Module character inner amenability of Banach algebras

Abstract

In the present paper, we introduce the notion of module $(\phi, \varphi )$-inner amenability and module character inner amenability for a Banach algebra A which is a Banach module over another Banach algebra $\mathfrak {A}$ with compatible actions. We characterize module $(\phi ,\varphi )$-inner amenability and prove some hereditary properties.

Introduction and preliminaries

Lau [10] introduced a wide class of Banach algebras, called F-algebras, and studied the notion of left amenability for these algebras. In [12], Nasr-Isfahani introduced the concept of inner amenability for Lau algebras. A Lau algebra A was said to be inner amenable if there exists a topological inner invariant mean on the $W^*$-algebra $A^*$, that is, a positive linear functional m of norm 1 on $A^*$, such that $m(f.a) = m(a.f)$ for all $f\in A^*$ and all $a\in P_1(A) = \{a \in A : \Vert a\Vert = 1\}$ (or equivalently, for all $a\in A$). Commutative Lau algebras, such as the Fourier algebra A(G) of a locally compact group G, are examples of inner amenable algebras. In addition, the group algebra $L^1(G)$ of any locally compact group G is inner amenable.

Recently, Jabbari et al. [8] have introduced the notion of $\varphi$-inner amenability for a Banach algebra A, where $\varphi \in \Delta (A)$, the character space of A. A Banach algebra A was said to be $\varphi$-inner amenable if there exists a $m\in A^{**}$ satisfying $m(\varphi )=1$ and $m(f.a)=m(a.f) (a\in A, f\in A^*)$. A is said to be character inner amenable if and only if A is $\varphi$-inner amenable for every $\varphi \in \Delta (A)$.

In [6], Ebrahimi Vishki and Khoddami have investigated the character inner amenability for certain products of Banach algebras consist of projective tensor product $A\widehat{\otimes }B$, Lau product $A\times _{\theta }B$, where $\theta \in \Delta (B)$ and the module extension $A\oplus X$. For instance, they showed that the projective tensor product $A\widehat{\otimes }B$ is character inner amenable if and only if both A and B are character inner amenable.

Let $\mathfrak {A}$ and A be Banach algebras, such that A be a Banach $\mathfrak {A}$-bimodule with compatible actions

$$\begin{aligned} \alpha .(ab)=(\alpha .a)b , (ab).\alpha =a(b.\alpha ), \alpha .(\beta .a)=(\alpha \beta ).a,(a.\beta ).\alpha =a.(\beta \alpha ), \end{aligned}$$

for all $a,b\in A$ and $\alpha \in \mathfrak {A}$.

Let X be a Banach A-bimodule and a Banach $\mathfrak {A}$-bimodule with compatible left actions defined by

$$\begin{aligned} \alpha .(a.x)=(\alpha .a ).x,~ a.(\alpha .x)=(a.\alpha ).x, ~ (\alpha .x).a=\alpha .(x.a)~ (a\in A , \alpha \in \mathfrak {A}, x\in X), \end{aligned}$$

and similar for the right or two-sided actions. Then, we say that X is a Banach A-$\mathfrak {A}$-module.

Let $A\widehat{\otimes }A$ be the projective tensor product of A and A which is a Banach A-bimodule and a Banach $\mathfrak {A}$-bimodule by the following actions:

$$\begin{aligned} \alpha .(a\otimes b)=(\alpha .a)\otimes b, c.(a\otimes b)=(ca)\otimes b (\alpha \in \mathfrak {A}, a,b,c\in A), \end{aligned}$$

similarly for the right actions. Let $I_{A\widehat{\otimes }A}$ be the closed ideal of $A\widehat{\otimes }A$ generated by elements of the form:

$$\begin{aligned} \{a.\alpha \otimes b-a\otimes \alpha .b |\alpha \in \mathfrak {A}, a,b\in A\}. \end{aligned}$$

(1)

Consider the map $\omega _A\in \mathcal {L}(A\widehat{\otimes }A,A)$ defined by $\omega _A(a\otimes b)=ab$ and extended by linearity and continuity. Let $J_A$ be the closed ideal of A generated by

$$\begin{aligned} \omega (I_{A\widehat{\otimes }A})=\{(a.\alpha )b-a(\alpha .b)\quad |a,b\in A, \alpha \in \mathfrak {A}\}. \end{aligned}$$

(2)

Then, the module projective tensor product $A\widehat{\otimes }_{\mathfrak {A}}A$, which is $(A\widehat{\otimes }A)/I_{A\widehat{\otimes }A}$ by [16], and the quotient Banach algebra $A/J_A$ are both Banach A-bimodules and Banach $\mathfrak {A}$-bimodules. In addition, $A/J_A$ is A-$\mathfrak {A}$-module with compatible actions when A acts on $A/J_A$ canonically.

Let A and $\mathfrak {A}$ be Banach algebras, such that A is a Banach $\mathfrak {A}$-bimodule with compatible actions. Let $\varphi \in \Delta (\mathfrak {A})\cup \{0\}$ and consider the set $\Omega _{A,\varphi }$ of linear continuous maps $\phi :A\rightarrow \mathfrak {A}$, such that

$$\begin{aligned} \phi (ab)=\phi (a)\phi (b), \phi (\alpha .a)=\phi (a.\alpha )=\varphi (\alpha )\phi (a)(a,b\in A, \alpha \in \mathfrak {A}). \end{aligned}$$

(3)

The concept of module $(\phi ,\varphi )$-amenability and module character amenability for Banach algebra A, where $\varphi \in \Delta (\mathfrak {A})$ and $\phi \in \Omega _A$ were introduced by Bodaghi and Amini in [4].

Our aim in this paper is to introduce and study module $(\phi ,\varphi )$-inner amenability and module character inner amenability of Banach algebras. We characterize $(\phi ,\varphi )$-inner amenability and prove some hereditary properties. Moreover, we investigate that module $(\phi ,\varphi )$-inner amenability for certain class of Banach algebras consists of projective tensor product $A\widehat{\otimes }B$, $A\oplus _{\infty }B$, and $A\oplus _p B$, the $l^p$-direct sum of A and B, where $1\le p<\infty$.

Characterization and hereditary properties

We commence this section with the following definition.

Definition 2.1

Let A be a Banach $\mathfrak {A}$-bimodule and let $\varphi \in \Delta (\mathfrak {A})$ and $\phi \in \Omega _A$. Then, A is called module $(\phi ,\varphi )$-inner amenable if there exists $m\in A^{**}$, such that $m(\varphi \circ \phi )=1$, $m(f.a)=m(a.f)$ and $m(\alpha .f)=m(f.\alpha )$ for all $a\in A, f\in A^*$ and $\alpha \in \mathfrak {A}$. A Banach $\mathfrak {A}$-bimodule A is called module character inner amenable if it is module $(\phi ,\varphi )$-inner amenable for each $\varphi \in \Delta (\mathfrak {A})$ and $\phi \in \Omega _A$.

We note that if $\mathfrak {A}=\mathbb {C}$ and $\varphi$ is the identity map, then the module $(\phi ,\varphi )$-inner amenability and module character inner amenability coincide with $\phi$-inner amenability and character inner amenability (see [8] and [6]).

The next theorem characterizes module $(\phi ,\varphi )$-inner amenability of Banach algebras that is analogue of Proposition 2.1 of [5] on module $(\phi ,\varphi )$-amenable Banach algebras.

Theorem 2.2

Let A be a Banach $\mathfrak {A}$-bimodule and let $\varphi \in \Delta ({\mathfrak {A}})$ and $\phi \in \Omega _A$. Then, the following statements are equivalent:

• (i) A is module $(\phi ,\varphi )$-inner amenable;

• (ii) There exists a bounded net $(a_i)_i$ in A such that $\Vert aa_i-a_ia\Vert \longrightarrow 0, \Vert \alpha .a_i-a_i.\alpha \Vert \longrightarrow 0~(a\in A, \alpha \in \mathfrak {A})$ and $\varphi \circ \phi (a_i)=1$ for all i;

• (iii) There exists a bounded net $(a_i)_i$ in A such that $\Vert aa_i-a_ia\Vert \longrightarrow 0, \Vert \alpha .a_i-a_i.\alpha \Vert \longrightarrow 0~(a\in A, \alpha \in \mathfrak {A})$ and $\varphi \circ \phi (a_i)\longrightarrow 1$.

Proof

(iii) $\Rightarrow$(i) Assume that a net $(a_i)_i$ exists. Let m be a $w^*$-cluster point of the net $(a_i)_i$ in $A^{**}$. Then, $\langle m,\varphi \circ \phi \rangle =\lim _i\langle \varphi \circ \phi ,a_i\rangle =1.$ For every $a\in A$ and $f\in A^*$, we have

$$\begin{aligned} \langle m,f.a\rangle&=\lim _i\langle f.a,a_i\rangle =\lim _i\langle f,aa_i\rangle \\&=\lim _i\langle f,aa_i-a_ia\rangle +\lim _i\langle f,a_ia\rangle \\&=\lim _i\langle a.f,a_i\rangle =\langle m,a.f\rangle , \end{aligned}$$

and similarly, we have $\langle m,f.\alpha \rangle =\langle m,\alpha .f\rangle (\alpha \in {\mathfrak {A}})$. Therefore, A is module $(\phi ,\varphi )$-inner amenable.

(i)$\Rightarrow$ (ii) Suppose that A is module $(\phi ,\varphi )$-inner amenable. Then, there exists $m\in A^{**}$ such that $m(\varphi \circ \phi )=1$, $m(f.a)=m(a.f)$, and $m(\alpha .f)=m(f.\alpha )$ for all $a\in A, f\in A^*$ and $\alpha \in \mathfrak {A}$. Choose a net $(u_{\beta })_{\beta }$ in A with $u_{\beta }\longrightarrow m$ in the $w^*$-topology of $A^{**}$ and $\Vert u_{\beta }\Vert \le \Vert m\Vert$ for all $\beta$. Since $\langle \varphi \circ \phi ,u_{\beta }\rangle \longrightarrow \langle \varphi \circ \phi ,m\rangle =1$, passing to a subnet and replacing $u_{\beta }$ by $\big (1/\varphi \circ \phi (u_{\beta })\big )u_{\beta },$ we may assume that $\varphi \circ \phi (u_{\beta })=1$ and $\Vert u_{\beta }\Vert \le \Vert m\Vert +1$ for all $\beta$. Consider the product space $A^A$ endowed with the product of norm topological. Define a linear map $T:A\rightarrow A^A$ by $T(b)=\big (ab-ba+\alpha .b-b.\alpha \big )_{a\in A},$ for all $b\in A$ and $\alpha \in \mathfrak {A}$. Let

$$\begin{aligned} B=\{b\in A:\Vert b\Vert \le \Vert m\Vert +1~ \mathrm \; {and}\; ~ \varphi \circ \phi (b)=1\}\subseteq A. \end{aligned}$$

Clearly, B is convex and so T(B) is a convex subset of $A^A$. For every $f\in A^*$, we have

$$\begin{aligned} \langle f,au_{\beta }-u_{\beta }a+\alpha .u_{\beta }-u_{\beta }.\alpha \rangle&=\langle f,au_{\beta }\rangle -\langle f,u_{\beta }a\rangle +\langle f,\alpha .u_{\beta }\rangle +\langle f,u_{\beta }.\alpha \rangle \\&= \langle f.a,u_{\beta }\rangle -\langle a.f,u_{\beta }\rangle +\langle f.\alpha ,u_{\beta }\rangle +\langle \alpha .f,u_{\beta }\rangle \\&\rightarrow \langle m,f.a\rangle -\langle m,a.f\rangle +\langle m,f.\alpha \rangle -\langle m,\alpha .f \rangle \\&=0. \end{aligned}$$

This product of weak topologies coincide with topology on $A^A$ (see Theorem 4.3 of [17]). By Mazur’s theorem, $0\in \overline{T(B)}^w=\overline{T(B)}^{\Vert .\Vert }.$ Therefore, there exists a bounded net $(a_i)_i$ in A, such that $\varphi \circ \phi (a_i)=1$ and

$$\begin{aligned} \Vert aa_i-a_ia\Vert \longrightarrow 0, \Vert \alpha .a_i-a_i.\alpha \Vert \longrightarrow 0 (a\in A, \alpha \in \mathfrak {A}). \end{aligned}$$

(ii)$\Rightarrow$ (iii) It is clear. $\square$

Definition 2.3

We say that the Banach algebra $\mathfrak {A}$ acts trivially on A from the left (right) if there is a multiplicative linear functional f on $\mathfrak {A}$, such that $\alpha .a=f(\alpha )a$ (resp. $a.\alpha =f(\alpha )a$) for all $\alpha \in \mathfrak {A}$ and $a\in A$.

For the proof of the following result, we refer to Lemma 3.13 of [1].

Lemma 2.4

Let $\mathfrak {A}$ acts on A trivially from the left or right and $A/J_A$ has a right bounded approximate identity, then for each $\alpha \in \mathfrak {A}$ and $a \in A$ we have $f(\alpha )a-a.\alpha \in J_A$.

Let $\varphi \in \Delta (\mathfrak {A})$ and $\phi \in \Omega _A$. Clearly, $\phi \big ((a.\alpha )b-a(\alpha .b)\big )=0(\alpha \in \mathfrak {A}, a,b\in A)$, and hence, $\phi =0$ on $J_A$ and $\tilde{\phi }:A/J_A\longrightarrow \mathfrak {A}$ given by $\tilde{\phi }(a+J_A)=\phi (a)$ is well defined. Then, $\tilde{\phi }\in \Omega _{A/J_A}$.

Proposition 2.5

Let A be a Banach $\mathfrak {A}$-bimodule, and let $\mathfrak {A}$ acts on A trivially from the left and $A/J_{A}$ has a bounded approximate identity. Then $A/J_{A}$ is $\varphi \circ \tilde{\phi }$-inner amenable for every $\varphi \in \Delta (\mathfrak {A})$ and $\phi \in \Omega _A$.

Proof

Let $(e_{\alpha }+J_A)_{\alpha }$ be a bounded approximate identity of $A/J_{A}$ and let $\varphi \in \Delta (\mathfrak {A})$ and $\phi \in \Omega _A$. Then, $\varphi \circ \tilde{\phi }(e_{\alpha }+J_A)\longrightarrow 1$. Clearly

$$\begin{aligned} \Vert (a+J_A)(e_{\alpha }+J_A)-(e_{\alpha }+J_A)(a+J_A)\Vert \longrightarrow 0 \end{aligned}$$

for all $a+J_A\in A/J_{A}$. By Proposition 2.2 of [6], $A/J_{A}$ is $\varphi \circ \tilde{\phi }$-inner amenable. $\square$

Note that in the above proposition, both left and right actions of $\mathfrak {A}$ on $A/J_{A}$ are trivial, by Lemma 2.4. Therefore, for every $\alpha \in \mathfrak {A}$, we have

$$\begin{aligned} \Vert \alpha .(e_{\alpha }+J_A)-(e_{\alpha }+J_A).\alpha \Vert&=\Vert \alpha .e_{\alpha }+J_A-e_{\alpha }.\alpha +J_A\Vert \\&=\Vert f(\alpha )(e_{\alpha }+J_A)-f(\alpha )(e_{\alpha }+J_A)\Vert =0. \end{aligned}$$

Then, the net $(e_{\alpha }+J_A)_{\alpha }$ satisfies condition (iii) of Theorem 2.2, and hence, $A/J_{A}$ is module $(\tilde{\phi },\varphi )$-inner amenable.

Remark 2.6

A inverse semigroup is a discrete semigroup S, such that for each $s\in S$, there is a unique element $s^*\in S$ with $ss^*s=s$ and $s^*ss^*=s^*$. An element $e\in S$ is called an idempotent if $e^2=e^*=e$. The set of idempotent elements of S is denoted by $E_S$. Define the relation $\le$ on $E_S$ by $e\le d \Leftrightarrow ed=e~ (e,d\in E_S).$ Then, $E_S$ is a commutative subsemigroup of S, and $l^1(E_S)$ may be regarded as a subalgebra of $l^1(S)$.

Let s be an inverse semigroup with the set of idempotents $E_S$. We let $l^1(E_S)$ acts on $l^1(S)$ by multiplication from the right and trivially from the left, that is

$$\begin{aligned} \delta _e.\delta _s=\delta _s~ \delta _s.\delta _e=\delta _{se}=\delta _s*\delta _e~ (e\in E_S, s\in S). \end{aligned}$$

By these actions, $l_1(S)$ becomes a Banach $l_1(E_S)$-module. In this case

$$\begin{aligned} J_{l^1(S)}= \{\delta _{set}-\delta _{st} | e\in E_S , s,t\in S\}. \end{aligned}$$

We consider an equivalence relation on S as follows $s\approx t \Leftrightarrow \delta _s-\delta _t\in J_{l^1(S)}~(s,t\in S).$ For inverse semigroup S, the quotient semigroup $S/\approx$ is discrete group and so $l^1(S/\approx )$ has an identity (see [2, 13]). Indeed, $S/\approx$ is homomorphic to the maximal group homomorphic image $G_S$ of S (see [11, 14]). It is also shown in Theorem 3.3 of [15] that $l^1(S)/J_{l^1(S)}\cong l^1(S/\approx )=l^1(G_S)$ is a commutative $l^1(E_S)$-bimodule with the following actions:

$$\begin{aligned} \delta _e.\delta _{[s]}=\delta _{[s]} ,\delta _{[s]}.\delta _e=\delta _{[se]}(s\in S , e\in E_S), \end{aligned}$$

where [s] denotes the equivalence class of s in $G_S$.

It is shown in [4] that the maps $\varphi$ and $\phi$ satisfying (3) exist for $l^1(S).$

Example 2.7

Let S be an inverse semigroup with the set of idempotents $E_S$. Consider $l^1(S)$ as a Banach module over $l^1(E_S)$ with the trivial left action and natural right action. Then, by Proposition 2.5, $l^1(G_S)$ is $\varphi \circ \tilde{\phi }$-inner amenable (module $(\tilde{\phi },\varphi )$-inner amenable) for all $\varphi \in \Delta (l^1(E_S))$ and $\phi \in \Omega _{l^1(S)}$.

Example 2.8

Let A be a commutative Banach algebra and commutative $\mathfrak {A}$-bimodule (i.e., $\alpha .a=a.\alpha ~(a\in A, \alpha \in \mathfrak {A}))$. Let $\varphi \in \Delta (\mathfrak {A}), \phi \in \Omega _{A}$ and let $a\in A$ be such that $\varphi \circ \phi (a)=1$. put $m=\widehat{a}.$ Then, $m(\varphi \circ \phi )=\widehat{a}(\varphi \circ \phi )=\varphi \circ \phi (a)=1$ and clearly, $m(f.a)=m(a.f)$ and $m(\alpha .f)=m(f.\alpha )$ for all $a\in A, \alpha \in \mathfrak {A}.$ Therefore, A is module $(\phi ,\varphi )$-inner amenable. In particular, if S is a commutative inverse semigroup, then $l^1(S)$ is commutative and commutative $l^1(E_S)$-bimodule. Therefore, $l^1(S)$ is module $(\phi ,\varphi )$-inner amenable for all $\varphi \in \Delta (l^1(E_S))$ and $\phi \in \Omega _{l^1(S)}$.

Example 2.9

Let $S=(\mathbb {N},\wedge )$ be the inverse semigroup of positive integers with the minimum operation. Let $A=l^1(S), \mathfrak {A}=l^1(E_S)$ and $\mathfrak {A}$ acts on A by the following actions:

$$\begin{aligned} \delta _e.\delta _s= \delta _s.\delta _e=\delta _{se}\quad (e\in E_S, s\in S). \end{aligned}$$

A is module amenable (see page 42 of [3]). By Theorem 2.1 of [4], A is module character amenable. Let $\varphi \in \Delta (\mathfrak {A})$ and $\phi \in \Omega _{A}$. Then, there exist $m\in A^{**}$, such that $m(f.a)=\varphi \circ \phi (a)m(f), m(f.\alpha )=\varphi (\alpha )m(f)$ and $m(\varphi \circ \phi )=1$ for every $f\in A^*, a\in A$ and $\alpha \in \mathfrak {A}$. Let $(a_{\alpha })_{\alpha }$ be a net in A converging to m in the $w^*$-topology of $A^{**}$. Since A is commutative and commutative $\mathfrak {A}$-bimodule, for every $a\in A, f\in A^*$

$$\begin{aligned} m(f.a)&=\lim _{\alpha }\langle f.a,a_{\alpha }\rangle =\lim _{\alpha }\langle f,aa_{\alpha }\rangle \\&=\lim _{\alpha }\langle f,a_{\alpha }a\rangle =\lim _{\alpha }\langle a.f,a_{\alpha }\rangle \\&=m(a.f). \end{aligned}$$

Similarly, for every $\alpha \in \mathfrak {A}$ and $f\in A^*$, we have

$$\begin{aligned} m(f.\alpha )=m(\alpha .f). \end{aligned}$$

Thus, A is module $(\phi ,\varphi )$-inner amenable. Therefore, A is module character inner amenable.

The proof of the following proposition is adapted from that of Proposition 2.3 of [4].

Proposition 2.10

Let A and B be Banach $\mathfrak {A}$-bimodules and let h be an $\mathfrak {A}$-module homomorphism with dense range. If $\phi \in \Omega _B, \varphi \in \Delta (\mathfrak {A})$ and A is module $(\phi \circ h,\varphi )$-inner amenable, then B is module $(\phi ,\varphi )$-inner amenable.

Proof

Let $m\in A^{**}$ be such that $m\big (\varphi \circ (\phi \circ h)\big )=1, m(f.a)=m(a.f)$ and $m(f.\alpha )=m(\alpha .f)$ for all $a\in A, f\in A^*$ and $\alpha \in \mathfrak {A}$. Define $m_B\in B^{**}$ by $m_B(g)=m(g\circ h) (g\in B^*)$. We show that $m_B\big (g.b\big )=m_B\big (b.g\big )~ (b\in B).$ For see this let $b\in B$ be such that $h(a)=b$. One can easily check that $(g.h(a))\circ h=(g\circ h).a$ and $(h(a).g)\circ h=a.(g\circ h).$ Hence for every $g\in B^*$

$$\begin{aligned} m_B(g.b)&=m_B\big (g.h(a)\big )=m\big ((g.h(a))\circ h\big )\\&=m\big ((g\circ h).a\big )=m\big (a.(g\circ h)\big )\\&=m\big ((h(a).g)\circ h\big )=m_B\big (h(a).g\big )\\&=m_B(b.g). \end{aligned}$$

By density of the range of h and the continuity of h, we conclude that $m_B\big (g.b\big )=m_B\big (b.g\big ) (b\in B).$ In addition, for every $\alpha \in \mathfrak {A}$, we have

$$\begin{aligned} m_B(g.\alpha )&=m\big ((g.\alpha )\circ h\big )=m\big ((g\circ h).\alpha \big )\\&=m\big (\alpha .(g\circ h)\big )=m\big ((\alpha .g)\circ h)\big )\\&=m_B(\alpha .g). \end{aligned}$$

Furthermore, $m_B(\varphi \circ \phi )=m\big ((\varphi \circ \phi )\circ h\big )=m\big (\varphi \circ (\phi \circ h)\big )=1.$ Therefore, B is module $(\phi ,\varphi )$-inner amenable.$\square$

Corollary 2.11

Let A and B be Banach $\mathfrak {A}$-bimodules and let h be an $\mathfrak {A}$-module homomorphism with dense range. Then the module character inner amenability of A implies the module character inner amenability of B. In particular, if A is module character inner amenable, then so is $A/J_A$.

The proof idea of the following result is taken from the proof of Lemma 2.6 of [4].

Proposition 2.12

Let A be a Banach $\mathfrak {A}$-bimodule and I be a closed ideal and $\mathfrak {A}$-submodule of A, and let $\varphi \in \Delta (\mathfrak {A})$ and $\phi \in \Omega _A$ be such that $\phi |_I\ne 0.$ If A is module $(\phi ,\varphi )$-inner amenable, then I is module $(\phi |_I,\varphi )$-inner amenable.

Proof

Let $m\in A^{**}$ satisfy $m(\varphi \circ \phi )=1$, $m(f.a)=m(a.f)$ and $m(\alpha .f)=m(f.\alpha )$ for all $a\in A, f\in A^*$ and $\alpha \in \mathfrak {A}$. By a similar argument as in the proof of Lemma 3.1 of [9], one can define a bounded linear functional n on $I^*$ by, $n(g)=m(f)$ for all $g\in I^*$, where f is an arbitrary element of $A^*$ extending g. Now, for every $g\in I^*, a\in I$ and $\alpha \in \mathfrak {A}$, we have

$$\begin{aligned} n(g.a)=m(f.a)=m(a.f)=n(a.g), \end{aligned}$$

and

$$\begin{aligned} n(g.\alpha )=m(f.\alpha )=m(\alpha .f)=n(\alpha .g). \end{aligned}$$

In addition, $n(\varphi \circ \phi |_I)=m (\varphi \circ \phi )=1.$ Therefore, I is module $(\phi |_I,\varphi )$-inner amenable. $\square$

We need to recall the following remark from [4] to give the next result:

Remark 2.13

Let $\mathfrak {A}$ be a Banach algebra and $\mathfrak {A}^{\#}$ be the unitization of $\mathfrak {A}$ which is $\mathfrak {A}^{\#}=\mathfrak {A}\oplus \mathbb {C}$ is a unital Banach algebra which contains $\mathfrak {A}$ as a closed ideal. Let A be a Banach $\mathfrak {A}$-module. Then, A is a Banach $\mathfrak {A}^{\#}$-bimodule with the following module actions:

$$\begin{aligned} (\alpha ,\lambda ).a=\alpha .a+\lambda a,~~~a.(\alpha ,\lambda )=a.\alpha +\lambda a~~~~~(\lambda \in \mathbb {C}, \alpha \in \mathfrak {A}, a\in A). \end{aligned}$$

Let $A^{\sharp }=(A\oplus \mathfrak {A}^{\#},\cdot )$, where the multiplication $\cdot$ is defined through

$$\begin{aligned} (a,u)\cdot (b,v)=(ab+a.v+u.b,uv)~~~~~(a,b\in A, u,v\in \mathfrak {A}^{\#}). \end{aligned}$$

Then, with the actions defined by

$$\begin{aligned} u.(a,v)=(u.a,uv),~~~(a,v).u=(a.u,vu)~~~~~(a\in A, u,v\in \mathfrak {A}^{\#}), \end{aligned}$$

$A^{\sharp }$ is a unital $\mathfrak {A}^{\#}$-module Banach algebra with the identity $e_{A^{\sharp }}=(0,e_{\mathfrak {A}^{\#}})$ , where $e_{\mathfrak {A}^{\#}}=(0,1).$ Now, suppose that $\phi \in \Omega _A$ and $\varphi ^{\#}$ is the extension of $\varphi$ on $\mathfrak {A}^{\#}$ defined by $\varphi ^{\#}(\alpha ,\lambda )=\varphi (\alpha )+\lambda ~(a\in A, \alpha \in \mathfrak {A}, \lambda \in \mathbb {C}$. If $u=(\alpha ,\lambda )\in \mathfrak {A}^{\#}$, it is easy to see that

$$\begin{aligned} \phi (a.u)=\phi (u.a)=\varphi ^{\#}(u)\phi (a)(a\in A). \end{aligned}$$

(4)

Define $\phi ^{\sharp }:A^{\sharp }\longrightarrow \mathfrak {A}^{\#}$ by

$$\begin{aligned} \phi ^{\sharp }(a,u)=(\phi (a),\varphi ^{\#}(u))(a\in A, u\in \mathfrak {A}^{\#}). \end{aligned}$$

(5)

Using (4), one can show that $\phi ^{\sharp }$ is multiplicative and

$$\begin{aligned} \phi ^{\sharp }(u.(a,v))=\phi ^{\sharp }((a,v).u)=\varphi ^{\#}(u)\phi ^{\sharp }(a,v)(a\in A, u,v\in \mathfrak {A}^{\#}). \end{aligned}$$

Therefore, $\phi ^{\sharp }$ is an extension of $\phi$, such that $\phi ^{\sharp }(0,u)=\varphi ^{\#}(u)$ is the extension $h_0=\tilde{0}$ of the zero function given by (5).

The proof of the following theorem is inspired by the proof of Proposition 2.7 of [4].

Proposition 2.14

Let A be a Banach $\mathfrak {A}$-bimodule, and let $\varphi \in \Delta (\mathfrak {A})$ and $\phi \in \Omega _A$. Then, A is module $(\phi ,\varphi )$-inner amenable if and only if $A^{\sharp }$ is module $(\phi ^{\sharp },\varphi ^{\#})$-inner amenable.

Proof

Let $A^{\sharp }$ be module $(\phi ^{\sharp },\varphi ^{\#})$-inner amenable. Since the image of $\phi ^{\sharp }|_A$ is included in $\mathfrak {A}$, by 2.12, we conclude that A is module $(\phi ,\varphi )$-inner amenable.

Conversely, suppose that A is module $(\phi ,\varphi )$-inner amenable. Then, there exists a $m\in A^{**}$, such that $m(\varphi \circ \phi )=1$, $m(f.a)=m(a.f)$, and $m(\alpha .f)=m(f.\alpha )$ for all $a\in A, f\in A^*$ and $\alpha \in \mathfrak {A}$. By Remark 2.13, we may identify the dual space $(A^{\sharp })^{*}$ with $A^*\oplus \mathbb {C}h_0$, where $h_0|_A=0$ and $h_0(e_{A^{\sharp }})=1.$ Define $n\in (A^{\sharp })^{**}$ by $n(f)=m(f)~(f\in A^*)$ and $n(h_0)=0$. Since A is an ideal and $\mathfrak {A}$-submodule of $A^{\sharp }$, it follows that $h_0.a=0$ and $h_0.\alpha =0$ for all $a\in A$ and $\alpha \in \mathfrak {A}$. A simple computation shows that

$$\begin{aligned} n\big ((f+\lambda h_0).(a+\lambda 'e_{A^{\sharp }})\big )=n\big ((a+\lambda 'e_{A^{\sharp }}).(f+\lambda h_0)\big ), \end{aligned}$$

and

$$\begin{aligned} n\big ((f+\lambda h_0).u\big )=n\big (u.(f+\lambda h_0)\big ), \end{aligned}$$

for all $f\in A^*, a\in A, u\in \mathfrak {A}^{\#}$ and $\lambda ,\lambda '\in \mathbb {C}$. For $f\in A^*$, consider the map $\bar{f}:A^{\sharp }\longrightarrow \mathbb {C}$ defined by $\bar{f}(a,u)=f(a)+\tilde{\varphi }(u)~(a\in A,u\in \mathfrak {A}).$ Thus, $\varphi ^{\#}\circ \phi ^{\sharp }=\overline{\varphi \circ \phi }$, and hence

$$\begin{aligned} n(\varphi ^{\#}\circ \phi ^{\sharp })=n(\overline{\varphi \circ \phi })=n(\varphi \circ \phi +\lambda h_0)=m(\varphi \circ \phi )=1. \end{aligned}$$

Therefore, $A^{\sharp }$ is module $(\phi ^{\sharp },\varphi ^{\#})$-inner amenable.

Module inner amenability of certain Banach algebras

Let $A\widehat{\otimes }B$ be the projective tensor product of two Banach algebras A and B. For every $f\in A^*$ and $g\in B^*$, let $f\otimes g$ denote the element of $(A\widehat{\otimes }B)^*$ satisfying, $f\otimes g(a\otimes b)=f(a)g(b)(a\in A, b\in B)$. In addition, note that $A\widehat{\otimes }B$ is a Banach $\mathfrak {A}\widehat{\otimes }\mathfrak {A}$-bimodule with the following actions:

$$\begin{aligned} (\alpha \otimes \beta ).(a\otimes b)=(\alpha .a)\otimes (\beta .b) (a\in A, b\in B, \alpha ,\beta \in \mathfrak {A}), \end{aligned}$$

and similarly for right action. For $\varphi _1,\varphi _2\in \Delta (\mathfrak {A}), \psi \in \Omega _A (=\Omega _{A,\varphi _1})$ and $\phi \in \Omega _A(=\Omega _{A,\varphi _2})$, define $(\phi \otimes \psi ):A\widehat{\otimes }B\rightarrow \mathfrak {A}\widehat{\otimes }\mathfrak {A}$ by $(\phi \otimes \psi )(a\otimes b)=\phi (a)\otimes \psi (b) (a\in A, b\in B)$. Clearly, $\phi \otimes \psi \in \Omega _{A\widehat{\otimes }B}(=\Omega _{A\widehat{\otimes }B,\varphi _1\otimes \varphi _2})$ and $\varphi _1\otimes \varphi _2\in \Delta (\mathfrak {A}\widehat{\otimes }\mathfrak {A})$. In addition, if $\overline{\varphi }\in \Delta (\mathfrak {A}\widehat{\otimes }\mathfrak {A})$, then $\overline{\varphi }=\varphi _1\otimes \varphi _2$, where $\varphi _1, \varphi _2\in \Delta (\mathfrak {A})$ (see [4]).

The technique of proof of the following theorem (one side) is similar to that of Theorem 2.8 of [4].

Theorem 3.1

Let A and B be Banach $\mathfrak {A}$-bimodules, and let $\varphi _1, \varphi _2\in \Delta (\mathfrak {A})$ and $\phi \in \Omega _A, \psi \in \Omega _B$. Then $A\widehat{\otimes }B$ is module $(\phi \otimes \psi , \varphi _1\otimes \varphi _2)$-inner amenable (as $\mathfrak {A}\widehat{\otimes }\mathfrak {A}$ module) if and only if A is module $(\phi ,\varphi _1)$-inner amenable and B is module $(\psi ,\varphi _2)$-inner amenable.

Proof

Suppose that $A\widehat{\otimes }B$ is module $(\phi \otimes \psi , \varphi _1\otimes \varphi _2)$-inner amenable. Then, there exists $m\in (A\widehat{\otimes }B)^{**}$, such that $m\big ( (\varphi _1\otimes \varphi _2)\circ (\phi \otimes \psi )\big )=1$ and

$$\begin{aligned} m\big (f\otimes g.(a\otimes b)\big )=m\big ((a\otimes b).f\otimes g\big ), m\big (f\otimes g.(\alpha \otimes \beta )\big )=m\big ((\alpha \otimes \beta ).f\otimes g\big ), \end{aligned}$$

for all $a\otimes b\in A\widehat{\otimes }B, f\in A^*, g\in B^*$ and $\alpha \otimes \beta \in \mathfrak {A}\widehat{\otimes }\mathfrak {A}$. Define $m_A:A^*\rightarrow \mathbb {C}$ by $m_A(f)=m\big (f\otimes (\varphi _2\circ \psi )\big )(f\in A^*)$. Therefore, $m_A(\varphi _1\circ \phi )=m\big ((\varphi _1\circ \phi ) \otimes (\varphi _2\circ \psi )\big )=m\big ( (\varphi _1\otimes \varphi _2)\circ (\phi \otimes \psi )\big )=1.$ Choose $b_0\in A$, such that $\varphi _2\circ \psi (b_0)=1$. Therefore, for every $a\in A$ and $f\in A^*$, we have

$$\begin{aligned} m_A(f.a)&=m\big ((f.a)\otimes (\varphi _2\circ \psi )\big )\\&=m\big ((f.a)\otimes (\varphi _2\circ \psi ).b_0\big )\\&=m\big (f\otimes (\varphi _2\circ \psi ).(a\otimes b_0)\big )\\&=m\big ((a\otimes b_0).f\otimes (\varphi _2\circ \psi )\big )\\&=m\big ((a.f)\otimes b_0.(\varphi _2\circ \psi )\big )\\&=m_A(a.f). \end{aligned}$$

Similarly, for every $\alpha \in \mathfrak {A}$, if we take $\beta \in \mathfrak {A}$, such that $\varphi _2(\beta )=1$, then

$$\begin{aligned} m_A(f.\alpha )&=m\big ((f.\alpha )\otimes (\varphi _2\circ \psi )\big )\\&=m\big ((f.\alpha )\otimes (\varphi _2\circ \psi ).\beta \big )\\&=m\big (f\otimes (\varphi _2\circ \psi ).(\alpha \otimes \beta )\big )\\&=m\big ((\alpha \otimes \beta ).f\otimes (\varphi _2\circ \psi )\big )\\&=m\big ((\alpha .f)\otimes \beta .(\varphi _2\circ \psi )\big )\\&=m\big ((\alpha .f)\otimes (\varphi _2\circ \psi )\big )\\&=m_A(\alpha .f), \end{aligned}$$

for all $f\in A^*$. Therefore, A is module $(\varphi ,\phi _1)$-inner amenable. Similarly, one can prove that B is module $(\psi ,\phi _2)$-inner amenable.

For the converse, let A is module $(\phi ,\varphi _1)$-inner amenable and B is module $(\psi ,\varphi _2)$-inner amenable. Then, by Theorem 2.2, there exist bounded nets $(a_i)_i$ in A and $(b_j)_j$ in B with bounds $M_1$ and $M_2$, respectively, such that

$$\begin{aligned} \varphi _1\circ \phi (a_i)=1, \Vert aa_i-a_ia\Vert \longrightarrow 0, \Vert \alpha .a_i-a_i.\alpha \Vert \longrightarrow 0(a\in A, \alpha \in \mathfrak {A}), \end{aligned}$$

and

$$\begin{aligned} \varphi _2\circ \psi (b_j)=1, \Vert bb_j-b_jb\Vert \longrightarrow 0, \Vert \alpha .b_j-b_j.\alpha \Vert \longrightarrow 0(b\in B, \alpha \in \mathfrak {A}). \end{aligned}$$

Consider the bounded net $(a_i\otimes b_j)_{(i,j)}$ in $A\widehat{\otimes }B$. Therefore, $\big ((\varphi _1\otimes \varphi _2)\circ (\phi \otimes \psi )\big )(a_i\otimes b_j)=\varphi _1\circ \phi (a_i)\varphi _2\circ \psi (b_j)=1$. Let $\mathfrak {F} =\sum _{l=1}^N\alpha _l\otimes \beta _l\in \mathfrak {A}\widehat{\otimes }\mathfrak {A}$, then

$$\begin{aligned} \Vert \mathfrak {F}.&(a_i\otimes b_j)-(a_i\otimes b_j).\mathfrak {F}\Vert \\&=\Vert \sum _{l=1}^N\big [(\alpha _l.a_i-a_i.\alpha _l)\otimes \beta _l.b_j+a_i.\alpha _l\otimes (\beta _l.b_j-b_j.\beta _l)\big ]\Vert \\&\le \sum _{l=1}^N M_2\Vert \beta _l\Vert \Vert \alpha _l.a_i-a_i.\alpha _l\Vert +\sum _{l=1}^N M_1\Vert \alpha _l\Vert \Vert \beta _l.b_j-b_j.\beta _l\Vert \longrightarrow 0. \end{aligned}$$

Now, let $\mathfrak {G}\in \mathfrak {A}\widehat{\otimes }\mathfrak {A}$, so there exist sequences $(\alpha _l)_l\subseteq \mathfrak {A}$ and $(\beta _l)_l\subseteq \mathfrak {A}$, such that $\mathfrak {G}=\sum _{l=1}^{\infty }\alpha _l\otimes \beta _l$ with $\sum _{l=1}^{\infty }\Vert \alpha _l\Vert \Vert \beta _l\Vert <\infty .$ By the same argument as in the proof of the Theorem 3.1 of [6], for every $\mathfrak {G}\in \mathfrak {A}\widehat{\otimes }\mathfrak {A}$, one can show that $\Vert \mathfrak {G}.(a_i\otimes b_j)-(a_i\otimes b_j).\mathfrak {G}\Vert \longrightarrow 0$. Similarly, we may show that $\Vert G(a_i\otimes b_j)-(a_i\otimes b_j)G\Vert \longrightarrow 0$ for all $G\in A\widehat{\otimes }B$. Therefore, Proposition 2.2 implies that $A\widehat{\otimes }B$ is module $(\phi \otimes \psi , \varphi _1\otimes \varphi _2)$-inner amenable. $\square$

Let A and B be Banach algebras, it is well known that $A\oplus _{\infty }B$ and $A\oplus _p B$, the $l^p$-direct sum of A and B, are Banach algebras with respect to the canonical multiplication defined by

$$\begin{aligned} (a,b)(c,d):= (ac,bd)\quad (a,c\in A, b,d\in B), \end{aligned}$$

and norms $\Vert (a,b)\Vert =\max \{\Vert a\Vert ,\Vert b\Vert \}$ and $\Vert (a,b)\Vert =(\Vert a\Vert ^p+\Vert b\Vert ^p)^{\frac{1}{p}}~(a\in A, b\in B).$ Furthermore, if A and B are two Banach $\mathfrak {A}$-bimodules, then $A\oplus _{\infty }B$ and $A\oplus _p B$ are Banach $\mathfrak {A}$-bimodules under the module actions

$$\begin{aligned} \alpha .(a,b)=(\alpha .a,\alpha .b),\quad (a,b).\alpha =(a.\alpha ,b.\alpha )\quad (a\in A, b\in B, \alpha \in \mathfrak {A}). \end{aligned}$$

Before stating the next theorem, we note that if for every $\varphi \in \Delta (\mathfrak {A}), \phi \in \Omega _A$ and $\psi \in \Omega _B$, we define $(0,\psi ):A\oplus _pB\rightarrow \mathfrak {A}$ and $(\phi ,0):A\oplus _pB\rightarrow \mathfrak {A}$ by

$$\begin{aligned} (0,\psi )(a,b)=\psi (b), (\phi ,0)(a,b)=\phi (a)(a\in A, b\in B), \end{aligned}$$

where $1\le p\le \infty ,$ then $(0,\psi )$ and $(\phi ,0)\in \Omega _{A\oplus _pB}(=\Omega _{A\oplus _pB,\varphi })$.

Theorem 3.2

Let A and B be two $\mathfrak {A}$-bimodule Banach algebras, $\varphi \in \Delta ({\mathfrak {A}}), \phi \in \Omega _A, \psi \in \Omega _B$ and $1\le p\le \infty .$ Then the following statements are valid:

• (i) $A\oplus _pB$ is module $\big ((\phi ,0),\varphi \big )$-inner amenable if and only if A is module $(\phi ,\varphi )$-inner amenable.

• (ii) $A\oplus _p B$ is module $\big ((0,\psi ),\varphi \big )$-inner amenable if and only if B is module $(\psi ,\varphi )$-inner amenable.

Proof

(i) Assume that $A\oplus _pB$ is module $\big ((\phi ,0),\varphi \big )$-inner amenable. By Theorem 2.2, there exists a net $(a_i,b_i)_i$ in $A\oplus _pB$, such that $\varphi \circ (\phi ,0)(a_i,b_i)\longrightarrow 1$ and

$$\begin{aligned} \Vert (a,b).(a_i,b_i)-(a_i,b_i).(a,b)\Vert \longrightarrow 0, \Vert \alpha .(a_i,b_i)-(a_i,b_i).\alpha \Vert \longrightarrow 0, \end{aligned}$$

for all $(a,b)\in A\oplus _pB$ and $\alpha \in \mathfrak {A}$. Consider the bounded net $(a_i)_i$ in A. One can easily show that $\Vert aa_i-a_ia\Vert \longrightarrow 0$ and $\Vert \alpha .a_i-a_i.\alpha \Vert \longrightarrow 0$ for all $a\in A, \alpha \in \mathfrak {A}$. In addition, it is clear that $\varphi \circ \phi (a_i)\longrightarrow 1$. Therefore, Theorem 2.2 implies that A is module $(\phi ,\varphi )$-inner amenable.

Conversely, suppose that A is module $(\phi ,\varphi )$-inner amenable. Then, there exists a bounded net $(a_i)_i$ in A, such that $\varphi \circ \phi (a_i)\longrightarrow 1$, $\Vert aa_i-a_ia\Vert \longrightarrow 0$ and $\Vert \alpha .a_i-a_i.\alpha \Vert \longrightarrow 0$ for all $a\in A, \alpha \in \mathfrak {A}$. Clearly, the bounded net $(a_i,0)_i\subset A\oplus _p B$ satisfies in the condition (iii) of Theorem 2.2. Therefore, $A\oplus _p B$ is module $\big ((\phi ,0),\varphi \big )$-inner amenable.

Similarly, we can prove (ii). $\square$

Corollary 3.3

Let A and B be two $\mathfrak {A}$-bimodule Banach algebras and $1\le p\le \infty .$ Then $A\oplus _pB$ is module $\big ((\phi ,0),\varphi \big )$-inner amenable and module $\big ((0,\psi ),\varphi \big )$-inner amenable for every $\varphi \in \Delta ({\mathfrak {A}}), \phi \in \Omega _A$ and $\psi \in \Omega _B$ if and only if both A and B are module character inner amenable.

We note that for two Banach algebras A and B, a direct verification shows that

$$\begin{aligned} \Delta (A\oplus _p B)=(\Delta (A)\times \{0\})\cup (\{0\}\times \Delta (B)), 1\le p\le \infty . \end{aligned}$$

Now, if we take $\mathfrak {A}=\mathbb {C}$ and $\varphi$ is the identity map in the above corollary, then we obtain that $A\oplus _pB$ is character inner amenable if and only if both A and B are character inner amenable. Therefore, the above corollary generalizes Proposition 4.2 of [6].

Let A be a Banach algebra and X be a Banach A-bimodule. The $l^1$-direct sum of A and X, denoted by $A\oplus _1X$, with the product defined by

$$\begin{aligned} (a,x)(a',x')=(aa',a.x'+x.a')(a,a'\in A, x,x'\in X), \end{aligned}$$

is a Banach algebra that is called the module extension Banach algebra of A and X.

If A is $\mathfrak {A}$-bimodule and X is a Banach A-$\mathfrak {A}$-module, then $A\oplus _1 X$ is Banach $\mathfrak {A}$-bimodules under the module actions:

$$\begin{aligned} \alpha .(a,x)=(\alpha .a,\alpha .x),(a,x).\alpha =(a.\alpha ,x.\alpha )(a\in A, x\in X, \alpha \in \mathfrak {A}). \end{aligned}$$

(6)

Let A and B be Banach algebras and let X be a Banach A, B-module; that is, a left A-module and a right B-module satisfying $\Vert axb\Vert \le \Vert a\Vert \Vert x\Vert \Vert b\Vert ,~(a\in A, b\in B, x\in X).$ The corresponding triangular Banach algebra

$$\begin{aligned} \tau =\Big \{ \left( \begin{array}{ccc} a &{} x \\ 0 &{} b \\ \end{array} \right) :a\in A, x\in X, b\in B \Big \}, \end{aligned}$$

is equipped with the usual $2\times 2$-matrix operations and the norm

$$\begin{aligned}\Vert \left( \begin{array}{ccc} a &{} x \\ 0 &{} b \\ \end{array} \right) \Vert =\Vert a\Vert +\Vert x\Vert +\Vert b\Vert . \end{aligned}$$

This Banach algebra were introduced by Forrest and Marcoux in [7]. Note that $\tau$ can be identified with the module extension $(A\oplus _1B)\oplus _1 X$, in which X is considered as a $A\oplus _1B$-module under the operations:

$$\begin{aligned} (a,b).x=ax, ~x.(a,b)=xb (a\in A, b\in B, x\in X). \end{aligned}$$

Furthermore, if A and B are two Banach $\mathfrak {A}$-bimodules and X is a Banach $A\oplus _1B$-$\mathfrak {A}$-module, then $\tau$ is Banach $\mathfrak {A}$-bimodules under the module actions defined as (6).

Let $\phi \in \Omega _A$ and define $\tilde{\phi }:A\oplus _1X\longrightarrow \mathfrak {A}$ by $\tilde{\phi }(a,x)=\phi (a)~(a\in A, x\in X)$. Then, $\tilde{\phi }\in \Omega _{A\oplus _1X}$.

Using Theorem 2.2, we can routinely prove the following proposition and so we omit its proof.

Proposition 3.4

Let A be $\mathfrak {A}$-bimodule and X be a Banach A-$\mathfrak {A}$-module and let $\varphi \in \Delta (\mathfrak {A})$ and $\phi \in \Omega _A$. Then $A\oplus _1X$ is module $(\tilde{\phi },\varphi )$-inner amenable if and only if there exists a bounded net $(a_i,x_i)_i$ in $A\oplus _1X$ satisfying

• (i) $\varphi \circ \phi (a_i)\longrightarrow 1$ and $\Vert aa_i-a_ia\Vert \longrightarrow 0, \Vert \alpha .a_i-a_i.\alpha \Vert \longrightarrow 0$ and $\Vert \alpha .x_i-x_i.\alpha \Vert \longrightarrow 0$ for all $a\in A$ and $\alpha \in \mathfrak {A},$

• (ii) $\Vert x.a_i-a_i.x\Vert \longrightarrow 0$ for all $x\in X$, and

• (iii) $\Vert a.x_i-x_i.a\Vert \longrightarrow 0$ for all $a\in A$.

Corollary 3.5

Let A be $\mathfrak {A}$-bimodule and X be a Banach A-$\mathfrak {A}$-module and let $\varphi \in \Delta (\mathfrak {A})$ and $\phi \in \Omega _A$. If $A\oplus _1X$ is module $(\tilde{\phi },\varphi )$-inner amenable, then A is module $(\phi ,\varphi )$-inner amenable.

Corollary 3.6

Let A and B be $\mathfrak {A}$-bimodules and X be a Banach $A\oplus _1B$-$\mathfrak {A}$-module. If $\tau$ is module character inner amenable, then so are A and B.

Proof

Suppose that $\tau$ is module character inner amenable. By Corollary 3.5, $A\oplus _1B$ is module character inner amenable. Therefore, corollary 3.3 implies that A and B are module character inner amenable. $\square$