$\varphi$-contractibility and $\varphi$-Connes amenability coincide with some older notions

TL;DR

This paper demonstrates that several notions of $\

Contribution

It shows that various definitions of $\

Findings

Different definitions of $\

Older concepts are equivalent to newer ones

Abstract

It is shown that various definitions of $\varphi$-Connes amenability as introduced independently in \cite{Gh-Ja, mah, Sh-Am}, are just rediscovering existing notions and presenting them in different ways. It is also proved that even $\varphi$-contractibility as defined in \cite{Sangani}, is equivalent to an older and simpler concept.

Equations

$$\mathfrak{A}\longrightarrow E\ \ ,\ \ \ a\longmapsto\left\{\begin{array}[]{ll}a\ .\ x\\ x\ .\ a\end{array}\right.\ ,$$

$$(\ast )\phi (u)=1\text{and}ua=au=\phi (a)u(a\in \mathfrak{A}).$$

$$⟨f,a.t⟩$$

$$=\phi (a)⟨D^{\ast }(f),u⟩-⟨Da,f⟩=\phi (a)⟨f,t⟩-⟨Da,f⟩.$$

$$\pi :{\mathfrak{A}}^{♯}\hat{\otimes }P⟶P;a\otimes x⟼a\cdot x(a\in {\mathfrak{A}}^{♯},x\in P)$$

Full text

1. Introduction

The precious and fertile notion of amenability was introduced by B. E. Johnson in [4]. A generalization of amenability depending on homomorphisms was introduced and studied by E. Kaniuth, A. T. Lau and J. Pym [5], and independently by M. S. Monfared [7]. For a Banach algebra $\mathfrak{A}$, we write $\Delta(\mathfrak{A})$ for the set of all homomorphisms from $\mathfrak{A}$ onto $\mathbb{C}$. Let $\varphi\in\Delta(\mathfrak{A})$. An element $m\in\mathfrak{A}^{**}$ is called a right [left] $\varphi$-mean if $m(\varphi)=1$ and $m(f\cdot a)=\varphi(a)m(f)$ [$m(a\cdot f)=\varphi(a)m(f)$] for $a\in\mathfrak{A}$ and $f\in\mathfrak{A}^{*}$. A Banach algebra is right [left] $\varphi$-amenable if it has a right [left] $\varphi$-mean [5, 7]. We call $\mathfrak{A}$ $\varphi$-amenable if it is both left and right $\varphi$-amenable.

Later in [3], the authors introduced the concept of $\varphi$-contractibility. Let $\mathfrak{A}$ be a Banach algebra and $E$ be a Banach $\mathfrak{A}$-bimodule. A continuous linear operator $D:\mathfrak{A}\longrightarrow E$ is a derivation if it satisfies $D(ab)=D(a)\cdot b+a\cdot D(b)$ for all $a,b\in\mathfrak{A}$. Given $x\in E$, the inner derivation $ad_{x}:\mathfrak{A}\longrightarrow E$ is defined by $ad_{x}(a)=a\cdot x-x\cdot a$. Let $\varphi\in\Delta(\mathfrak{A})$. We write $\mathbb{M}_{\varphi}$ $[_{\varphi}\mathbb{M}]$ for the set of all Banach $\mathfrak{A}$-bimodules $E$ such that the right [left] module action of $\mathfrak{A}$ on $E$ is given by $x\cdot a:=\varphi(a)x$ $[a\cdot x:=\varphi(a)x]$ for $a\in\mathfrak{A}$, $x\in E$. Precisely, $\mathfrak{A}$ is right [left] $\varphi$-contractible if for each Banach $\mathfrak{A}$-bimodule $E\in\mathbb{M}_{\varphi}$ $[E\in$ ${}_{\varphi}\mathbb{M}]$, every derivation $D:\mathfrak{A}\longrightarrow E$ is inner. We say $\mathfrak{A}$ is $\varphi$-contractible if it is both left and right $\varphi$-contractible.

Recently and motivated by the above notions, several authors have defined and studied the concept of $\varphi$-Connes amenability, where $\varphi$ is a $w^{*}$-continuous homomorphism on a dual Banach algebra [1, 6, 9]. However, [6] was released into the public domain over three years ago.

In this brief note, we take a look at $\varphi$-contractibility and $\varphi$-Connes amenability in minute detail. While the authors put a lot of effort into studying these concepts, their effort may have been wasted here. Considering all three types of the notion of $\varphi$-Connes amenability introduced in [1, 6, 9], we shall see that none of them are new. In fact they coincide with both $\varphi$-amenability and $\varphi$-contractibility. Next, we shall prove that the concept of $\varphi$-contractibility is also equivalent to an existing notion. On closer inspection, saying that a Banach algebra $\mathfrak{A}$ is $\varphi$-contractible is equivalent to saying that the one-dimensional Banach $\mathfrak{A}$-bimodule $\mathbb{C}_{\varphi}$ is projective. Although this concept goes back to Helemskii’s works in the 1970s (see his book [2], or alternatively a paper of White [10]), most of the authors who have studied $\varphi$-contractibility seem unaware of this fact.

2. $\varphi$-Connes amenability

Let $\mathfrak{A}$ be a Banach algebra. A Banach $\mathfrak{A}$-bimodule $E$ is dual if there is a closed submodule $E_{*}$ of $E^{*}$ such that $E=(E_{*})^{*}$. We say $E_{*}$ the predual of $E$. A Banach algebra is dual if it is dual as a Banach $\mathfrak{A}$-bimodule. We write $\mathfrak{A}=(\mathfrak{A}_{*})^{*}$ if we wish to stress that $\mathfrak{A}$ is a dual Banach algebra with predual $\mathfrak{A}_{*}$.

We start with the definition of $\varphi$-Connes amenability in the sense of [1]. Let $\mathfrak{A}=(\mathfrak{A}_{*})^{*}$ be a dual Banach algebra, and let $\varphi\in\Delta(\mathfrak{A})\cap\mathfrak{A}_{*}$. A dual Banach $\mathfrak{A}$-bimodule $E\in$ ${}_{\varphi}\mathbb{M}$ is normal if the module action $a\longmapsto x\cdot a$ of $\mathfrak{A}$ on $E$ is $w^{*}$-continuous. A dual Banach algebra $\mathfrak{A}=(\mathfrak{A}_{*})^{*}$ is left $\varphi$-Connes amenable if for every normal dual Banach $\mathfrak{A}$-bimodule $E\in$ ${}_{\varphi}\mathbb{M}$, every $w^{*}$-continuous derivation $D:\mathfrak{A}\longrightarrow E$ is inner. Although they consider just left $\varphi$-Connes amenable Banach algebras, there are similar definitions for right $\varphi$-Connes amenable and $\varphi$-Connes amenable Banach algebras. The authors show that (left) $\varphi$-Connes amenability of $\mathfrak{A}$ is equivalent to the existence a (left) $\varphi$-mean [1, Theorem 2.3].

Theorem 2.1**.**

Suppose that $\mathfrak{A}=(\mathfrak{A}_{*})^{*}$ is a dual Banach algebra and $\varphi\in\Delta(\mathfrak{A})\cap\mathfrak{A}_{*}$. Then the following statement are equivalent:

$(i)$ $\mathfrak{A}$ is $\varphi$-Connes amenable in the sense of [1];

$(ii)$ $\mathfrak{A}$ is $\varphi$-contractible;

$(iii)$ $\mathfrak{A}$ is $\varphi$-amenable.

Proof. The implications $(ii)\Longrightarrow(iii)$ and $(iii)\Longrightarrow(i)$ are immediate.

$(i)\Longrightarrow(ii)$ Take a $\varphi$-mean $m\in\mathfrak{A}^{**}$. Consider the $\mathfrak{A}$-bimodule inclusion map $\imath:\mathfrak{A}_{*}\longrightarrow\mathfrak{A}^{*}$. Taking adjoints, we obtain a $w^{*}$-$w^{*}$-continuous $\mathfrak{A}$-bimodule map $\xi:\mathfrak{A}^{**}\longrightarrow\mathfrak{A}$. Now put $u=\xi(m)\in\mathfrak{A}$. It is easily checked that $\varphi(u)=1$ and $ua=au=\varphi(a)u$, for all $a\in\mathfrak{A}$. Therefore by Theorem 3.1 below, $\mathfrak{A}$ is $\varphi$-contractible.

$(ii)\Longrightarrow(i)$ Again by Theorem 3.1, there is an element $u\in\mathfrak{A}$ satisfying $\varphi(u)=1$ and $ua=ua=\varphi(a)u$, for all $a\in\mathfrak{A}$. It is readily seen that $u$ is a $\varphi$-mean and whence $\mathfrak{A}$ is $\varphi$-Connes amenable. ∎

Now, we consider the definition of $\varphi$-Connes amenability in the sense of [9]. Let $\mathfrak{A}=(\mathfrak{A}_{*})^{*}$ be a dual Banach algebra, and let $\varphi$ be a non-zero $w^{*}$-continuous multiplicative linear functional on $\mathfrak{A}$. The authors in [9, Definition 2.1] say that $\mathfrak{A}$ is (left) $\varphi$-Connes amenable if there exists $m\in\mathfrak{A}$ such that $m(\varphi)=1$ and $am=\varphi(a)m$, for every $a\in\mathfrak{A}$. Then by Theorem 3.1, left $\varphi$-Connes amenability is nothing but right $\varphi$-contractibility. Hence the following is straightforward.

Theorem 2.2**.**

Suppose that $\mathfrak{A}$ is a dual Banach algebra and $\varphi$ be a non-zero $w^{*}$-continuous multiplicative linear functional on $\mathfrak{A}$. Then the following statement are equivalent:

$(i)$ $\mathfrak{A}$ is $\varphi$-Connes amenable (in the sense of [9]);

$(ii)$ $\mathfrak{A}$ is $\varphi$-contractible;

$(iii)$ $\mathfrak{A}$ is $\varphi$-amenable.

Let $\mathfrak{A}$ be a dual Banach algebra and let $E$ be a Banach $\mathfrak{A}$-bimodule. From [8] we write $\sigma wc(E)$ for the set of all elements $x\in E$ such that the maps

[TABLE]

are $w^{*}$-weak continuous.

We conclude by looking at the definition of $\varphi$-Connes amenability from [6]. Suppose that $\mathfrak{A}$ is a dual Banach algebra and $\varphi$ is a homomorphism from $\mathfrak{A}$ onto $\mathbb{C}$. Then it is an easy observation that $\varphi$ is $w^{*}$-continuous if and only if $\varphi\in\sigma wc(\mathfrak{A}^{*})$. Suppose that $\mathfrak{A}$ is a dual Banach algebra and $\varphi$ is a $w^{*}$-continuous homomorphism from $\mathfrak{A}$ onto $\mathbb{C}$. We call $\mathfrak{A}$ (right) $\varphi$-Connes amenable if $\mathfrak{A}$ admits a (right) $\varphi$-Connes mean $m$, i.e., there exists a bounded linear functional $m$ on $\sigma wc(\mathfrak{A}^{*})$ satisfying $m(\varphi)=1$ and $m(f\ .\ a)=\varphi(a)m(f)$ for all $a\in\mathfrak{A}$ and $f\in\sigma wc(\mathfrak{A}^{*})$. Similarly, we may consider left $\varphi$-Connes amenability. Meanwhile, $\mathfrak{A}$ is $\varphi$-Connes amenable if it is both left and right $\varphi$-Connes amenable.

Theorem 2.3**.**

Suppose that $\mathfrak{A}=(\mathfrak{A}_{*})^{*}$ is a dual Banach algebra and $\varphi$ be a non-zero $w^{*}$-continuous multiplicative linear functional on $\mathfrak{A}$. Then the following statement are equivalent:

$(i)$ $\mathfrak{A}$ is $\varphi$-Connes amenable (in the sense of [6]);

$(ii)$ $\mathfrak{A}$ is $\varphi$-contractible;

$(iii)$ $\mathfrak{A}$ is $\varphi$-amenable.

Proof. Only $(i)\Longleftrightarrow(ii)$ needs the proof. Let $\iota:\mathfrak{A}\longrightarrow\sigma wc(\mathfrak{A}^{*})^{*}$ be the $\mathfrak{A}$-bimodule map obtained by composing the canonical inclusion $\mathfrak{A}\longrightarrow\mathfrak{A}^{**}$ with the quotient map $\mathfrak{A}^{**}\longrightarrow\sigma wc(\mathfrak{A}^{*})^{*}$, so that $\langle\iota(a),\psi\rangle=\psi(a)$ for all $a\in\mathfrak{A}$ and $\psi\in\sigma wc(\mathfrak{A}^{*})$.

$(i)\Longrightarrow(ii)$ Since $\mathfrak{A}$ is a dual Banach algebra, $\mathfrak{A}_{*}$ is an $\mathfrak{A}$-bimodule and $\mathfrak{A}_{*}\subseteq\sigma wc(\mathfrak{A}^{*})$ [8, Corollary 4.6]. Therefore taking adjoints gives us a $w^{*}$-$w^{*}$-continuous $\mathfrak{A}$-bimodule map $\xi:\sigma wc(\mathfrak{A}^{*})^{*}\longrightarrow\mathfrak{A}$. Notice that $\xi\circ\iota(a)=a$ for all $a\in\mathcal{A}$. By the assumption, there exists a $\varphi$-Connes mean $m\in\sigma wc(\mathfrak{A}^{*})^{*}$. Setting $u=\xi(m)\in\mathfrak{A}$, we observe that $\mathfrak{A}$ is $\varphi$-contractible by Theorem 3.1.

$(ii)\Longrightarrow(i)$ Take $u\in\mathfrak{A}$ satisfying $\varphi(u)=1$ and $ua=au=\varphi(a)u$, for all $a\in\mathfrak{A}$. Then $\iota(u)$ is a $\varphi$-Connes mean on $\sigma wc(\mathfrak{A}^{*})$. ∎

3. $\varphi$-contractibility

It was shown that right [left] $\varphi$-contractibility of $\mathfrak{A}$ is equivalent to the existence of a right [left] $\varphi$-diagonal for $\mathfrak{A}$, i. e., an element $m\in\mathfrak{A}\widehat{\otimes}\mathfrak{A}$ such that $\varphi(\pi(m))=1$ and $a\cdot m=\varphi(a)m$ $[m\cdot a=\varphi(a)m]$ for $a\in\mathfrak{A}$, where $\pi:\mathfrak{A}\widehat{\otimes}\mathfrak{A}\longrightarrow\mathfrak{A}$ is the bounded linear map determined by $\pi(a\otimes b)=ab$. If $m$ is both left and right $\varphi$-diagonal, it called $\varphi$-diagonal.

The following is likely to be well-known, but since we could not locate a reference, we include a proof.

Theorem 3.1**.**

Suppose that $\mathfrak{A}$ is a Banach algebra and $\varphi\in\Delta(\mathfrak{A})$. Then $\mathfrak{A}$ is $\varphi$-contractible if and only if there exists an element $u\in\mathfrak{A}$ satisfying

[TABLE]

Proof. Let $u\in\mathfrak{A}$ satisfies the conditions in $(*)$. Let $D:\mathfrak{A}\longrightarrow E$ be a derivation for a Banach $\mathfrak{A}$-bimodule $E\in\mathbb{M}_{\varphi}$. It is routinely checked that $D^{*}(f\ .\ a)=D^{*}(f)\ .\ a-\langle Da,f\rangle\varphi$, for each $f\in E^{*}$ and $a\in\mathfrak{A}$. Put $t:=Du\in E$. For $f\in E^{*}$ and $a\in\mathfrak{A}$ we have

[TABLE]

Therefore $a\ .\ t=\varphi(a)t-Da$, $a\in\mathfrak{A}$, and hence $D=ad_{-t}$. Whence $\mathfrak{A}$ is right $\varphi$-contractible. A similar argument shows that $\mathfrak{A}$ is also left $\varphi$-contractible.

Conversely, suppose that $m\in\mathfrak{A}\widehat{\otimes}\mathfrak{A}$ is a $\varphi$-diagonal for $\mathfrak{A}$. Put $u:=\pi(m)\in\mathfrak{A}$. Now, it is easily checked that $u$ has the desired properties in $(*)$.∎

Let $\mathfrak{A}$ be a Banach algebra with the unitization $\mathfrak{A}^{\sharp}$, and let $P$ be a left Banach $\mathfrak{A}$-module. We recall that $P$ is a projective left $\mathfrak{A}$-module if the multiplication map

[TABLE]

has a right inverse which is also a left $\mathfrak{A}$-module homomorphism. Similar definitions hold for projective right $\mathfrak{A}$-modules and projective $\mathfrak{A}$-bimodules.

For $\varphi\in\Delta(\mathfrak{A})$, the space $\mathbb{C}_{\varphi}=\{\alpha\varphi\ :\ \alpha\in\mathbb{C}\}$ is a Banach $\mathfrak{A}$-bimodule with module actions $a\cdot\varphi=\varphi\cdot a:=\varphi(a)\varphi$, $(a\in\mathfrak{A})$.

Theorem 3.2**.**

Suppose that $\mathfrak{A}$ is a Banach algebra and $\varphi\in\Delta(\mathfrak{A})$. Then $\mathfrak{A}$ is $\varphi$-contractible if and only if $\mathbb{C}_{\varphi}$ is a projective Banach $\mathfrak{A}$-bimodule.

Proof. Without loss of generality, we may assume that $\mathfrak{A}$ is unital. Let $\mathbb{C}_{\varphi}$ be projective as a left $\mathfrak{A}$-module. Then there exists a bounded linear map $\rho:\mathbb{C}_{\varphi}\longrightarrow\mathfrak{A}\widehat{\otimes}\mathbb{C}_{\varphi}$ satisfying $\pi\rho(\varphi)=\varphi$ and $a\cdot\rho(\varphi)=\varphi(a)\rho(\varphi)$ for each $a\in\mathfrak{A}$. We have $\rho(\varphi)=\sum_{n=1}^{\infty}a_{n}\otimes\varphi$, where $a_{n}\in\mathfrak{A}$ $(n=1,2,...)$ with $\sum_{n=1}^{\infty}||a_{n}||<\infty$. Putting $u:=\sum_{n=1}^{\infty}a_{n}\in\mathfrak{A}$, we observe that $\varphi(u)=1$ and $au=\varphi(a)u$, $a\in\mathfrak{A}$. Now, by Theorem 3.1 $\mathfrak{A}$ is right $\varphi$-contractible.

Conversely, let $\mathfrak{A}$ be right $\varphi$-contractible. Take $u\in\mathfrak{A}$ with $\varphi(u)=1$ and $au=\varphi(a)u$ for all $a\in\mathfrak{A}$. Then it is easy to verify that the map $\rho:\mathbb{C}_{\varphi}\longrightarrow\mathfrak{A}\widehat{\otimes}\mathbb{C}_{\varphi}$ defined by $\rho(\varphi):=u\otimes\varphi$ is a left $\mathfrak{A}$-module homomorphism which is a right inverse of $\pi$. Whence $\mathbb{C}_{\varphi}$ a is projective left $\mathfrak{A}$-module.

Similarly, one can see that $\mathbb{C}_{\varphi}$ is a projective right $\mathfrak{A}$-module if and only if $\mathfrak{A}$ is left $\varphi$-contractible. ∎