A note on left $\phi$-biflat Banach algebras

TL;DR

This paper investigates the property of left $$-biflatness in Banach algebras, establishing its connection to amenability of groups and characterizing it for semigroup algebras, thus advancing understanding of algebraic structures in functional analysis.

Contribution

It introduces and characterizes the concept of left $$-biflatness in Banach algebras, linking it to amenability and biflatness in specific algebraic contexts.

Findings

Segal algebra $S(G)$ is left $$-biflat iff $G$ is amenable.

Left $$-biflatness of $ell^{1}(S)$ characterized by biflatness of $S$.

Provides new insights into the structure of Banach algebras related to group and semigroup properties.

Abstract

In this paper, we study the notion of $\phi$-biflatness for some Banach algebras, where $\phi$ is a non-zero multiplicative linear functional. We show that the Segal algebra $S(G)$ is left $\phi$-biflat if and only if $G$ is amenable. Also, we characterize left $\phi$-biflatness of semigroup algebra $\ell^{1}(S)$ in the term of biflatness, where $S$ is a Clifford semigroup.

Equations

$$\rho (ab)=\varphi (b)\rho (a)=a\cdot \rho (b)(\rho (ab)=\varphi (a)\rho (b)=\rho (a)\cdot b)$$

$$\tilde{\varphi }\circ {\pi }_A^{\ast \ast }\circ \rho (a)=\varphi (a),$$

$$(a\cdot f)(x)=f(x\cdot a),(f\cdot a)(x)=f(a\cdot x)(a\in A,x\in X,f\in X^{\ast }).$$

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

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

$$\xi (l)=(i{d}_A\otimes q{)}^{\ast \ast }\circ \rho (l)=\underset{n}{\lim }(i{d}_A\otimes q{)}^{\ast \ast }\circ \rho (a_n{l}_n)=\underset{n}{\lim }\varphi (l_n)(i{d}_A\otimes q{)}^{\ast \ast }\circ \rho (a_n)=0,$$

$$\theta :=(i{d}_A\otimes \overline{\varphi }{)}^{\ast \ast }\circ \overline{\xi }:\frac{A}{\ker \varphi }\to (A{\otimes }_p\frac{A}{\ker \varphi }{)}^{\ast \ast },$$

$$\tilde{\varphi }\circ (i{d}_A\otimes \overline{\varphi }{)}^{\ast \ast }=(\varphi \otimes \overline{\varphi }{)}^{\ast \ast },(\varphi \otimes \overline{\varphi }{)}^{\ast \ast }\circ \xi (a)=\tilde{\varphi }\circ {\pi }_A^{\ast \ast }\circ \rho (a),(a\in A),$$

$$\begin{array}{rl}\tilde{\varphi }\circ \theta (a+\ker \varphi )=\tilde{\varphi }\circ (i{d}_A\otimes \overline{\varphi }{)}^{\ast \ast }\circ \overline{\xi }(a+\ker \varphi )& =(\varphi \otimes \overline{\varphi }{)}^{\ast \ast }\circ \xi (a)\\ & =\tilde{\varphi }\circ {\pi }_A^{\ast \ast }\circ \rho (a)\\ & =\varphi (a),\end{array}$$

$$am=a\theta (a_0+\ker \varphi )=\theta (a{a}_0+\ker \varphi )=\theta (\varphi (a)a_0+\ker \varphi )=\varphi (a)\theta (a_0+\ker \varphi )=\varphi (a)m$$

$$\tilde{\varphi }(m)=\tilde{\varphi }\circ \theta (a_0+\ker \varphi )=\varphi (a_0)=1,$$

$${\psi }^{\ast \ast }\circ \rho {|}_A:A\to (A{\otimes }_{p}A{)}^{\ast \ast }$$

$${\psi }^{\ast \ast }\circ \rho {|}_A(ab)=\varphi (b){\psi }^{\ast \ast }\circ \rho {|}_A(a)=a\cdot {\psi }^{\ast \ast }\circ \rho {|}_A(b)$$

$$\widetilde{\stackrel{~}{\varphi }}\circ {\pi }_A^{\ast \ast \ast \ast }\circ \rho (a)=\tilde{\varphi }(a),(a,b\in A)$$

$$am=\varphi (a)m,\widetilde{\stackrel{~}{\varphi }}(m)=1,(a\in A).$$

$$\eta (ab)=a\cdot \eta (b)=\varphi (b)\eta (a),\tilde{\varphi }\circ {\pi }_A^{\ast \ast }\circ \eta (a)=\varphi (a)(a,b\in A).$$

$$\gamma (ab)=a\cdot \gamma (b)=\varphi (b)\gamma (a),\tilde{\varphi }\circ {\pi }_{A^{\ast \ast }}^{\ast \ast }\circ \gamma (a)=\varphi (a)(a,b\in A).$$

$$\Delta (S(G))=\{{\varphi }_{{|}_{S(G)}}|\varphi \in \Delta (L^1(G))\},$$

$$|||f|||=||f|{|}_{L^1(G)}+||f|{|}_{A(G)},$$

$$s\le t\iff s=s{s}^{\ast }t(s,t\in S).$$

Full text

1. Introduction and preliminaries

A Banach algebra $A$ is called amenable, if there exists an element $M\in(A\otimes_{p}A)^{**}$ such that $a\cdot M=M\cdot a$ and $\pi^{**}_{A}(M)a=a$ for each $a\in A.$ It is well-known that an amenable Banach algebra has a bounded approximate identity. For the history of amenability see [12].

In homological theory, the notion of biflatness is an amenability-like property. In fact a Banach algebra $A$ is biflat if there exists a Banach $A$-bimodule $\rho$ from $A$ into $(A\otimes_{p}A)^{**}$ such that $\pi^{**}_{A}\circ\rho(a)=a,$ for each $a\in A.$ It is well-known that a Banach algebra $A$ with a bounded approximate identity is biflat if and only if $A$ is amenable.

Kanuith et. al. in [9], defined a version of amenability with respect to a non-zero multiplicative functional $\phi$. Indeed a Banach algebra $A$ is called left $\phi$-amenable if there exists an element $m\in A^{**}$ such that $am=\phi(a)m$ and $\tilde{\phi}(m)=1$ for every $a\in A.$ We shall mention that the Segal algebra $S(G)$ is left $\phi$-amenable if and only if $G$ is amenable, for further information see [1],[8] and [7].

Motivated by these considerations, Essmaili et. al. in [2] defined a biflat-like property related to a multiplicative linear functional, they called it condition $W$ (which we call it here right $\phi$-biflatness).

Definition 1.1*.*

[2] Let $A$ be a Banach algebra and $\phi\in\Delta(A)$. The Banach algebra $A$ is called left $\phi$-biflat (right $\phi$-biflat or satisfies condition $W$), if there exists a bounded linear map $\rho:A\rightarrow(A\otimes_{p}A)^{**}$ such that

[TABLE]

and

[TABLE]

for each $a,b\in A,$ respectively.

They showed that a symmetric Segal algebra $S(G)$ (on a locally compact group $G$) is right $\phi$-biflat if and only if $G$ is amenable [2, Theorem 3.4]. As a consequence of this result in [2, Corollary 3.5] authors charactrized the right $\phi$-biflatness of Lebesgue-Fourier algebra $\mathcal{LA}(G)$, Weiner algebra $M_{1}$ and Feichtinger’s Segal algebra $S_{0}(G)$ over a unimodular locally compact group.

In this paper, we extend [2, Theorem 3.4] for any Segal algebra (in left $\phi$-biflat case). In fact we show that the Segal algebra $S(G)$ is left $\phi$-biflat if and only if $G$ is amenable. Using this tool we charactrize left $\phi$-biflatness of the Lebesgue-Fourier algebra $\mathcal{LA}(G)$. Also we characterize left $\phi$-biflatness of second dual of Segal algebra $S(G)^{**}$ in the term of amenability $G.$ We study left $\phi$-biflatness of some semigroup algebras.

We remark some standard notations and definitions that we shall need in this paper. Let $A$ be a Banach algebra. If $X$ is a Banach $A$-bimodule, then $X^{*}$ is also a Banach $A$-bimodule via the following actions

[TABLE]

Throughout, the character space of $A$ is denoted by $\Delta(A)$, that is, all non-zero multiplicative linear functionals on $A$. Let $\phi\in\Delta(A)$. Then $\phi$ has a unique extension $\tilde{\phi}\in\Delta(A^{**})$ which is defined by $\tilde{\phi}(F)=F(\phi)$ for every $F\in A^{**}$.

Let $A$ be a Banach algebra. 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. Left $\phi$-biflatness

In this section we give two criterion which show the relation of left $\phi$-biflatness and left $\phi$-amenability.

Lemma 2.1**.**

Suppose that $A$ is a left $\phi$-biflat Banach algebra with $\overline{A\ker\phi}^{||\cdot||}=\ker\phi.$ Then $A$ is left $\phi$-amenable.

Proof.

Let $A$ be left $\phi$-biflat. Then there exists a bounded linear map $\rho:A\rightarrow(A\otimes_{p}A)^{**}$ such that $\rho(ab)=a\cdot\rho(b)=\phi(b)\rho(a)$ and $\tilde{\phi}\circ\pi^{**}_{A}\circ\rho(a)=\phi(a)$ for all $a\in A.$ We finish the proof in three steps:

Step1:There exists a bounded left $A$-module morphism $\xi:A\rightarrow(A\otimes_{p}\frac{A}{\ker\phi})^{**}$ which $\xi(l)=0,$ for each $l\in\ker\phi.$ To see this, we denote $id_{A}:A\rightarrow A$ for the identity map. Also we denote $q:A\rightarrow\frac{A}{\ker\phi}$ for the qoutient map. Put $\xi:=(id_{A}\otimes q)^{**}\circ\rho:A\rightarrow(A\otimes_{p}\frac{A}{\ker\phi})^{**}$, where $id_{A}\otimes q(a\otimes b)=id_{A}(a)\otimes q(b)$ for every $a,b\in A.$ Clearly $id_{A}\otimes q:A\otimes_{p}A\rightarrow A\otimes_{p}\frac{A}{\ker\phi}$ is a bounded left $A$-module morphism, it folloows that $(id_{A}\otimes q)^{**}$ is also a bounded left $A$-module morphism. So $\xi:A\rightarrow(A\otimes_{p}\frac{A}{\ker\phi})^{**}$ is a bounded left $A$-module morphism. Let $l$ be an arbitrary element of $\ker\phi$. Since $\overline{A\ker\phi}^{||\cdot||}=\ker\phi$, there exist two sequences $(a_{n})$ in $A$ and $(l_{n})$ in $\ker\phi$ such that $a_{n}l_{n}\xrightarrow{||\cdot||}l.$

[TABLE]

the last equality holds because $(l_{n})$ is in $\ker\phi.$

Step2:There exists a bounded left $A$-module morphism $\eta:\frac{A}{\ker\phi}\rightarrow A^{**}$ such that $\tilde{\phi}\circ\eta(a+\ker\phi)=\phi(a)$ for each $a\in A.$ To see this, in step1 we showed that $\xi(\ker\phi)=\{0\}$. It induces a map $\overline{\xi}:\frac{A}{\ker\phi}\rightarrow(A\otimes_{p}\frac{A}{\ker\phi})^{**}$ which is defined by $\overline{\xi}(a+\ker\phi)=\xi(a)$ for each $a\in A.$ Define

[TABLE]

where $\overline{\phi}$ is a character on $\frac{A}{\ker\phi}$ given by $\overline{\phi}(a+\ker\phi)=\phi(a)$ for each $a\in A.$ Clearly $\theta$ is a bounded left $A$-module morphism. On the other hand we know that $\frac{A}{\ker\phi}\cong\mathbb{C}$ and $A\otimes_{p}\frac{A}{\ker\phi}\cong A$. Thus the composition of $\tilde{\phi}$ and $\theta$ can be defined. Since

[TABLE]

we have

[TABLE]

for each $a\in A.$

Step3: We prove that $A$ is left $\phi$-amenable. To see that, choose an element $a_{0}$ in $A$ such that $\phi(a_{0})=1.$ Put $m=\theta(a_{0}+\ker\phi)\in A^{**}$. Since $aa_{0}-\phi(a)a_{0}\in\ker\phi,$ we have $aa_{0}+\ker\phi=\phi(a)a_{0}+\ker\phi.$ Consider

[TABLE]

and

[TABLE]

for every $a\in A.$ It implies that $A$ is left $\phi$-amenable. ∎

Theorem 2.2**.**

Let $A$ be a Banach algebra with a left approximate identity and $\phi\in\Delta(A).$ Then $A^{**}$ is left $\tilde{\phi}$-biflat if and only if $A$ is left $\phi$-biflat.

Proof.

Suppose that $A^{**}$ is left $\tilde{\phi}$-biflat. Then there exists a bounded linear map $\rho:A^{**}\rightarrow(A^{**}\otimes_{p}A^{**})^{**}$ such that $\tilde{\tilde{\phi}}\circ\pi^{**}_{A^{**}}\circ\rho(a)=\tilde{\phi}(a)$ for all $a\in A^{**}.$ On the other hand, there exists a bounded linear map $\psi:A^{**}\otimes_{p}A^{**}\rightarrow(A\otimes_{p}A)^{**}$ such that for $a,b\in A$ and $m\in A^{**}\otimes_{p}A^{**}$, the following holds;

1. (i)

$\psi(a\otimes b)=a\otimes b$, 2. (ii)

$\psi(m)\cdot a=\psi(m\cdot a)$,  $a\cdot\psi(m)=\psi(a\cdot m),$ 3. (iii)

$\pi_{A}^{**}(\psi(m))=\pi_{A^{**}}(m),$

see [4, Lemma 1.7]. Clearly

[TABLE]

is a bounded linear map which

[TABLE]

and

[TABLE]

Following the similar arguments as in the previous lemma (step 1), we can find a bounded left $A$-module morphism $\xi:A\rightarrow(A\otimes_{p}\frac{A}{\ker\phi})^{****}$ such that $\xi(\ker\phi)=\{0\}.$ Now following the same course as in the previos lemma (step 2) we can find a bounded linear map $\theta:\frac{A}{\ker\phi}\rightarrow A^{****}$ such that $\tilde{\tilde{\phi}}\circ\theta(a+\ker\phi)=\phi(a)$ for each $a\in A.$ Choose $a_{0}$ in $A$ which $\phi(a_{0})=1.$ Set $m=\theta(a_{0}+\ker\phi)$. It is easy to see that

[TABLE]

Applying Goldestine’s theorem, we can find a bounded net $(m_{\alpha})$ in $A^{**}$ such that $am_{\alpha}-\phi(a)m_{\alpha}\xrightarrow{w^{*}}0$ and $\tilde{\phi}(m_{\alpha})\rightarrow 1,$ for each $a\in A.$ On the other hand $(m_{\alpha})$ is a bounded net, therefore $(m_{\alpha})$ has a $w^{*}$-limit point, say $M$. It is easy to see that $aM=\phi(a)M$ and $\tilde{\phi}(M)=1$ for each $a\in A.$ Define $\eta:A\rightarrow(A\otimes_{p}A)^{**}$ by $\eta(a)=\phi(a)M\otimes M,$ for each $a\in A.$ It is easy to see that $\eta$ is a bounded linear map such that

[TABLE]

It follows that $A$ is left $\phi$-biflat.

Conversely, suppose that $A$ is left $\phi$-biflat. Since $A$ has a left approximate identity, we have $\overline{A\ker\phi}^{||\cdot||}=\ker\phi.$ So by previous lemma $A$ is left $\phi$-amenable. Applying [9, Proposition 3.4] $A^{**}$ is left $\tilde{\phi}$-amenable. Thus there exists an element $m\in A^{****}$ such that $am=\phi(a)m$ and $\tilde{\tilde{\phi}}(m)=1$ for each $a\in A^{**}.$ Define $\gamma:A\rightarrow(A^{**}\otimes_{p}A^{**})^{**}$ by $\gamma(a)=\phi(a)m\otimes m,$ for each $a\in A.$ It is easy to see that $\gamma$ is a bounded linear map such that

[TABLE]

It follows that $A^{**}$ is left $\tilde{\phi}$-biflat. ∎

3. Applications to Banach algebras related to a locally compact group

A linear subspace $S(G)$ of $L^{1}(G)$ is said to be a Segal algebra on $G$ if it satisfies the following conditions

1. (i)

$S(G)$ is dense in $L^{1}(G)$, 2. (ii)

$S(G)$ with a norm $||\cdot||_{S(G)}$ is a Banach space and $||f||_{L^{1}(G)}\leq||f||_{S(G)}$ for every $f\in S(G)$, 3. (iii)

for $f\in S(G)$ and $y\in G$, we have $L_{y}(f)\in S(G)$ the map $y\mapsto L_{y}(f)$ from $G$ into $S(G)$ is continuous, where $L_{y}(f)(x)=f(y^{-1}x)$, 4. (iv)

$||L_{y}(f)||_{S(G)}=||f||_{S(G)}$ for every $f\in S(G)$ and $y\in G$.

For various examples of Segal algebras, we refer the reader to [11].

It is well-known that $S(G)$ always has a left approximate identity. For a Segal algebra $S(G)$ it has been shown that

[TABLE]

see [1, Lemma 2.2].

Theorem 3.1**.**

Let $G$ be a locally compact group. Then the following statements are equivallent:

1. (i)

$S(G)^{**}$* is left $\tilde{\phi}$-biflat,* 2. (ii)

$S(G)$* is left $\phi$-biflat,* 3. (iii)

$G$* is an amenable group.*

Proof.

$(i)\Rightarrow(ii)$ Let $S(G)^{**}$ be left $\tilde{\phi}$-biflat. Since $S(G)$ has a left approximate identity. Then by Theorem 2.2, $S(G)$ is left $\phi$-biflat.

$(ii)\Rightarrow(iii)$ Suppose that $S(G)$ is left $\phi$-biflat. Since $S(G)$ has a left approximate identity, $\overline{S(G)\ker\phi}^{||\cdot||}=S(G).$ Applying Lemma 2.1, follows that $S(G)$ is left $\phi$-amenable. Now by [1, Corollary 3.4] $G$ is amenable.

$(iii)\Rightarrow(i)$ Let $G$ be amenable. So by [1, Corollary 3.4] $S(G)$ is left $\phi$-amenable. Thus $S(G)$ is left $\phi$-biflat. Using Theorem 2.2, $S(G)^{**}$ is left $\tilde{\phi}$-biflat. ∎

Let $G$ be a locally compact group. Define $\mathcal{LA}(G)=L^{1}(G)\cap A(G)$, where $A(G)$ is the Fourier algebra over $G$. For $f\in\mathcal{LA}(G)$ put

[TABLE]

with this norm and the convoloution product $\mathcal{LA}(G)$ becomes a Banach algebra called Lebesgue-Fourier algebra. In fact $\mathcal{LA}(G)$ is a Segal algebra in $L^{1}(G)$, see [3]. Following corollary is an easy consequence of previous theorem:

Corollary 3.2**.**

Let $G$ be a locally compact group. Then the following statements are equivallent:

1. (i)

$\mathcal{LA}(G)^{**}$* is left $\tilde{\phi}$-biflat,* 2. (ii)

$\mathcal{LA}(G)$* is left $\phi$-biflat,* 3. (iii)

$G$* is an amenable group.*

Let $G$ be a locally compact group and let $\hat{G}$ be its dual group, which consists of all non-zero continuous homomorphism $\zeta:G\rightarrow\mathbb{T}$. It is well-known that $\Delta(L^{1}(G))=\{\phi_{\zeta}:\zeta\in\hat{G}\}$, where $\phi_{\zeta}(f)=\int_{G}\overline{\zeta(x)}f(x)dx$ and $dx$ is a left Haar measure on $G$, for more details, see [5, Theorem 23.7].

Corollary 3.3**.**

Let $G$ be a locally compact group. Then the following statements are equivallent:

1. (i)

${L^{1}(G)}^{**}$* is left $\tilde{\phi}$-biflat,* 2. (ii)

$L^{1}(G)$* is left $\phi$-biflat,* 3. (iii)

$G$* is an amenable group.*

Proof.

Clear. ∎

A discrete semigroup $S$ is called inverse semigroup, if for each $s\in S$ there exists an element $s^{*}\in S$ such that $ss^{*}s=s^{*}$ and $s^{*}ss^{*}=s$. There is a partial order on each inverse semigroup $S$, that is,

[TABLE]

Let $(S,\leq)$ be an inverse semigroup. For each $s\in S$, set $(x]=\{y\in S|\,y\leq x\}$. $S$ is called uniformly locally finite if $\sup\{|(x]|\,:x\in S\}<\infty$. Suppose that $S$ is an inverse semigroup and $e\in E(S)$, where $E(S)$ is the set of all idempotents of $S$. Then $G_{e}=\{s\in S|ss^{*}=s^{*}s=e\}$ is a maximal subgroup of $S$ with respect to $e$. An inverse semigroup $S$ is called Clifford semigroup if for each $s\in S$ there exists $s^{*}\in S$ such that $ss^{*}=s^{*}s.$ See [6] as a main reference of semigroup theory.

Proposition 3.4**.**

Let $S=\cup_{e\in E(S)}G_{e}$ be a Clifford semigroup such that $E(S)$ is uniformly locally finite. Then the followings are equivalent:

1. (i)

${\ell^{1}(S)}^{**}$* is left $\tilde{\phi}$-biflat, for each $\phi\in\Delta(\ell^{1}(S))$.* 2. (ii)

$\ell^{1}(S)$* is left $\phi$-biflat, for each $\phi\in\Delta(\ell^{1}(S))$.* 3. (iii)

Each $G_{e}$ is an amenable group. 4. (iv)

$\ell^{1}(S)$* is biflat.*

Proof.

$(i)\Rightarrow(ii)$ Suppose that $\ell^{1}(S)^{**}$ is left $\phi$-biflat for all $\phi\in\Delta(\ell^{1}(S))$. By [10, Theorem 2.16], $\ell^{1}(S)\cong\ell^{1}-\oplus_{e\in E(S)}\ell^{1}(G_{e})$. Since each $\ell^{1}(G_{e})$ has an identity, $\ell^{1}(S)\cong\ell^{1}-\oplus_{e\in E(S)}\ell^{1}(G_{e})$ has an approximate identity. Applying Theorem 2.2 gives that $\ell^{1}(S)$ is left $\phi$-biflat.

$(ii)\Rightarrow(iii)$ Suppose that $\ell^{1}(S)$ is left $\phi$-biflat for each $\phi\in\Delta(\ell^{1}(S))$. Since $\ell^{1}(S)\cong\ell^{1}-\oplus_{e\in E(S)}\ell^{1}(G_{e})$ has an approximate identity, Lemma 2.1 implies that $\ell^{1}(S)$ is left $\phi$-amenable for each $\phi\in\Delta(\ell^{1}(S))$. We know that each $\ell^{1}(G_{e})$ is a closed ideal in $\ell^{1}(S)$, so every non-zero multiplicative linear functional $\phi\in\Delta(\ell^{1}(G_{e}))$ can be extended to $\ell^{1}(S)$. Thus by [9, Lemma 3.1] left $\phi$-amenability of $\ell^{1}(S)$ implies that each $\ell^{1}(G_{e})$ is left $\phi$-amenable. Using [1, Corollary 3.4] each $G_{e}$ is amenable.

$(iv)\Rightarrow(i)$ It is clear by [10, Theorem 3.7]. ∎