $w^*$-topology.

Contribution

It develops the concepts of pointwise Connes amenability and $w^*$-approximate Connes amenability for dual Banach algebras, analyzing their properties in weighted sequence and semigroup algebras.

Findings

Characterization of pointwise amenability for $ell^1( abla)$ and $ell^1( abla, abla)$

Analysis of pointwise Connes amenability in weighted semigroup algebras

Investigation of diagonals related to pointwise amenability in these algebras

Abstract

We shall develop two notions of pointwise amenability, namely pointwise Connes amenability and pointwise $w^*$-approximate Connes amenability, for dual Banach algebras which take the $w^*$-topology into account. We shall study these concepts for the Banach sequence algebras $\ell^1(\omega)$ and the weighted semigroup algebras $ \ell^{1}(\mathbb{N}_{\wedge},\omega)$. For a weight $\omega$ on a discrete semigroup $S$, we shall investigate pointwise amenability/Connes amenability of $\ell^1(S,\omega)$ in terms of diagonals.

Equations

$$|<\psi ,a-au>|+|<\psi ,a-ua>|<ϵ,(\psi \in \mathcal{K}).$$

$$a_i-u{a}_i=(e-u_n)...(e-u_{i+1})((e-u_i)...(e-u_1)a_i)=0,$$

$$A_n=\{a\in \mathcal{A}:ua=a\text{for some}u\in \mathcal{A}\text{with}\Vert u\Vert \le n\}.$$

$$\mathcal{A}\longrightarrow E,\quad a\longmapsto\left\{\begin{array}[]{c}a\cdot x\\ x\cdot a\end{array},\right.$$

$${\pi }^{\ast }(f)=(⟨{\delta }_{gh},f⟩\Omega (g,h){)}_{(g,h)\in S\times S}\in {\ell }^{\infty }(S\times S).$$

$$⟨(f_{gh}\Omega (g,h){)}_{(g,h)\in S\times S},M⟩=⟨f,{\pi }^{\ast \ast }(M)⟩=⟨f,e_{\mathcal{A}}⟩=f_{e_S}.$$

$$⟨{\delta }_g\otimes {\delta }_h,{\delta }_k.T-T.{\delta }_k⟩=⟨{\delta }_g,T({\delta }_{hk})⟩\Omega (h,k)-⟨{\delta }_{kg},T({\delta }_h)⟩\Omega (k,g).$$

$$⟨(f(hk,g)\Omega (h,k)-f(h,kg)\Omega (k,g){)}_{(g,h)\in S\times S},M⟩$$

$$=⟨{\delta }_k.T-T.{\delta }_k,M⟩$$

$$=⟨T,{\delta }_k.M-M.{\delta }_k⟩=0,$$

$$\underset{n}{\lim }\underset{m}{\lim }⟨{\delta }_{k_n{g}_m},T({\delta }_{h_m})⟩,\underset{n}{\lim }\underset{m}{\lim }\Omega (k_n,g_m)$$

$$\underset{n}{\lim }\underset{m}{\lim }⟨{\delta }_{g_m},T({\delta }_{h_m{k}_n})⟩,\underset{n}{\lim }\underset{m}{\lim }\Omega (h_m,k_n)$$

$$J_h(f):=(f_{hg}\Omega (h,g)\omega (h)\Omega (g^{-1},h^{-1})\omega (h^{-1}){)}_{g\in G}(f=(f_g{)}_g\in {\ell }^{\infty }(G)).$$

$$⟨(f_{gh}\Omega (g,h){)}_{(g,h)},M⟩=⟨(f_{e_G}\Omega (g,g^{-1}){)}_g,N⟩=⟨(\Omega (g,g^{-1}){)}_g,N⟩=1.$$

$$\psi ((f(hk,g)\Omega (h,k)-f(h,kg)\Omega (k,g){)}_{(g,h)})=(f(g^{-1}k,g)\Omega (g^{-1},k)-f(g^{-1},kg)\Omega (k,g){)}_g.$$

$$⟨(f(hk,g)\Omega (h,k)-f(h,kg)\Omega (k,g){)}_{(g,h)},M⟩$$

$$=⟨\psi (F)-J_k(\psi (F)),N⟩$$

$$=⟨\psi (F),N-J_k^{\ast }(N)⟩=0,$$

Full text

Keywords: pointwise amenability, pointwise Connes amenability, Beurling algebras.

MSC 2010: Primary: 46H25; Secondary: 16E40, 43A20.

1. Introduction

The key concept of amenability for Banach algebras introduced by B. E. Johnson [7]. The pointwise variant of amenability introduced by H. G. Dales, F. Ghahramani and R. J. Loy, however this appeared formally in [3].

Let $\mathcal{A}$ be a Banach algebra. Then the projective tensor product $\mathcal{A}\hat{\otimes}\mathcal{A}$ naturally is a Banach $\mathcal{A}$-bimodule and the map $\pi:\mathcal{A}\hat{\otimes}\mathcal{A}\longrightarrow\mathcal{A}$ defined by $\pi(a\otimes b)=ab$, $a,b\in\mathcal{A}$ is a linear continuous $\mathcal{A}$-bimodule homomorphism. Let $E$ be a Banach $\mathcal{A}$-bimodule. A derivation is a bounded linear map $D:\mathcal{A}\longrightarrow E$ satisfying $D(ab)=Da\cdot b+a\cdot Db\;\;(a,b\in\mathcal{A})$. A Banach algebra $\mathcal{A}$ is pointwise amenable at $a_{0}\in\mathcal{A}$ if, for each Banach $\mathcal{A}$-bimodule $E$, every derivation $D:\mathcal{A}\to E^{*}$ is* pointwise inner * at $a_{0}$, that is, there exist $\phi\in E^{*}$ such that $D(a_{0})=a_{0}\cdot\phi-\phi\cdot a_{0}$ [3].

Let $\mathcal{A}$ be a Banach algebra. A Banach $\mathcal{A}$-bimodule $E$ is dual if there is a closed submodule $E_{*}$ of $E^{*}$ such that $E=(E_{*})^{*}$. We call $E_{*}$ the predual of $E$. A Banach algebra $\mathcal{A}$ is dual if it is dual as a Banach $\mathcal{A}$-bimodule. We write $\mathcal{A}=(\mathcal{A}_{*})^{*}$ if we wish to stress that $\mathcal{A}$ is a dual Banach algebra with predual $\mathcal{A}_{*}$. For a dual Banach algebra $\mathcal{A}$, a dual Banach $\mathcal{A}$-bimodule $E$ is normal if the module actions of $\mathcal{A}$ on $E$ are $w^{*}$-continuous. The notion of Connes amenability for dual Banach algebras, which is another modification of the notion amenability systematically introduced by V. Runde [12]. A dual Banach algebra $\mathcal{A}$ is Connes amenable if every $w^{*}$-continuous derivation from $\mathcal{A}$ into a normal, dual Banach $\mathcal{A}$-bimodule is inner.

The concept of $w^{*}$-approximately Connes amenability introduced by the first author in [10]. One may see also [5, 6, 9, 11, 13, 14], for more information on Connes amenability and other related notions.

The purpose of this paper is to study pointwise Connes amenability of dual Banach algebras as well as their pointwise $w^{*}$-approximate Connes amenability. The organization of the paper is as follows. In section 2, some basic properties are given. It is shown that every commutative pointwise Connes amenable dual Banach algebras must be unital.

In section 3, it is proved that the Banach sequence algebra $\ell^{1}(\omega)$ is pointwise $w^{*}$-approximately Connes amenable while it is not pointwise Connes amenable, where $\omega$ is a weight function. It is also shown that the same is true for the class of weighted semigroup algebras of the form $\ell^{1}(\mathbb{N}_{\wedge},\omega)$, provided $\lim_{n}\omega(n)=\infty$.

In section 4, the relation between pointwise amenability/Connes amenability of weighted semigroup algebras $\ell^{1}(S,\omega)$ and the existence of some specified diagonals is studied. For a discrete group $G$, the special case $\ell^{1}(G,\omega)$ is also considered.

In this note, we introduce and study the pointwise variants of Connes amenability and $w^{*}$-approximate Connes amenability. We show that every commutative pointwise Connes amenable dual Banach algebras must be unital (Theorem 2.5). For a weight $\omega$ on $\mathbb{N}$, we describe the Banach sequence algebra $\ell^{1}(\omega)$. We show that $\ell^{1}(\omega)$ is pointwise $w^{*}$-approximately Connes amenable, while it is not pointwise Connes amenable (Theorem 3.1). Indeed, $\ell^{1}(\omega)$ clearly shows the difference between pointwise $w^{*}$-approximate Connes amenability and both pointwise Connes amenability and $w^{*}$-approximate Connes amenability. We consider the class of weighted semigroup algebras of the form $\ell^{1}(\mathbb{N}_{\wedge},\omega)$, where $\omega:\mathbb{N}\longrightarrow[1,\infty)$ is any function such that $\lim_{n}\omega(n)=\infty$. We show that $\ell^{1}(\mathbb{N}_{\wedge},\omega)$ is not pointwise Connes amenable, however it is pointwise $w^{*}$-approximately Connes amenable (Theorem 3.2).

2. Connes amenability; pointwise versions

From [10], we recall that a dual Banach algebra $\mathcal{A}$ is $w^{*}$-approximately Connes amenable if, for every normal, dual Banach $\mathcal{A}$-bimodule $E$, every $w^{*}$-continuous derivation $D:\mathcal{A}\to E$ is $w^{*}$-approximately inner, that is, there exists a net $(x_{n})\subseteq E$ such that $D(a)$=$w^{*}$-$\lim_{n}(a\cdot\phi_{n}-\phi_{n}\cdot a)$.

We first introduce the pointwise versions of ($w^{*}$-approximate) Connes amenability.

Definition 2.1**.**

Let $\mathcal{A}$ be a dual Banach algebra. Then:

$(i)$ $\mathcal{A}$ is pointwise Connes amenable at $a_{0}\in\mathcal{A}$ if for every normal, dual Banach $\mathcal{A}$-bimodule $E$, every $w^{*}$-continuous derivation $D:\mathcal{A}\to E$ is pointwise inner at $a_{0}$.

$(ii)$ $\mathcal{A}$ is pointwise $w^{*}$-approximately Connes amenable at $a_{0}\in\mathcal{A}$ if for every normal, dual Banach $\mathcal{A}$-bimodule $E$, every $w^{*}$-continuous derivation $D:\mathcal{A}\to E$ is pointwise $w^{*}$-approximately inner at $a_{0}$, that is, there exists a net $(x_{n})\subseteq E$ such that $D(a_{0})=w^{*}-\lim_{n}\left(a_{0}\cdot x_{n}-x_{n}\cdot a_{0}\right)$.

$(iii)$ $\mathcal{A}$ is * pointwise ($w^{*}$-approximately) Connes amenable* if $\mathcal{A}$ is pointwise ($w^{*}$-approximately) Connes amenable at each $a\in A$.

Let $\mathcal{A}$ be a Banach algebra. From [8], we recall that $\mathcal{A}$ has * left (right) approximate units* if, for each $a\in\mathcal{A}$ and $\varepsilon>0$ there exists $u\in\mathcal{A}$ such that $\|a-ua\|<\epsilon$ ($\|a-au\|<\epsilon$), and $\mathcal{A}$ has approximate units if, for each $a\in\mathcal{A}$ and $\varepsilon>0$, there exists $u\in\mathcal{A}$ such that $\|a-ua\|+\|a-au\|<\epsilon$. The appropriate approximate units have bound $m$ if the element $u$ can be chosen such that $\|u\|\leq m$. The algebra $\mathcal{A}$ has a * bounded* (left or right) approximate units if it has (left or right) approximate units of bound $m$ for some $m\geq 1$.

Definition 2.2**.**

A dual Banach algebra $\mathcal{A}=(\mathcal{A}_{*})^{*}$ has * left (right) $w^{*}$-approximate units* if, for each $a\in\mathcal{A}$ and $\varepsilon>0$, and for each finite subset $\mathcal{K}\subseteq\mathcal{A}_{*}$, there is $u\in\mathcal{A}$ such that $|\langle\psi,a-ua\rangle|<\varepsilon$ ($|\langle\psi,au-a\rangle|<\varepsilon$) for $\psi\in\mathcal{K}$. We say $\mathcal{A}$ has $w^{*}$ -approximate units if, for each $a\in\mathcal{A}$ and $\varepsilon>0$ and for each finite subset $\mathcal{K}\subseteq\mathcal{A}_{*}$, there is $u\in\mathcal{A}$ such that

[TABLE]

The appropriate $w^{*}$-approximate units have bounded $m$ if the element $u$ can be chosen such that $\|u\|\leq m$. The dual Banach algebra $\mathcal{A}$ has bounded (left or right) $w^{*}$-approximate units if it has (left or right) $w^{*}$-approximate units of bound $m$ for some $m\geq 1$.

The following lemma is useful in considerations of identities.

Lemma 2.3**.**

Let $\mathcal{A}$ be a Banach algebra and take $m\geq 1$. Suppose that $\mathcal{A}$ has pointwise left identity of bound $m$ ($i.e.$ for every $a\in\mathcal{A}$ there exists $u\in\mathcal{A}$ with $\|u\|\leq m$ such that $ua=a$). Then, for each $a_{1},...,a_{n}\in\mathcal{A}$ there exists $u\in\mathcal{A}$ with $\|u\|\leq m$ such that $ua_{i}=a_{i},1\leq i\leq n$.

Proof.

Take $a_{1},...,a_{n}\in\mathcal{A}$ . Successively choose $u_{1},...,u_{n}\in\mathcal{A}$ with $\|u_{i}\|\leq m$ , $1\leq i\leq n$, and $(e-u_{i})...(e-u_{1})a_{i}=0$ $(1\leq i\leq n)$. Here, notice that we use $e$ as a symbol. For instance, by $(e-u)a$ we mean $a-au$. Define $u\in\mathcal{A}$ by $e-u=(e-u_{n})...(e-u_{1})$. Then for each $1\leq i\leq n$ we have

[TABLE]

as required. ∎

Theorem 2.4**.**

Let $\mathcal{A}$ be a dual Banach algebra. Suppose that for each $a_{1},...,a_{n}\in\mathcal{A}$, there exists $u\in\mathcal{A}$ with $\|u\|\leq m$ and $ua_{i}=a_{i},$ $1\leq i\leq n$. Then $\mathcal{A}$ has left identity.

Proof.

Let $\mathcal{F}$ be the family of all non-empty, finite subset of $\mathcal{A}$, ordered by inclusion. Therefore $\mathcal{F}$ is a directed set. For each $F\in\mathcal{F}$ choose $e_{F}\in\mathcal{A}$ with $\|e_{F}\|\leq m$ and $a=e_{F}a$, $a\in F$. Since $(e_{F})_{F\in\mathcal{F}}$ is a bounded net in a dual Banach algebra, there is $e\in\mathcal{A}$ such that $e=\lim e_{F}$ . Now, it is clear that $e$ is a left identity for $\mathcal{A}$. ∎

Theorem 2.5**.**

Let $\mathcal{A}=(\mathcal{A}_{*})^{*}$ be a commutative pointwise Connes amenable dual Banach algebra . Then $\mathcal{A}$ has an identity.

Proof.

By [10, Proposition 4.2], for every $a\in\mathcal{A}$ there is a bounded net $(u_{i})\subseteq\mathcal{A}$ such that $au_{i}\xrightarrow{w^{*}}a$ in $\mathcal{A}$. Because $\mathcal{A}$ is a dual space, passing to a subnet, we may suppose that there is $u\in\mathcal{A}$ such that $u_{i}\longrightarrow u$. Hence, for every $a\in\mathcal{A}$, there is $u\in\mathcal{A}$ such that $au=u$. For each $n=1,2,....$, we define

[TABLE]

An argument similar to [8, Theorem 9.7], shows that $\mathcal{A}$ has pointwise identity of bound $m$ for some $m\geq 1$. Now, Lemma 2.3 and Theorem 2.4 yield that $\mathcal{A}$ possesses an identity. ∎

3. Examples

We recall that a weight on a discrete semigroup $S$ is a function $\omega:S\longrightarrow[1,\infty)$ such that $\omega(gh)\leq\omega(g)\omega(h)$, for all $g,h\in S$. Then $\ell^{1}(S,\omega)$ is the Banach space of all complex functions $f=(a_{g})_{g\in S}$ on $S$ with the norm $||(a_{g})_{g}||=\sum_{g\in S}|a_{g}|\omega(g)<\infty$.

It is known that $\ell^{1}(\omega):=\ell^{1}(\mathbb{N},\omega)$ is a dual Banach algebra under pointwise multiplication with the predual $c_{0}(\omega^{-1})$ and without identity element, see for instance [2].

Theorem 3.1**.**

Let $\omega$ be a weight on $\mathbb{N}$. Then:

$(i)$* $\ell^{1}(\omega)$ is not Connes amenable;*

$(ii)$* $\ell^{1}(\omega)$ is not pointwise Connes amenable;*

$(iii)$* $\ell^{1}(\omega)$ is not $w^{*}$-approximately Connes amenable;*

$(iv)$* $\ell^{1}(\omega)$ is pointwise $w^{*}$-approximately Connes amenable.*

Proof.

Since $\ell^{1}(\omega)$ does not have an identity, the clause $(i)$ is immediate by [12, Proposition 4.1]. The commutativity of $\ell^{1}(\omega)$ together with Theorem 2.5 imply $(ii)$. The clause $(iii)$ is exactly [10, Theorem 3.3]. Finally, it was shown in [3, Corollary 1.8.5] that $\ell^{1}(\omega)$ is pointwise-approximately amenable. Therefore, automatically it is pointwise $w^{*}$-approximately Connes amenable. ∎

Next, we consider the semigroup $\mathbb{N}_{\wedge}$ which is $\mathbb{N}$ with the semigroup operation $m\wedge n:=min\{m,n\}$, $(m,n\in\mathbb{N})$. It is clear that any function $\omega:\mathbb{N}\longrightarrow[1,\infty)$ is a weight on the semigroup $\mathbb{N}_{\wedge}$. Then $\ell^{1}(\mathbb{N}_{\wedge},\omega)$ is a commutative Banach algebra with the convolution product $\delta_{m}\star\delta_{n}=\delta_{m\wedge n}$, where $\delta_{n}$ stands for the characteristic function of $\{n\}$ for $n\in\mathbb{N}$. It is well known that $\ell^{1}(\mathbb{N}_{\wedge},\omega)$ is a dual Banach algebra, whenever $\lim_{n}\omega(n)=\infty$ and with the predual $c_{0}(\omega^{-1})$ [3, Proposition 3.1.1].

Suppose that $\lim_{n}\omega(n)=\infty$. Then, because of the lack of identity [3, Propositions 3.3.1 and 3.3.2], $\ell^{1}(\mathbb{N}_{\wedge},\omega)$ is not Connes amenable. The same reason, using Theorem 2.5, implies that $\ell^{1}(\mathbb{N}_{\wedge},\omega)$ is not pointwise Connes amenable. It was shown in [3, Theorem 3.7.1] that $\ell^{1}(\mathbb{N}_{\wedge},\omega)$ is pointwise approximately amenable, and therefore it is pointwise $w^{*}$-approximately Connes amenable as well. We summarize these facts as follows.

Theorem 3.2**.**

Let $\omega$ be a function on $\mathbb{N}$ such that $\lim_{n}\omega(n)=\infty$. Then:

$(i)$* $\ell^{1}(\mathbb{N}_{\wedge},\omega)$ is not Connes amenable;*

$(ii)$* $\ell^{1}(\mathbb{N}_{\wedge},\omega)$ is not pointwise Connes amenable;*

$(iii)$* $\ell^{1}(\mathbb{N}_{\wedge},\omega)$ is pointwise $w^{*}$-approximately Connes amenable.*

At the end, it should be remarked that $w^{*}$-approximate Connes amenability of $\ell^{1}(\mathbb{N}_{\wedge},\omega)$ is an open question for the authors.

4. Relations with diagonals

Let $\mathcal{A}=(\mathcal{A}_{*})^{*}$ be a dual Banach algebra and let $E$ be a Banach $\mathcal{A}$-bimodule. We write $\sigma wc(E)$ for the set of all elements $x\in E$ such that the map

[TABLE]

are $w^{*}$-weak continuous. It was shown that $\pi^{*}(\mathcal{A}_{*})\subseteq\sigma wc(\mathcal{A}\hat{\otimes}\mathcal{A})^{*}$ [14, Corollary 4.6]. Taking adjoints, we can extend $\pi$ to an $\mathcal{A}$-bimodule homomorphism $\pi_{\sigma wc}$ from $\sigma wc((\mathcal{A}\hat{\otimes}\mathcal{A})^{*})^{*}$ to $\mathcal{A}$.

Let $E$ be a Banach space. We then have the canonical map $\imath_{E}:E\longrightarrow E^{**}$ defined by $\langle\mu,\imath_{E}(x)\rangle=\langle x,\mu\rangle$ for $\mu\in E^{*}$, $x\in E$. For Banach spaces $E$ and $F$, we write $\mathcal{L}(E,F)$ for the Banach space of bounded linear maps between $E$ and $F$. It is standard that $(E\hat{\otimes}F)^{*}=\mathcal{L}(F,E^{*})$. For a Banach algebra $\mathcal{A}$, then we obtain a bimodule structure on $\mathcal{L}(\mathcal{A},\mathcal{A}^{*})=(\mathcal{A}\hat{\otimes}\mathcal{A})^{*}$ through $(a\ .\ T)(b)=T(ba)\ ,\ \ \ (T\ .\ a)(b)=T(b)\ .\ a$, for $a,b\in\mathcal{A}$, and for $T\in\mathcal{L}(\mathcal{A},\mathcal{A}^{*})$.

Throughout, we use the term *unital * for a semigroup (or an algebra) $X$ with an identity element $e_{X}$.

The following characterizations will be needed in the sequel.

Theorem 4.1**.**

Let $\mathcal{A}$ be a unital Banach algebra. Then $\mathcal{A}$ is pointwise amenable if and only if for each $a\in\mathcal{A}$ there exists $M\in(\mathcal{A}\hat{\otimes}\mathcal{A})^{**}$ such that $a\ .\ M=M\ .\ a$, and $\pi^{**}(M)=e_{\mathcal{A}}.$

Proof.

The proof is a small variation of the standard argument in . ∎

Theorem 4.2**.**

Let $\mathcal{A}$ be a unital dual Banach algebra. Then:

$(i)$* $\mathcal{A}$ is pointwise Connes amenable if and only if for each $a\in\mathcal{A}$ there exists $M\in\sigma wc((\mathcal{A}\hat{\otimes}\mathcal{A})^{*})^{*}$ such that $a\ .\ M=M\ .\ a$, and $\pi_{\sigma wc}(M)=e_{\mathcal{A}}.$*

$(ii)$* $\mathcal{A}$ is pointwise Connes amenable if and only if for each $a\in\mathcal{A}$ there exists $M\in(\mathcal{A}\hat{\otimes}\mathcal{A})^{**}$ such that $\langle T,a\ .\ M-M\ .\ a\rangle=0$ for each $T\in\sigma wc(\mathcal{L}(\mathcal{A},\mathcal{A}^{*}))$, and $\imath_{\mathcal{A}_{*}}^{*}\pi^{**}(M)=e_{\mathcal{A}}.$*

Proof.

The clause $(i)$ is analogous to [14, Theorem 4.8]. Because $\sigma wc((\mathcal{A}\hat{\otimes}\mathcal{A})^{*})^{*}$ is a quotient of $(\mathcal{A}\hat{\otimes}\mathcal{A})^{**}$, the clause $(ii)$ is just a re-statement of $(i)$. ∎

For a discrete semigroup $S$, we recall that $\ell^{1}(S)\hat{\otimes}\ell^{1}(S)=\ell^{1}(S\times S)$, where $\delta_{g}\otimes\delta_{h}$ is identified with $\delta_{(g,h)}$ for $g,h\in S$. Thus we have $\mathcal{L}(\ell^{1}(S),\ell^{\infty}(S))=(\ell^{1}(S)\hat{\otimes}\ell^{1}(S))^{*}=\ell^{1}(S\times S)^{*}=\ell^{\infty}(S\times S)$, where $T\in\mathcal{L}(\ell^{1}(S),\ell^{\infty}(S))$ is identified with $(T_{(g,h)})_{(g,h)\in S\times S}\in\ell^{\infty}(S\times S)$, while $T_{(g,h)}:=\langle\delta_{h},T(\delta_{g})\rangle$. Let $\omega$ be a weight on $S$. If $S$ is unital then, without loss of generality, we put $\omega(e_{S})=1$. The Banach space $\ell^{1}(S,\omega)$ with the convolution product is a Banach algebra, called a Beurling algebra. We consider $\ell^{1}(S,\omega)$ as the Banach space $\ell^{1}(S)$ with the product $\delta_{g}\star_{\omega}\delta_{h}:=\delta_{gh}\Omega(g,h)$, where $\Omega(g,h):=\frac{\omega(gh)}{\omega(g)\omega(h)}$, $(g,h\in S)$, and extend $\star_{\omega}$ to $\ell^{1}(S)$ by linearity and continuity. A semigroup $S$ is weakly cancellative if, for each $s\in S$, the maps $L_{s}$ and $R_{s}$, defined by $L_{s}(t)=st$ and $R_{s}(t)=ts$, are finite-to-one. In this case, $\ell^{1}(S,\omega)$ is a dual Banach algebra with predual $c_{0}(S)$ [4, Proposition 5.1]

Theorem 4.3**.**

Let $S$ be a discrete unital semigroup, let $\omega$ be a weight on $S$ and let $\mathcal{A}:=\ell^{1}(S,\omega)$. Consider the following statements:

$(1)$* $\mathcal{A}$ is pointwise amenable.*

$(2)$* For each $k\in S$ there exists $M\in(\mathcal{A}\hat{\otimes}\mathcal{A})^{**}=\ell^{\infty}(S\times S)^{*}$ such that:*

$(i)$* $\langle(f(hk,g)\Omega(h,k)-f(h,kg)\Omega(k,g))_{(g,h)\in S\times S},M\rangle=0$ for each bounded function $f:S\times S\longrightarrow\mathbb{C}$;*

$(ii)$* $\langle(f_{gh}\Omega(g,h))_{(g,h)\in S\times S},M\rangle=f_{e_{S}}$ for each bounded family $(f_{g})_{g\in S}$.*

$(3)$* $\mathcal{A}$ is pointwise amenable at $\delta_{k}$ for each $k\in S$.*

Then we have $(1)\Longrightarrow(2)\Longrightarrow(3)$.

Proof.

$(1)\Longrightarrow(2)$ Suppose that $\mathcal{A}$ is pointwise amenable and that $k\in S$. Take $M\in\ell^{\infty}(S\times S)^{*}$ as in Theorem 4.1. For each bounded family $(f_{g})_{g\in S}$, we have

[TABLE]

Therefore

[TABLE]

Next, for every $T\in\mathcal{L}(\mathcal{A},\mathcal{A}^{*})=\ell^{\infty}(S\times S)$, we see that

[TABLE]

Let $f:S\times S\longrightarrow\mathbb{C}$ be a bounded function and take $T\in\mathcal{L}(\mathcal{A},\mathcal{A}^{*})=\ell^{\infty}(S\times S)$ defined by $\langle\delta_{h},T(\delta_{g})\rangle=f(g,h)$. Hence

[TABLE]

as required. Likewise for the implication $(2)\Longrightarrow(3)$. ∎

The following is [4, Proposition 5.5], in which $\mathcal{W}(\mathcal{A},\mathcal{A}^{*})$ stands for the collection of weakly compact operators in $\mathcal{L}(\mathcal{A},\mathcal{A}^{*})$, and the set of weakly almost periodic elements in $\mathcal{W}(\mathcal{A},\mathcal{A}^{*})$ is denoted by $WAP(\mathcal{W}(\mathcal{A},\mathcal{A}^{*}))$.

Theorem 4.4**.**

Let $S$ be a weakly cancellative semigroup, let $\omega$ be a weight on $S$, and let $\mathcal{A}:=\ell^{1}(S,\omega)$. Let $T\in\mathcal{L}(\mathcal{A},\mathcal{A}^{*})$ be such that $T(\mathcal{A})\subseteq\imath_{c_{0}(S)}(c_{0}(S))$ and $T^{*}(\imath_{\mathcal{A}}(\mathcal{A}))\subseteq\imath_{c_{0}(S)}(c_{0}(S))$. Then $T\in\mathcal{W}(\mathcal{A},\mathcal{A}^{*})$, and $T\in WAP(\mathcal{W}(\mathcal{A},\mathcal{A}^{*}))$ if and only if, for each sequence $(k_{n})$ of distinct elements of $S$, and each sequence $(g_{m},h_{m})$ of distinct elements of $S\times S$ such that the repeated limits

[TABLE]

[TABLE]

all exist, we have at least one repeated limit in each row is zero.

Theorem 4.5**.**

Let $S$ be a discrete, weakly cancellative semigroup, let $\omega$ be a weight on $S$ and let $\mathcal{A}:=\ell^{1}(S,\omega)$ be unital. Consider the following statements:

$(1)$* $\mathcal{A}$ is pointwise Connes amenable.*

$(2)$* For each $k\in S$ there exists $M\in(\mathcal{A}\hat{\otimes}\mathcal{A})^{**}=\ell^{\infty}(S\times S)^{*}$ such that:*

$(i)$* $\langle(f(hk,g)\Omega(h,k)-f(h,kg)\Omega(k,g))_{(g,h)\in S\times S},M\rangle=0$ for each bounded function $f:S\times S\longrightarrow\mathbb{C}$ which is such that the map $T\in\mathcal{L}(\mathcal{A},\mathcal{A}^{*})$, defined by $\langle\delta_{h},T(\delta_{g})\rangle=f(g,h)$, for $g,h\in S$, satisfies the conclusions of Theorem 4.4;*

$(ii)$* $\langle(f_{gh}\Omega(g,h))_{(g,h)\in S\times S},M\rangle=\langle f,e_{\mathcal{A}}\rangle$ for each family $(f_{g})_{g\in S}\in c_{0}(S)$.*

$(3)$* $\mathcal{A}$ is pointwise Connes amenable at $\delta_{k}$ for each $k\in S$.*

Then we have $(1)\Longrightarrow(2)\Longrightarrow(3)$.

Proof.

Using Theorem 4.2 $(ii)$ in place of Theorem 4.1, this follows as Theorem 4.3. ∎

Let $G$ be a discrete group and let $h\in G$. Define $J_{h}:\ell^{\infty}(G)\longrightarrow\ell^{\infty}(G)$ by

[TABLE]

It is clear that $||J_{h}(f)||\leq||f||\omega(h)\omega(h^{-1})$, so that $J_{h}$ is bounded.

The following is the pointwise variant of [4, Definition 5.10].

Definition 4.6**.**

Let $\omega$ be a weight on a discrete group $G$, and let $h_{0}\in G$. Then $G$ is pointwise $\omega$-amenable at $h_{0}$ if there exists $N\in\ell^{\infty}(G)^{*}$ such that $\langle(\Omega(g,g^{-1})_{g\in G},N\rangle=1$ and $J_{h_{0}}^{*}(N)=N.$ We say that $G$ is pointwise $\omega$-amenable if it is pointwise $\omega$-amenable at each $h\in G$.

Theorem 4.7**.**

Let $G$ be a discrete group, let $\omega$ be a weight on $G$ and let $\mathcal{A}=\ell^{1}(G,\omega)$. Consider the following statements:

$(1)$* $\mathcal{A}$ is pointwise amenable.*

$(2)$* $\mathcal{A}$ is pointwise Connes-amenable.*

$(3)$* $G$ is pointwise $\omega$-amenable.*

$(4)$* $\mathcal{A}$ is pointwise amenable at $\delta_{k}$ for each $k\in G$.*

Then we have $(1)\Longrightarrow(2)\Longrightarrow(3)\Longrightarrow(4)$.

Proof.

The implication $(1)\Longrightarrow(2)$ is trivial, and $(2)\Longrightarrow(3)$ is a more or less verbatim of the proof of [4, Theorem 5.11 $(1)\Longrightarrow(3)$] .

$(3)\Longrightarrow(4)$: Let $k\in G$, and let $N\in\ell^{\infty}(G)^{*}$ be given as in $(3)$. Define $\psi:\ell^{\infty}(G\times G)\longrightarrow\ell^{\infty}(G)$ by $\langle\delta_{g},\psi(F)\rangle:=F(g,g^{-1})$, for each $F\in\ell^{\infty}(G\times G)$ and $g\in G$. Put $M:=\psi^{*}(N)$. Then it suffices to show that $M$ has desired properties in Theorem 4.3 $(2)$. First, for a bounded family $(f_{g})_{g\in G}$, we see that

[TABLE]

Next, for an arbitrary bounded function $f:G\times G\longrightarrow\mathbb{C}$ , it is clear that

[TABLE]

Define $F:G\times G\longrightarrow\mathbb{C}$, by $F(g,h):=f(hk,g)\Omega(h,k)$, for each $g,h\in G$. Hence, it is readily seen that $F$ is bounded and $||F||_{\infty}\leq||f||_{\infty}$. Therefore

[TABLE]

as required. ∎

Remark 4.8**.**

It seems to be a right conjecture that pointwise amenability of $\ell^{1}(G,\omega)$ coincides with its pointwise Connes amenability, however we are not able to prove (or disprove) it. In fact, we can not establish the implication $(4)\Longrightarrow(1)$ in Theorem 4.7, because we can see no reason that pointwise amenability at elements $a$ and $b$ gives any information about $a+b$.