A note on essentially left $\phi$-contractible Banach algebras

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Abstract

In this note, we show that \cite[Corollary 3.2]{sad} is not always true. In fact, we characterize essential left $\phi$-contractibility of the the group algebras in the term of compactness of its related locally compact group. Also we show that for any compact commutative group $G$, $L^{2}(G)$ is always essentially left $\phi$-contractible. We discuss essential left $\phi$-contractibility of some Fourier algebras.

Equations

$$D(a)=a\cdot x_0-x_0\cdot a(a\in A),$$

$$a\cdot (b\otimes c)=ab\otimes c,(b\otimes c)\cdot a=b\otimes ca(a,b,c\in A).$$

$$T(a\cdot x)=a\cdot T(x),T(x\cdot a)=T(x)\cdot a,(a\in A,x\in X).$$

$${\varphi }_{\rho }(f)={\int }_G\overline{\rho (x)}f(x)dx,f\in L^1(G),$$

$$\begin{array}{rl}f\ast \rho (x)={\int }_{G}f(y)\rho (y^{-1}x)dy& ={\int }_{G}f(y)\rho (y^{-1})\rho (x)dy\\ & =\rho (x){\int }_{G}f(y)\overline{\rho (y)}dy\\ & =\rho (x){\varphi }_{\rho }(f).\end{array}$$

$$\begin{array}{rl}{\varphi }_{\rho }(\rho )={\int }_G\rho (x)\overline{\rho (x)}dy& ={\int }_G\rho (x)\rho (x^{-1})dy\\ & ={\int }_{G}1dx=1,\end{array}$$

$$\{u_{\beta }=\sum _{i\in \beta }{e}_i:\beta \in \Lambda \},$$

Full text

Keywords: Group algebra, Essential left $\phi$-contractible, Banach algebra.

1. **Introduction and preliminaries **

Johnson introduced and studied the notion of amenability for Banach algebras. A Banach algebra $A$ is called amenable, if every continuous (bounded linear) derivation $D$ from $A$ into $X^{*}$ is inner, that is, $D$ has a form

[TABLE]

for some $x_{0}\in X^{*}$, where $X$ is a Banach $A$-bimodule. For the history of amenability of Banach algebras, see [10].

Ghahramani and Loy in [2] defined a generalized notion of amenability for Banach algebras called essential amenability, that is, every continuous derivation $D$ from $A$ into $X^{*}$ is inner, where $X$ is a neo-unital Banach $A$-bimodule ($X=A\cdot X\cdot A$).

Kanuith et. al. in [5] defined and investigated the notion of left $\phi$-amenability for a Banach algebra $A$, where $\phi$-is a non-zero multiplicative linear functional. Indeed a Banach algebra $A$ is left $\phi$-amenable if every derivation $D:A\rightarrow X^{*}$ is inner, where $X$ is a Banach $A$-bimodule with the left module action $a\cdot x=\phi(a)x$ for all $a\in A,x\in X$. It is known that for a locally compact group $G$, the group algebra $L^{1}(G)$ is left $\phi$-amenable if and only if $G$ is amenable. Also the Fourier algebra $A(G)$ is always left $\phi$-amenable, see [12], [5].

Motivated by these considerations Nasr-isfahani et. al. in [6] introduced the concept of essential left $\phi$-amenability for Banach algebras. A Banach algebra $A$ is called essentially left $\phi$-amenable if every derivation $D:A\rightarrow X^{*}$ is inner, where $X$ is a neo-unital Banach $A$-bimodule with the left module action $a\cdot x=\phi(a)x$ for all $a\in A,x\in X$. Nasr-isfahani et. al. studied some Banach algebras related to a locally compact groups under the concept of essential left $\phi$-amenability.

Recently R. Sadeghi Nahrkhalaji defined the concept of essential left $\phi$-contractible for Banach algebras. A Banach algebra $A$ is called essentially left $\phi$-contractible if every continuous derivation $D:A\rightarrow X^{*}$ is inner, where $X$ is a neo-unital Banach $A$-bimodule with the right module action $x\cdot a=\phi(a)x$ for all $a\in A,x\in X$, see [11]. R. Sadeghi Nahrkhalaji studied the essentially left $\phi$-contractibility of some Banach algebras related to a locally compact group. Also some hereditary properties of this new notion are given in [11].

In this paper, we study essentially left $\phi$-contractibility of Banach algebras.we show that [11, Corollary 3.2] is not always true. In fact, we characterize essential left $\phi$-contractibility of the the group algebras in the term of compactness of its related locally compact group. Also we show that for any compact commutative group $G$, $L^{2}(G)$ is always essentially left $\phi$-contractible. We discuss essential left $\phi$-contractibility of some Fourier algebras.

We give some notations and definitions that we use in this paper frequently. Suppose that $A$ is a Banach algebra. Throughout this manuscript, the character space of $A$ is denoted by $\Delta(A)$, that is, all non-zero multiplicative linear functionals (characters) on $A$.

The projective tensor product $A\otimes_{p}A$ is a Banach $A$-bimodule via the following actions

[TABLE]

The product morphism $\pi_{A}:A\otimes_{p}A\rightarrow A$ is given by $\pi_{A}(a\otimes b)=ab,$ for every $a,b\in A.$ Let $X$ and $Y$ be Banach $A-$bimodules. The map $T:X\rightarrow Y$ is called $A-$bimodule morphism, if

[TABLE]

2. Essential left $\phi$-contractibility

Note that the Cohen-Hewit factorization is valid, whenever the Banach algebra $A$ has a ”bounded” left approximate identity, see [4, Theorem 1.1.4, p2]. Then in [11, Proposition 2.3] to show that $A\otimes_{p}A$ is neo-unital, $A$ must have a bounded approximate identity. So we state the correct version of [11, Proposition 2.3] here.

Theorem 2.1**.**

Let $A$ be a Banach algebra with a bounded approximate identity and $\phi\in\Delta(A).$ Then $A$ is left $\phi$-contractible if and only if $A$ is essentially left $\phi$-contractible.

Proof.

See the proof of [11, Proposition 2.3]. ∎

Let $G$ be a locally compact group and $L^{1}(G)$ be its associated group algebra. We denote $\widehat{G}$ for the dual group of $G$, that is, the set of all non-zero continuous homomorphisms $\rho$ from $G$ into $\mathbb{T}=\{z\in\mathbb{C}:|z|=1\}$. It is known that every non-zero multiplicative linear functional on $L^{1}(G)$ has the form $\phi_{\rho}$ for some $\rho\in\widehat{G},$ where

[TABLE]

where $dx$ is denoted for the Haar measure. For more information about the characters of group algebra see [3, Theorem 23.7].

We should remind that a Banach algebra is left $\phi$-contractible if and only if there exists an element $m\in A$ such that $am=\phi(a)m$ and $\phi(m)=1$ for all $a\in A$. For knowing more about left $\phi$-contractibility of a Banach algebra and its hereditary properties through the homological approach, see [7].

Theorem 2.2**.**

Let $G$ be a locally compact group. Then $L^{1}(G)$ is essentially $\phi$-contractible if and only if $G$ is compact.

Proof.

Let $G$ be a compact group. Then each continuous homomorphism $\rho:G\rightarrow\mathbb{T}$ belongs to $L^{\infty}(G)$. On the other hand $L^{\infty}(G)\subseteq L^{1}(G)$. So $\rho\in L^{1}(G).$ Consider

[TABLE]

It follows that $f\ast\rho=\phi_{\rho}(f)\rho.$ Also

[TABLE]

here we consider the normalized Haar measure on $G$. Thus by [7, Theorem 2.1] $L^{1}(G)$ is left $\phi_{\rho}$-contractible. So $L^{1}(G)$ is essentially left $\phi$-contractible

Conversely, suppose that $L^{1}(G)$ is essentially left $\phi_{\rho}$-contractible. Since $L^{1}(G)$ has a bounded approximate identity, by [11, Proposition 2.3], essential left $\phi_{\rho}$-contractibility of $L^{1}(G)$ implies the left $\phi_{\rho}$-contractibility of $L^{1}(G)$. Applying [1, Theorem 3.3] follows that $G$ is compact. ∎

In the following theorem we show that $ii\Rightarrow iv$ of [11, Corollary 3.2] is not valid just only for a finite group $G$. Suppose that $G$ is a locally compact group. It is well-known that $L^{2}(G)$ is a Banach algebra with convolution if and only if $G$ is compact.

Theorem 2.3**.**

Let $G$ be a compact commutative group. Then $L^{2}(G)$ is essentially left $\phi_{\rho}$-contractible, for each $\phi_{\rho}\in\Delta(L^{2}(G))$.

Proof.

Let $L^{2}(G)$ be essentially left $\phi_{\rho}$-contractible. It is known by Plancherel Theorem [9, Theorem 1.6.1] that $L^{2}(G)$ is isometrically isomorphic to $\ell^{2}(\widehat{G})$, where $\ell^{2}(\widehat{G})$ is equipped with the pointwise multiplication. Zhang in [13, Example] showed that $\ell^{2}(\widehat{G})$ is approximately biprojective, that is, there exists a net $\rho_{\alpha}:\ell^{2}(\widehat{G})\rightarrow\ell^{2}(\widehat{G})\otimes_{p}\ell^{2}(\widehat{G})$ of $\ell^{2}(\widehat{G})$-bimodule morphisms such that $\pi_{\ell^{2}(\widehat{G})}\circ\rho_{\alpha}(a)\rightarrow a,$ for each $a\in\ell^{2}(\widehat{G})$, where $\pi_{\ell^{2}(\widehat{G})}:\ell^{2}(\widehat{G})\otimes_{p}\ell^{2}(\widehat{G})\rightarrow\ell^{2}(\widehat{G})$ is the product morphism given by $\pi_{\ell^{2}(\widehat{G})}(a\otimes b)=ab$ for all $a,b\in\ell^{2}(\widehat{G}).$ On the other hand suppose that $\Lambda$ is the collection of all finite subsets of $\widehat{G}$. Clearly with the inclusion $\Lambda$ becomes an ordered set. One can see that

[TABLE]

here $e_{i}$ is an element of $\ell^{2}(G)$ equal to $1$ at $i$ and [math] elsewhere, forms a central approximate identity for $\ell^{2}(\widehat{G})$. Then for each $\phi_{\rho}\in\Delta(L^{2}(G))$ we can find an element $x_{0}\in L^{2}(G)$ such that $ax_{0}=x_{0}a$ and $\phi_{\rho}(x_{0})=1$ for each $a\in L^{2}(G).$ Applying [8, Lemma 3.5] follows that $L^{2}(G)$ is $\phi_{\rho}$-contractible, for each $\phi_{\rho}\in\Delta(L^{2}(G))$. Therefore $L^{2}(G)$ is essentially left $\phi_{\rho}$-contractible, for each $\phi_{\rho}\in\Delta(L^{2}(G))$. ∎

Theorem 2.4**.**

Let $G$ be an amenable group and $A(G)$ be the Fourier algebra on $G$. Then $A(G)$ is essentially left $\phi$-contractible if and only if $G$ is discrete.

Proof.

Let $A(G)$ be essentially left $\phi$-contractible. Since $G$ is amenable by Leptin’s Theorem ([10, Theorem 7.1.3]), amenability of $G$ implies that $A(G)$ has a bounded approximate identity. By [11, Proposition 2.3] essentially left $\phi$-contractibility of $A(G)$ gives that $A(G)$ is left $\phi$-contractible. Using [1, Theorem 3.5] shows that $G$ is discrete.

Converse is clear by [1, Theorem 3.5] ∎