On left $\phi$-biprojectivity and left $\phi$-biflatness of certain Banach algebras

TL;DR

This paper investigates the properties of left -biflatness and left -biprojectivity in Banach algebras, establishing conditions under which these properties hold for various algebraic constructions and their relation to the structure of groups.

Contribution

It introduces new results linking left -biprojectivity of the second dual to left -biflatness of the algebra and characterizes these properties for matrix algebras and tensor products.

Findings

Left -biprojectivity of $A^{**}$ implies left -biflatness of $A$.

Characterization of -biprojectivity for $M(G)\u2297_{p}A(G)$ and $M(G)\u2297_{p}L^1(G)$ based on group properties.

Conditions under which tensor products of Banach algebras are left -biprojective, depending on the finiteness or compactness of the group.

Abstract

In this paper, we study left $\phi$-biflatness and left $\phi$-biprojectivity of some Banach algebras, where $\phi$ is a non-zero multiplicative linear function. We show that if the Banach algebra $A^{**}$ is left $\phi$-biprojective, then $A$ is left $\phi$-biflat. Using this tool we study left $\phi$-biflatness of some matrix algebras. We also study left $\phi$-biflatness and left $\phi$-biprojectivity of the projective tensor product of some Banach algebras. We prove that for a locally compact group $G$, $M(G)\otimes_{p} A(G)$ is left $\phi\otimes \psi$-biprojective if and only if $G$ is finite. We show that $M(G)\otimes_{p} L^1(G)$ is left $\phi\otimes \psi$-biprojective if and only if $G$ is compact.

Equations

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

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

$$\rho (ab)=a\cdot \rho (b)=\varphi (b)\rho (a),\varphi \circ {\pi }_A\circ \rho (a)=\varphi (a),(a,b\in A).$$

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

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

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

$$a\cdot {\rho }_{\alpha }(b)\stackrel{w^{\ast }}{\to }a\cdot \rho (b)=\rho (ab),{\rho }_{\alpha }(ab)\stackrel{w^{\ast }}{\to }\rho (ab),\varphi (b){\rho }_{\alpha }(a)\stackrel{w^{\ast }}{\to }\varphi (b)\rho (a)=\rho (ab).$$

$$\varphi \circ {\pi }_A\circ {\rho }_{\alpha }(a)=\tilde{\varphi }\circ {\pi }_A^{\ast \ast }\circ {\rho }_{\alpha }(a)={\pi }_A^{\ast \ast }\circ {\rho }_{\alpha }(a)(\varphi )\to {\pi }_A^{\ast \ast }\circ \rho (a)(\varphi )=\tilde{\varphi }\circ {\pi }_A^{\ast \ast }\circ \rho (a)=\varphi (a).$$

$$a\cdot {\rho }_{\alpha }(b)\stackrel{w^{\ast }}{\to }a\cdot \rho (b)=\rho (ab),{\rho }_{\alpha }(ab)\stackrel{w^{\ast }}{\to }\rho (ab),\varphi (b){\rho }_{\alpha }(a)\stackrel{w^{\ast }}{\to }\varphi (b)\rho (a).$$

$$a\cdot {\rho }_{\alpha }(b)-{\rho }_{\alpha }(ab)\stackrel{w^{\ast }}{\to }0,\varphi (b){\rho }_{\alpha }(a)-\varphi (b){\rho }_{\alpha }(a)\stackrel{w^{\ast }}{\to }0.$$

$$M=\{(a_1\cdot T(b_1)-T(a_1{b}_1),a_2\cdot T(b_2)-T(a_2{b}_2),...,a_n\cdot T(b_n)-T(a_n{b}_n)):T\in B(A,A{\otimes }_{p}A)\},$$

$$||a_i\cdot {\xi }_{(ϵ,F,G)}(b_i)-{\xi }_{(ϵ,F,G)}(a_i{b}_i)||<ϵ,||{\xi }_{(ϵ,F,G)}(a_i{b}_i)-\varphi (b_i){\xi }_{(ϵ,F,G)}(a_i)||<ϵ$$

$$a\cdot {\xi }_{(ϵ,F,G)}-{\xi }_{(ϵ,F,G)}(ab)\to 0,\varphi (b){\xi }_{(ϵ,F,G)}(a)-{\xi }_{(ϵ,F,G)}(ab)\to 0$$

$$\varphi \circ {\pi }_A\circ {\xi }_{(ϵ,F,G)}(a)-\varphi (a)\to 0.$$

$$a\cdot {\rho }_{\alpha }(b)-{\rho }_{\alpha }(ab)\to 0,\varphi (b){\rho }_{\alpha }(a)-\rho (ab)\to 0,\varphi \circ {\pi }_A\circ {\rho }_{\alpha }(a)-\varphi (a)\to 0.$$

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

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

$$\tilde{\varphi }\circ {\pi }_A^{\ast \ast }\circ \eta (a)=\tilde{\varphi }\circ {\pi }_{A^{\ast \ast }}\circ \psi \circ \rho (a)=\tilde{\varphi }\circ {\pi }_{A^{\ast \ast }}\circ \rho (a)=\varphi (a),(a\in A).$$

$$||(a_{i,j}{)}_{i,j\in I}||=\sum _{i,j\in I}||a_{i,j}||<\infty .$$

$$J=\{(a_{i,j}{)}_{i,j\in I}\in U{P}_I(A)|a_{i,j}=0\text{for}j\ne i_0\}.$$

$$j{j}_{\alpha }-{\psi }_{\varphi }(j)j_{\alpha }\to 0,{\psi }_{\varphi }(j_{\alpha })=1(j\in J).$$

$$\text{( a∗b)(n)=}\{\begin{array}{l}a(1)b(1)ifn=1\\ a(1)b(n)+b(1)a(n)+a(n)b(n)ifn>1,\end{array}$$

$$\rho (xy)=x\cdot \rho (y)=\widetilde{\varphi \otimes \psi }(y)\rho (x),\widetilde{\varphi \otimes \psi }\circ {\pi }_{A{\otimes }_{p}B}^{\ast \ast }\circ \rho (x)=\varphi \otimes \psi (x)(x,y\in A{\otimes }_{p}B).$$

$$a_1{a}_2\otimes x_0=a_1{a}_2\otimes x_0=a_1{a}_2\otimes x_0^2=(a_1\otimes x_0)(a_2\otimes x_0).$$

$$\begin{array}{rl}\rho (a_1{a}_2\otimes x_0)=\rho ((a_1\otimes x_0)(a_2\otimes x_0))& =(a_1\otimes x_0)\cdot \rho (a_2\otimes x_0)\\ & =a_1(e\otimes x_0)\cdot \rho (a_2\otimes x_0)\\ & =a_1\rho (e{a}_2\otimes x_0^2),\end{array}$$

$$\rho (a_1{a}_2\otimes x_0)=\rho ((a_1\otimes x_0)(a_2\otimes x_0))=\varphi \otimes \psi (a_2\otimes x_0)\rho (a_1\otimes x_0)=\varphi (a_2)\rho (a_1\otimes x_0)$$

$$\widetilde{\varphi \otimes \psi }\circ {\pi }_{A{\otimes }_{p}B}^{\ast \ast }\circ \rho (a_1\otimes x_0)=\varphi \otimes \psi (a_1\otimes x_0)=\varphi (a_1),$$

$${\pi }_A^{\ast \ast }\circ {\xi }^{\ast \ast }=(i{d}_A\otimes \psi {)}^{\ast \ast }\circ {\pi }_{A{\otimes }_{p}B}^{\ast \ast }.$$

$$a\cdot \theta (b)=a\cdot {\xi }^{\ast \ast }\circ \rho (b)={\xi }^{\ast \ast }\circ \rho (ab)=\varphi (b){\xi }^{\ast \ast }\circ \rho (a)=\varphi (b)\theta (a),(a,b\in A).$$

$$\begin{array}{rl}\tilde{\varphi }\circ {\pi }_A^{\ast \ast }\circ \theta (a)=\tilde{\varphi }\circ {\pi }_A^{\ast \ast }\circ {\xi }^{\ast \ast }\circ \rho (a\otimes x_0)& =\tilde{\varphi }\circ (i{d}_A\otimes \psi {)}^{\ast \ast }\circ {\pi }_{A{\otimes }_{p}B}^{\ast \ast }\circ \rho (a\otimes x_0)\\ & =\widetilde{\varphi \otimes \psi }\circ {\pi }_{A{\otimes }_{p}B}^{\ast \ast }\circ \rho (a\otimes x_0)\\ & =\varphi \otimes \psi (a\otimes x_0)=\varphi (a),\end{array}$$

Full text

1. Introduction and Preliminaries

Banach homology theory have two important notions, biflatness and biprojectivity which have key role in studying the structure of Banach algebras. A Banach algebra $A$ is called biflat (biprojective), if there exists a bounded $A$-bimodule morphism $\rho:A\rightarrow(A\otimes_{p}A)^{**}$ ($\rho:A\rightarrow A\otimes_{p}A$) such that $\pi_{A}^{**}\circ\rho$ is the canonical embedding of $A$ into $A^{**}$ ($\rho$ is a right inverse for $\pi_{A}$), respectively. It is well known that for a locally compact group $G$, the group algebra $L^{1}(G)$ is biflat (biprojective) if and only if $G$ is amenable (compact), respectively. We have to mention that a biflat Banach algebra $A$ with a buonded approximate identity is amenable and vise versa, see [13].

A Banach algebra $A$ is called left $\phi$-amenable, if there exists a bounded net $(a_{\alpha})$ in $A$ such that $aa_{\alpha}-\phi(a)a_{\alpha}\rightarrow 0$ and $\phi(a_{\alpha})\rightarrow 1$ for all $a\in A,$ where $\phi\in\Delta(A).$ For a locally compact group $G$, the Fourier algebra $A(G)$ is always left $\phi$-amenable. Also the group algebra $L^{1}(G)$ is left $\phi$-amenable if and only if $G$ is amenable, for further information see [8] and [1].

Following this course, Essmaili et. al. in [3] introduced and studied a biflat-like property related to a multiplicative linear functional, they called it condition $W$ (which we call it here right $\phi$-biflatness). The Banach algebra $A$ is called left $\phi$-biflat, if there exists a bounded linear map $\rho:A\rightarrow(A\otimes_{p}A)^{**}$ such that

[TABLE]

and

[TABLE]

for each $a,b\in A$. We followed their work and showed that the Segal algebra $S(G)$ is left $\phi$-biflat if and only if $G$ is amenable see [16]. also we defined a notion of left $\phi$-biprojectivity for Banach algebras. In fact $A$ Banach algebra is left $\phi$-biprojective if there exists a bounded linear map $\rho:A\rightarrow A\otimes_{p}A$ such that

[TABLE]

We showed that the Lebesgue-Fourier algebra $LA(G)$ is left $\phi$-contractible if and only if $G$ is compact. Also the Fourier algebra $A(G)$ is left $\phi$-contractible if and only if $G$ is discrete, see [15].

In this paper, We show that if the Banach algebra $A^{**}$ is left $\phi$-biprojective, then $A$ is left $\phi$-biflat. Using this tool we study left $\phi$-biflatness of some matrix algebras. We also study left $\phi$-biflatness and left $\phi$-biprojectivity of the projective tensor product of some Banach algebras. We prove that for a locally compact group $G$, $M(G)\otimes_{p}A(G)$ is left $\phi\otimes\psi$-biprojective if and only if $G$ is finite. We show that $M(G)\otimes_{p}L^{1}(G)$ is left $\phi\otimes\psi$-biprojective if and only if $G$ is compact.

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]

For Banach algebras $A$ and $B$ with $\phi\in\Delta(A)$ and $\psi\in\Delta(B)$, we denote $\phi\otimes\psi$ for a multiplicative linear functional on $A\otimes_{p}B$ given by $\phi\otimes\psi(a\otimes b)=\phi(a)\psi(b)$ for each $a\in A$ and $b\in B.$ 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. Some general properties

Let $A$ be a Banach algebra and $\phi\in\Delta(A)$. $A$ is called approximate left $\phi$-biprojective if there exists a net of bounded linear maps from $A$ into $A\otimes_{p}A$, say $(\rho_{\alpha})_{\alpha\in I}$, such that

1. (i)

$a\cdot\rho_{\alpha}(b)-\rho_{\alpha}(ab)\xrightarrow{||\cdot||}0$, 2. (ii)

$\rho_{\alpha}(ba)-\phi(a)\rho_{\alpha}(b)\xrightarrow{||\cdot||}0$, 3. (iii)

$\phi\circ\pi_{A}\circ\rho_{\alpha}(a)-\phi(a)\rightarrow 0$,

for every $a,b\in A$, see [14].

Proposition 2.1**.**

Let $A$ be a left $\phi$-biflat Banach algebra. Then $A$ is approximate left $\phi$-biprojective.

Proof.

Since $A$ is left $\phi$-biflat, 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 $\tilde{\phi}\circ\pi^{**}_{A}\circ\rho(a)=\phi(a)$. Since $\rho\in B(A,(A\otimes_{p}A)^{**})$, there exists a net $\rho_{\alpha}\in B(A,A\otimes_{p}A)$ such that $\rho_{\alpha}\xrightarrow{W^{*}OT}\rho$. Thus for each $a\in A$ we have $\rho_{\alpha}(a)\xrightarrow{w^{*}}\rho(a)$. Then

[TABLE]

On the other hand, the map $\pi^{**}_{A}$ is a $w^{*}$-continuous map, so $\pi^{**}_{A}\circ\rho_{\alpha}(a)\xrightarrow{w^{*}}\pi^{**}_{A}\circ\rho(a)$, for each $a\in A.$ Then

[TABLE]

Also foe each $a,b\in A$, we have

[TABLE]

So

[TABLE]

Put $F=\{a_{1},a_{2},...,a_{n}\}$ and $G=\{b_{1},b_{2},...,b_{n}\}$ for finite subsets of $A.$ Define

[TABLE]

it is easy to see that $M$ is a convex subset of $\prod_{i=1}^{n}(A\otimes_{p}A)\oplus_{1}\prod_{i=1}^{n}\mathbb{C}$ and $(0,0,...,0)\in\overline{M}^{w}=\overline{M}^{||\cdot||}.$ It follows that, there exists a net $\xi_{(\epsilon,F,G)}\in B(A,A\otimes_{p}A)$ such that

[TABLE]

and $|\phi\circ\pi_{A}\circ\xi_{(\epsilon,F,G)}(a_{i})-\phi(a_{i})|<\epsilon,$ for each $i\in\{1,2,...,n\}$. It follow that the net $(\xi_{(\epsilon,F,G)})$, for each $a,b\in A$, satisfies

[TABLE]

and

[TABLE]

Therefore $A$ is approximately left $\phi-$biprojective. ∎

Lemma 2.2**.**

If $A$ is an approximately left $\phi$-biprojective with bounded net $\rho_{\alpha},$ then $A$ is left $\phi$-biflat.

Proof.

Let $A$ be approximately left $\phi$-biprojective with bounded net $\rho_{\alpha}.$ So $\rho_{\alpha}\in B(A,(A\otimes_{p}A)^{**})\cong(A\otimes_{p}(A\otimes_{p}A)^{*})^{*}$ has a $w^{*}$-limit-point, say $\rho.$ Since

[TABLE]

It follows that

[TABLE]

for each $a\in A.$ ∎

Proposition 2.3**.**

Let $A$ be a Banach algebra with an approximate identity and let $\phi\in\Delta(A)$. If $A^{**}$ is approximately biflat, then $A$ is left $\phi$-biflat.

Proof.

Since $A$ has an approximate identity $\overline{A\ker\phi}=\ker\phi.$ Thus by [11, Theorem 3.3] $A$ is left $\phi-$amenable. So there exists an element $m\in A^{**}$ such that $am=\phi(a)m$ and $\tilde{\phi}(m)=1$ for every $a\in A.$ Define $\rho:A\rightarrow A^{**}\otimes_{p}A^{**}$ by $\rho(a)=\phi(a)m\otimes m$. Claerly $\rho$ is a bounded linear map such that

[TABLE]

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]. Set $\eta=\psi\circ\rho:A\rightarrow(A\otimes_{p}A)^{**}$. It is easy to see that $a\cdot\eta(b)=\eta(ab)=\phi(b)\eta(a)$

[TABLE]

So $A$ is left $\phi$-biflat. ∎

Let $A$ be a Banach algebra and $I$ be a totally ordered set. By $UP_{I}(A)$ we denote the set of $I\times I$ upper triangular matrices which its entries come from $A$ and

[TABLE]

With matrix operations and $||\cdot||$ as a norm, $UP_{I}(A)$ becomes a Banach algebra.

Proposition 2.4**.**

Let $I$ be a totally ordered set with the greatest element. Also let $A$ be a Banach algebra with left identity and $\phi\in\Delta(A).$ Then $UP(I,A)^{**}$ is left $\psi_{\phi}$-biflat if and only if $|I|=1$ and $A$ is left $\phi$-biflat.

Proof.

Suppose $UP_{I}(A)$ is left $\psi_{\phi}$-biflat. Let $i_{0}\in I$ be the greatest element of $I$ with respect to $\leq$. Since $A$ has a left unit, $UP_{I}(A)$ has a left approximate identity. By [16, Lemma 2.1] left $\psi_{\phi}$-amenability of $UP_{I}(A)^{**}$ implies that $UP_{I}(A)$ is left $\psi_{\phi}$-amenable.

Define

[TABLE]

Clearly $J$ is a closed ideal of $UP_{I}(A)$ with $\psi_{i_{0}}|_{J}\neq 0$. Applying [6, Lemma 3.1] gives that $J$ is left $\psi_{i_{0}}$-amenable. So by [6, Theorem 1.4] there exists a bounded net $(j_{\alpha})$ in $J$ which satisfies

[TABLE]

Suppose in contradiction that $I$ has at least two elements. Let $a_{0}$ be an element in $A$ such that $\phi(a_{0})=1.$ Set j=\left(\begin{array}[]{ccccc}\cdots&0&\cdots&0&a_{0}\\ \cdots&0&\cdots&0&a_{0}\\ \colon&\colon&\colon&\colon&\colon\\ \cdots&0&\cdots&0&a_{0}\\ \colon&\colon&\colon&\colon&0\end{array}\right). Clearly for each $\alpha$ the net $j_{\alpha}$ has a form \left(\begin{array}[]{ccccc}\cdots&0&\cdots&0&j_{i}^{\alpha}\\ \cdots&0&\cdots&0&\cdots\\ \colon&\colon&\colon&\colon&\colon\\ \cdots&0&\cdots&0&j_{k}^{\alpha}\\ \colon&\colon&\colon&\colon&j_{i_{0}}^{\alpha}\end{array}\right), where $(j_{i}^{\alpha}),(j_{k}^{\alpha})$ and $(j_{i_{0}}^{\alpha})$ are some nets in $A$. Put $j$ and $j_{\alpha}$ in (2.1) we have $j_{i_{0}}^{\alpha}a_{0}\rightarrow 0$. Since $\phi$ is continuous, we have $\phi(j_{i_{0}}^{\alpha})\rightarrow 0.$ On the other hand $\psi_{\phi}(j_{\alpha})=\phi(j_{i_{0}}^{\alpha})=1$ which is a contradiction. So $I$ must be singleton and the proof is complete. ∎

Corollary 2.5**.**

Let $I$ be a totally ordered set with the greatest element. Also let $A$ be a Banach algebra with left identity and $\phi\in\Delta(A).$ Then $UP_{I}(A)^{**}$ is approximately biflat if and only if $|I|=1$ and $A$ is approximately biflat.

Example 2.6*.*

We give a Banach algebra which is not left $\phi$-biflat but it is approximate left $\phi$-biprojective. So the converse of Proposition 2.1 is not always true. Let denote $\ell^{1}$ for the set of all sequences $a=((a_{n}))$ of complex numbers equipped with $||a||=\sum^{\infty}_{n=1}|a_{n}|<\infty$ as its norm. With the following product:

[TABLE]

$A=(\ell^{1},||\cdot||)$ becomes a Banach algebra. Clearly $\Delta(\ell^{1})=\{\phi_{1},\phi_{1}+\phi_{n}\}$, where $\phi_{n}(a)=a(n)$ for every $a\in\ell^{1}$. We claim that $\ell^{1}$ is not left $\phi_{1}$-biflat but $\ell^{1}$ is approximately left $\tilde{\phi}_{1}$-biprojective for some $\phi\in\Delta(\ell^{1})$. We assume conversely that $\ell^{1}$ is left $\phi_{1}$-biflat. One can see that $(1,0,0,...)$ is a unit for $\ell^{1}$. Therefore by [16, Lemma 2.1] left $\phi_{1}$-biflatness of $\ell^{1}$ implies that ${\ell^{1}}$ is left $\phi_{1}$-amenable. On the other hand by [9, Example 2.9] $\ell^{1}$ is not left $\phi_{1}$-amenable which is a contradiction.

Applying [9, Example 2.9], gives that $\ell^{1}$ is approximate left $\phi_{1}$-amenable. So [14, Proposition 2.4] follows that that $\ell^{1}$ is approximate left $\phi_{1}$-biprojective.

3. Left $\phi$-biprojectivity of the projective tensor product Banach algebras

Theorem 3.1**.**

Let $A$ and $B$ be Banach algebras which $\phi\in\Delta(A)$ and $\psi\in\Delta(B)$. Suppose that $A$ has a unit and $B$ has an identity $x_{0}$ such that $\psi(x_{0})=1.$ If $A\otimes_{p}B$ is left $\phi\otimes\psi$-biflat, then $A$ is left $\phi$-amenable.

Proof.

Let $\rho:A\otimes_{p}B\rightarrow((A\otimes_{p}B)\otimes_{p}(A\otimes_{p}B))^{**}$ be a bounded linear map such that

[TABLE]

For idempotent $x_{0}\in B$ and elements $a_{1},a_{2}\in A$ we have

[TABLE]

We denote $e$ for the unit of $A$. So we have

[TABLE]

also

[TABLE]

and

[TABLE]

for each $a_{1},a_{2}\in A.$ Put $\xi:(A\otimes_{p}B)\otimes_{p}(A\otimes_{p}B)\rightarrow A\otimes_{p}A$ for a bounded linear map which is given by $\xi((a\otimes b)\otimes(c\otimes d)=\psi(bd)a\otimes c,$ for each $a,c\in A$ and $b,d\in B.$ Clearly

[TABLE]

Define $\theta:A\rightarrow(A\otimes_{p}A)^{**}$ by $\theta(a)=\xi^{**}\circ\rho(a\otimes x_{0})$. Clearly $\theta$ is a bounded linear map. We have

[TABLE]

Also

[TABLE]

for each $a\in A$. It follows that $A$ is left $\phi$-biflat. Since $A$ has a unit by … $A$ is left $\phi$-amenable. ∎

Note that previous theorem is also valid in the left $\phi$-biprojective case. In fact we have

Corollary 3.2**.**

Let $A$ and $B$ be Banach algebras which $\phi\in\Delta(A)$ and $\psi\in\Delta(B)$. Suppose that $A$ has a unit and $B$ has an identity $x_{0}$ such that $\psi(x_{0})=1.$ If $A\otimes_{p}B$ is left $\phi\otimes\psi$-biprojective, then $A$ is left $\phi$-contractible.

Proposition 3.3**.**

Suppose that $A$ is a Banach algebra and $\phi\in\Delta(A).$ Let $A^{**}$ be left $\tilde{\phi}$-biprojective. Then $A$ is left $\phi$-biflat.

Proof.

Let $A^{**}$ be $\tilde{\phi}$-biprojective. Then there exists a bounded linear map $\rho:A^{**}\rightarrow A^{**}\otimes_{p}A^{**}$ such that $\rho(ab)=a\cdot\rho(b)=\tilde{\phi}(b)\rho(a)$ and $\tilde{\phi}\circ\pi_{A^{**}}\circ\rho(a)=\tilde{\phi}(a)$, for each $a,b\in A^{**}$. 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]. Set $\eta=\psi\circ\rho|_{A}:A\rightarrow(A\otimes_{p}A)^{**}.$ Clearly $\eta$ is a bounded linear map which satisfies

[TABLE]

and

[TABLE]

Also we have

[TABLE]

for each $a\in A.$ It follows that $A$ is left $\phi$-biflat. ∎

Theorem 3.4**.**

Let $A$ and $B$ be banach algebra with $\phi\in\Delta(A)$ and $\psi\in\Delta(B).$ If $A$ left $\phi$-biprojective and $B$ is $\psi$-biprojective, then $A\otimes_{p}B$ is left $\phi\otimes\psi$-biprojective.

Proof.

Since $A$ left $\phi$-biprojective and $B$ is $\psi$-biprojective, there exist bounded linear map $\rho_{A}:A\rightarrow A\otimes_{p}A$ and $\rho_{B}:B\rightarrow B\otimes_{p}B$ such that

[TABLE]

and

[TABLE]

Let $\theta$ be an isometrical isomorphism from $(A\otimes_{p}A)\otimes_{p}(B\otimes_{p}B)$ into $(A\otimes_{p}B)\otimes_{p}(A\otimes_{p}B)$ which is given by $\theta(a_{1}\otimes a_{2}\otimes b_{1}\otimes b_{2})=a_{1}\otimes b_{1}\otimes a_{2}\otimes b_{2}$ for each $a_{1},a_{2}\in A$ and $b_{1},b_{2}\in B.$ Define $\rho=\theta\circ(\rho_{A}\otimes\rho_{B})$. So

[TABLE]

for each $a_{1},a_{2}\in A$ and $b_{1},b_{2}\in B.$ It follows that $\rho(xy)=x\cdot\rho(y)$ for each $x,y\in A\otimes_{p}B$. Also we have

[TABLE]

for each $a_{1},a_{2}\in A$ and $b_{1},b_{2}\in B.$ So for each $x,y\in A\otimes_{p}B$ we have $\phi\otimes\psi(x)\rho(y)=\rho(yx)$. Note that

[TABLE]

it implies that $\pi_{A\otimes_{p}B}\circ\theta=\pi_{A}\otimes\pi_{B}$. Then

[TABLE]

for each $a\in A$ and $b\in B.$ Therefore $(\phi\otimes\psi)\circ\pi_{A\otimes_{p}B}\circ\rho(x)=\phi\otimes\psi(x)$ for every $x\in A\otimes_{p}B.$ It follows that $A\otimes_{p}B$ is left $\phi\otimes\psi$-biprojective. ∎

Let $\hat{G}$ be the dual group of $G$ which consists of all non-zero continuous homomorphism $\rho:G\rightarrow\mathbb{T}$. It is well-known that every character (multiplicative linear functional) $\phi\in\Delta(L^{1}(G))$ has the form $\phi_{\rho}(f)=\int_{G}\overline{\rho(x)}f(x)dx$, where $dx$ is the normalized Haar measure and $\rho\in\hat{G}$, for more details see [5, Theorem 23.7]. Note that, since $L^{1}(G)$ is a closed ideal of the mearsure algebra $M(G)$, each character on $L^{1}(G)$ can be extended to $M(G).$ Note that for a locally compact group $G$, we denote $A(G)$ for the Fourier algebra. The character space $\Delta(A(G))$ consists of all point evaluations $\phi_{x}$ for each $x\in G,$ where

[TABLE]

see[6, Example 2.6].

Theorem 3.5**.**

Let $G$ be a locally compact group. Then $M(G)\otimes_{p}A(G)$ is left $\phi\otimes\psi$-biprojective if and only if $G$ is finite, where $\phi\in\Delta(L^{1}(G))$ and $\psi\in\Delta(A(G))$.

Proof.

Let $M(G)\otimes_{p}A(G)$ be left $\phi\otimes\psi$-biprojective. Let $e$ be the unit of $M(G)$ and $a_{0}$ be the element of $A(G)$ such that $\psi(a_{0})=1.$ Put $x_{0}=e\otimes a_{0}$. Clearly $xx_{0}=x_{0}x$ and $\phi\otimes\psi(x_{0})=1,$ for every $x\in M(G)\otimes_{p}A(G).$ Now applying [15, Lemma 2.2] $M(G)\otimes_{p}A(G)$ is left $\phi\otimes\psi$-contractible. Now using [10, Theorem 3.14] $M(G)$ is left $\phi$-contractible, so by [10, Theorem 6.2] $G$ is compact. Also by [10, Theorem 3.14] $A(G)$ is left $\psi$-contractible. Thus by [10, Proposition 6.6] $G$ is discrete. Therefore $G$ is finite.

Converse is clear. ∎

Theorem 3.6**.**

Let $G$ be a locally compact group. Then $M(G)\otimes_{p}L^{1}(G)$ is left $\phi\otimes\psi$-biprojective if and only if $G$ is compact, where $\phi,\psi\in\Delta(L^{1}(G))$.

Proof.

Suppose that $M(G)\otimes_{p}L^{1}(G)$ is left $\phi\otimes\psi$-biprojective. Let $e$ be the unit of $M(G)$ and $e_{\alpha}$ be a bounded approximate identity of $L^{1}(G).$ Clearly $e\otimes e_{\alpha}$ is a bounded approximate identity. Thus by [15, Lemma 2.2] $M(G)\otimes_{p}L^{1}(G)$ is left $\phi\otimes\psi$-contractible. So [10, Theorem 3.14] $L^{1}(G)$ is left $\psi$-contractible. Then by [10, Theorem 6.2] $G$ is compact.

For converse, suppose that $G$ is compact. Then by [10, Theorem 3.14] M(G) is left $\phi$-contractible and by [10, Theorem 3.14] $L^{1}(G)$ is left $\psi$-contractible. Applying [10, Theorem 3.14] $M(G)\otimes_{p}L^{1}(G)$ is left $\phi\otimes\psi$-contractible. So by [15, Lemma 2.1] $M(G)\otimes_{p}L^{1}(G)$ is left $\phi\otimes\psi$-biprojective. ∎

A Banach algebra $A$ is called left character biprojective (left character biflat) if $A$ is left $\phi$-biprojective (if $A$ is left $\phi$-biflat) for each $\phi\in\Delta(A),$ respectively.

Theorem 3.7**.**

Let $G$ be a locally compact group. Then $M(G)\otimes_{p}L^{1}(G)$ is left character biprojective if and only if $G$ is finite.

Proof.

Let $M(G)\otimes_{p}L^{1}(G)$ be left character biprojective. So $M(G)\otimes_{p}L^{1}(G)$ is left $\phi\otimes\psi$-biprojective for each $\phi\in\Delta(M(G))$ and $\psi\in\Delta(L^{1}(G))$. So by similar arguments as in previous Proposition, $M(G)$ left $\phi$-contractible for each $\phi\in\Delta(M(G))$. ̵ُ Since $M(G)$ is unital, by [10, Corollary 6.2] $G$ is finite.

Converse is clear. ∎

Theorem 3.8**.**

Let $G$ be a locally compact group. Then $M(G)\otimes_{p}L^{1}(G)$ is left character biflat if and only if $G$ is a discrete amenable group.

Proof.

Since $M(G)$ is unital and $L^{1}(G)$ has a bounded approximate identity, $M(G)\otimes_{p}L^{1}(G)$ has a bounded approximate identity. Thus by [16, Lemma 2.1] $M(G)\otimes_{p}L^{1}(G)$ is left $\phi\otimes\psi$-amenable for each $\phi\in\Delta(M(G))$ and $\psi\in\Delta(L^{1}(G))$. So by [6, Theorem 3.3] $M(G)$ is left $\phi$-amenable for each $\phi\in\Delta(M(G))$. Since $M(G)$ is unital, $M(G)$ character amenable. Therefore by the main result of [8], $G$ is discrete and amenable.

For converse, let $G$ be discrete and amenable. Then $M(G)\otimes_{p}L^{1}(G)=\ell^{1}(G)\otimes_{p}\ell^{1}(G)\cong\ell^{1}(G\times G)$. Applying Johnson’s theorem (see [13, Theorem 2.1.18]) that $\ell^{1}(G\times G)$ is an amenable Banach algebra. So by [13, Exercise 4.3.15] $\ell^{1}(G\times G)$ biflat. Then $\ell^{1}(G\times G)$ is left character biflat. ∎

Proposition 3.9**.**

Let $G$ be an amenable group. Then $A(G)\otimes_{p}L^{1}(G)$ is left $\phi\otimes\psi$-biprojective if and only if $G$ is finite.

Proof.

Since $G$ is amenable, Leptin’s Theorem [13, Theorem 7.1.3] gives that $A(G)$ has a bounded approximate identity. It is well-known that $L^{1}(G)$ has a bounded approximate identity. Then by [15, Proposition 2.4], left $\phi\otimes\psi$-biprojectivity of $A(G)\otimes_{p}L^{1}(G)$ implies that $A(G)\otimes_{p}L^{1}(G)$ is left $\phi\otimes\psi$-contractible. So using [10, Theorem 3.14] gives that $A(G)$ is left $\phi$-contractible. Then by [10, Proposition 6.6] $G$ is discrete. Also by [10, Theorem 3.14] $L^{1}(G)$ is left $\psi$-contractible. Then [10, Theorem 6.1] implies that $G$ is compact. It follows that $G$ is finite.

Converse is clear. ∎

Proposition 3.10**.**

Let $G$ be a locally compact group. Then $A(G)\oplus_{1}L^{1}(G)$ is left character biprojective if and only if $G$ is finite.

Proof.

Suppose that $A(G)\oplus_{1}L^{1}(G)$ is left character biprojective. Let $\phi\in\Delta(A(G))$. Choose an element $a_{0}\in A(G)$ such that $\phi(a_{0})=1.$ Clearly the element $x_{0}=(a_{0},0)$ belongs to $A(G)\oplus_{1}L^{1}(G)$ which $xx_{0}=x_{0}x$ and $\phi(x_{0})=1.$ Using [15, Lemma 2.2], left character biprojectivity of $A(G)\oplus_{1}L^{1}(G)$ implies that $A(G)\oplus_{1}L^{1}(G)$ is left $\phi$-contractible. Since $A(G)$ is a closed ideal in $A(G)\oplus_{1}L^{1}(G)$ and $\phi|_{A(G)}\neq 0$, by [10, Proposition 3.8]$A(G)$ is left $\phi$-contractible. So by [10, Proposition 6.6] $G$ is discrete. Thus $A(G)\oplus_{1}L^{1}(G)=A(G)\oplus_{1}\ell^{1}(G)$. We know that $\ell^{1}(G)$ has an identity $e$. Replacing $e$ with $a_{0}$ and $\psi$ with $\phi$ (for some $\psi\in\Delta(L^{1}(G))$) and following the same argument as above, we can see that $\ell^{1}(G)$ is left $\psi$-contractible. Thus by [10, Theorem 6.1] $G$ is compact. Therefore $G$ must be finite.

Converse is clear. ∎

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 [12].

A locally compact group $G$ is called $SIN$, if it contains a foundamental family of compact invariant neibourhoods of the identity, see [2, p. 86].

Proposition 3.11**.**

Let $G$ be a SIN group. Then $S(G)\otimes_{p}S(G)$ is left $\phi\otimes\psi$-biprojective if and only if $G$ is compact, for some $\phi\in\Delta(S(G))$.

Proof.

Let $S(G)\otimes_{p}S(G)$ be left $\phi\otimes\phi$-biprojective. Since $G$ is a $SIN$ group, the main result of [7] gives that $S(G)$ has a central approximate identity. It follows that there exists an element $x_{0}\in S(G)$ such that $xx_{0}=x_{0}x$ and $\phi(x_{0})=1,$ for each $x\in S(G)$. Set $u_{0}=x_{0}\otimes x_{0}$. It is easy to see that $uu_{0}=u_{0}u$ and $\phi\otimes\phi(u_{0})=1,$ for every $u\in S(G)\otimes_{p}S(G).$ Using [15, Lemma 2.2] left $\phi\otimes\phi$-biprojectivity of $S(G)\otimes_{p}S(G)$ follows that $S(G)\otimes_{p}S(G)$ is left $\phi\otimes\psi$-contractible. By [10, Theorem 3.14] $S(G)$ is left $\phi$-contractible. Thus [1, Theorem 3.3] gives that $G$ is compact.

For converse, suppose that $G$ is compact. Then by [1, Theorem 3.3] $S(G)$ is left $\phi$-contractible. So by [10, Theorem 3.14] $S(G)\otimes_{p}S(G)$ be left $\phi\otimes\phi$-contractible. Applying [15, Lemma 2.1] finishes the proof. ∎