On generalized biderivations of Banach algebras

1.   Introduction and preliminaries

Let $A$ be a unital Banach algebra with unit $e_{A}$ and $X$ be a unitary Banach $A$-bimodule in the sense that $e_{A}\cdot x = x\cdot e_{A} = x$ for every $x\in X$. We say that a linear map $d:A\rightarrow X$ is a derivation if $d(ab) = d(a)\cdot b+a\cdot d(b)$ for all $a, b\in A$. For each $x\in X$, the mapping $d_{x}:A\rightarrow X$, $d_{x}(a): = a\cdot x-x\cdot a$ is a bounded derivation, called an inner derivation.

We can define the right and left actions of $A$ on the dual space $X^{*}$ of $X$ via

for each $a\in A$, $x\in X$, $f\in X^{*}$.

A Banach algebra $A$ is called amenable if for each Banach $A$-bimodule $X$, the only bounded derivations from $A$ to $X^{*}$ are inner derivations. The notion of an amenable Banach algebra was introduced by Johnson in ^[11]. For more details about this notion, see ^[14]. A Banach algebra $A$ is weakly amenable, if every bounded derivation from $A$ to $A^{*}$ is an inner derivation. The concept of weak amenability of Banach algebras was introduced by Bade, Curtis, and Dales ^[1] for commutative Banach algebras and then by Johnson ^[12] for a general Banach algebra.

A bilinear mapping $D:A\times A\rightarrow X$ is called a biderivation if it is a derivation in each argument; that is, for every $b\in A$, the maps $a\mapsto D(a, b)$ and $a\mapsto D(b, a)$ are derivations. Consider the subspace $Z(A, X) = \{x\in X\mid a\cdot x = x\cdot a, \forall a\in A\}$ of $X$. Then, for each $x\in Z(A, X)$, the mapping $D_{x}:A\times A\rightarrow X$ defined by $D_{x}(a, b) = x[a, b] = x(ab-ba) \ \ (a, b\in A)$ is an example of a biderivation and called an inner biderivation. In ^[7], Bresar proved that all biderivations on noncommutative prime rings are inner. For more applications and details about biderivations, see ^[8]. Also see ^[5,9], where the structures of biderivations on triangular algebras and generalized matrix algebras were studied, along with the conditions under which these biderivations are inner.

Although derivations and biderivations, as well as inner derivations and inner biderivations, appear similar, there are fundamental differences between them. These differences become more evident when a biderivation is required to be an inner biderivation. One reason is that biderivations depend on two components, while another is that inner biderivations must involve elements from $Z(A, X)$. Similarly, amenability and weak amenability differ from biamenability and weak biamenability, as explained in ^[3,4].

In ^[4], Barootkoob and Mohammadzadeh introduced the concept of biamenability of Banach algebras and demonstrated that, although amenability and biamenability of Banach algebras share some superficial similarities, they exhibit notably distinct and, in some cases, contrasting properties. Specifically, it was shown that commutative Banach algebras and the unitization of Banach algebras are not biamenable, even if they are amenable. Furthermore, it was established that $B(H)$, the algebra of all bounded operators on an infinite-dimensional Hilbert space $H$, is biamenable but not amenable. A complete characterization of biderivations and inner biderivations on triangular Banach algebras has been provided in ^[2]. Additionally, a result concerning the weak biamenability of triangular Banach algebras was established in ^[2].

Bresar introduced the concept of generalized derivations in ^[6]. The notion of generalized amenability of Banach algebras was later investigated in ^[13,15], where the authors provided various results specifically for triangular Banach algebras.

Let $A$ be a unital Banach algebra and $X$ be a unitary Banach $A$-bimodule. A linear mapping $g:A\rightarrow X$ is said to be a generalized derivation if

for all $a, b\in A$. The generalized derivation $g_{x, y}:A\rightarrow X$ is called an inner generalized derivation if there exist $x, y\in X$ such that $g_{x, y}(a): = x\cdot a+a\cdot y$. Similar to the amenability of a Banach algebra, we say $A$ is generalized amenable if for each Banach $A$-bimodule $X$, the only bounded generalized derivations from $A$ to $X^{*}$ are inner generalized derivations.

A bilinear mapping $G:A\times A\rightarrow X$ is called a generalized biderivation if it is a generalized derivation in each argument; that is,

and

for all $a, b, c\in A$.

Example 1. Let $A = M_{2}(\mathbb{C})$ be the unital Banach algebra of all $2\times 2$ upper triangular matrices over the complex field $\mathbb{C}$. We define a map $G:A\times A\rightarrow A$ by

Then, we see that $G$ is a generalized biderivation.

Lemma 1. Let $A$ be a unital Banach algebra and $X$ be a unitary Banach $A$-bimodule. Suppose that $x, y\in X$. Then the mapping $G_{x, y}:A\times A\rightarrow X$ defined by

is a generalized biderivation.

Proof. It is clear that $G_{x, y}$ is bilinear. Also, we have

and

for all $a, b, c\in A$. Hence, $G_{x, y}$ is a generalized biderivation. □

We call the map of the form $G_{x, y}$ given in Lemma 1 an inner generalized biderivation. We denote by $GZ^{1}(A, X)$ and $GN^{1}(A, X)$ the linear spaces of all bounded generalized biderivations and inner generalized biderivations from $A\times A$ into $X$, respectively. Also we call the quotient space

the first generalized bicohomology group from $A\times A$ into $X$. Similar to the definition of biamenability ^[2,4] of Banach algebras, we now define the concept of generalized biamenability of unital Banach algebras as follows. The Banach algebra $A$ is said to be generalized biamenable if every bounded generalized biderivation $G: A\times A\rightarrow X^{*}$ is an inner generalized biderivation; i.e., $GH^{1}(A, X^{*}) = \{0\}$. A Banach algebra $A$ is called weakly generalized biamenable if every bounded generalized biderivation from $A\times A$ to $A^{*}$ is an inner generalized biderivation.

In this paper, we first prove some theorems on generalized biamenability of Banach algebras. And we also give some results characterizing all generalized biderivations and inner generalized biderivations on triangular Banach algebras.

2.   Results

In this section, let $A$ be a unital Banach algebra with the unit $e_{A}$ and $X$ be a unitary Banach $A$-bimodule.

Theorem 1. Let $A$ be a Banach algebra and consider $\mathbb{C}$ as a Banach $A$-bimodule. If there is a nonzero generalized derivation $g:A\rightarrow \mathbb{C}$, then generalized biamenability of $A$ implies generalized amenability of $A$.

Proof. Let $X$ be a Banach $A$-bimodule and $g^{\prime}:A\rightarrow X^{*}$ be a bounded generalized derivation. Then,

is a bounded generalized biderivation. Indeed, we have

and

for all $a, b, c\in A$. Since $\mathbb{C}$ is a Banach $A$-bimodule, we get

for all $a, b, c\in A$. Similarly, we have

and

for all $a, b, c\in A$. Hence, it is clear that

Then, we see that $G$ is a generalized biderivation. Hence, there are $f, h\in X^{*}$ such that

for all $a, b\in A$. Therefore, for every $b\in A$ and for some $a\in A$ such that $g(a)\neq 0$, we have

So, we get that $g^{\prime}$ is inner, and then $A$ is generalized amenable. □

A question naturally arises from the generalized biamenability of a Banach algebra, can we then determine whether another Banach algebra is generalized biamenable? The answer is affirmative in the following cases.

Theorem 2. If $\theta:A\rightarrow B$ is a continuous homomorphism of Banach algebras with the dense range and $A$ is generalized biamenable, then so is $B$.

Proof. Let $X$ be a Banach $B$-bimodule. Consider $X$ as an $A$-bimodule with module actions $a\cdot x = \theta(a)x$ and $x\cdot a = x\theta(a)$ for each $a\in A$ and $x\in X$. Now, for each $G\in GZ^{1}(B, X^{*})$, $G\circ (\theta\times\theta)\in GZ^{1}(A, X^{*})$ and generalized biamenability of $A$ implies that

for some $f, h\in X^{*}$. By density we conclude that

for all $a^{\prime}, b^{\prime}\in B$. □

An immediate consequence of this theorem is the following:

Corollary 1. Let $A$ be a generalized biamenable Banach algebra and $I$ be a closed ideal in $A$. Then the Banach algebra $A/I$ is generalized biamenable.

Proof. The canonical mapping $A\rightarrow A/I$ is a contractive, surjective homomorphism, and therefore continuous. □

3.   Results for triangular Banach algebras

Let $A$ and $B$ be Banach algebras, and suppose that $X$ is $A$-$B$-module; that is, $X$ is a Banach space, a left $A$-module, a right $B$-module, and the actions of $A$ and $B$ are continuous in that

for each $a\in A$, $x\in X$, $b\in B$. If $A$ has a unit $1_{A}$ and $B$ has a unit $1_{B}$, then $X$ is said to be unital in the sense that $1_{A}\cdot x = x\cdot 1_{B} = x$ for every $x\in X$. We define the corresponding triangular Banach algebra

with the usual $2\times 2$ matrix addition and multiplication. The norm on $T$ is

Moreover, if the Banach algebras $A$, $B$, and the $A$-$B$-module $X$ are unital, then $T$ is unital. The dual of triangular Banach algebra $T$ is

where $\left(\begin{array}{ll} f & h \\ 0 & g \end{array}\right)\left(\left(\begin{array}{ll} a & x \\ 0 & b \end{array}\right)\right): = f(a)+h(x)+g(b)$. $T^{*}$ is a triangular $T$-bimodule with respect to the following module actions

for every $\left(\begin{array}{ll} a & x \\ 0 & b \end{array}\right) \in T$ and $\left(\begin{array}{ll} f & h \\ 0 & g \end{array}\right) \in T^*$. Such algebras were introduced by Forrest and Marcoux in ^[10].

We will assume that the corner Banach algebras $A$ and $B$ are unital and that $X$ is a unital $A$-$B$-module and $T$ is associated triangular Banach algebra. We begin with the following theorem.

Theorem 3. Let $\delta_{A}:A\times A\rightarrow A^{*}$ and $\delta_{B}:B\times B\rightarrow B^{*}$ be bounded generalized biderivations. Then the bilinear mapping $G:T\times T\rightarrow T^{*}$ defined by

is a bounded generalized biderivation. Furthermore, $G$ is inner if and only if $\delta_{A}$ and $\delta_{B}$ are inner.

Proof. It is easy to verify that $G$ is a generalized biderivation. Also,

Hence, $G$ is bounded. Suppose that $G$ is an inner generalized biderivation. Then, there exist $\left(\begin{array}{cc} f_1 & h_1 \\ 0 & g_1 \end{array}\right), \left(\begin{array}{cc} f_2 & h_2 \\ 0 & g_2 \end{array}\right) \in T^*$ such that

In particular, we have that

Thus, $\delta_A\left(a, a^{\prime}\right) = a \cdot f_1 \cdot a^{\prime}+a^{\prime} \cdot f_2 \cdot a$. So $\delta_A$ is an inner generalized biderivation. Similarly, we can show that $\delta_{A}(a, a^{\prime}) = a\cdot f_{1}\cdot a^{\prime}+a^{\prime}\cdot f_{2}\cdot a$ is an inner generalized biderivation.

Conversely, if $\delta_{A}$ and $\delta_{B}$ are inner generalized biderivations, then there are $\delta_{A}:A\times A\rightarrow A^{*}$ and $\delta_{B}:B\times B\rightarrow B^{*}$ such that for each $f_{1}, f_{2}\in A^{*}$, $g_{1}, g_{2}\in B^{*}$, and for each $a, a^{\prime}\in A$, $\delta_{A}(a, a^{\prime}) = a\cdot f_{1}\cdot a^{\prime}+a^{\prime} \cdot f_{2}\cdot a$. Then, we have

Hence, $G$ is an inner generalized biderivation. □

Theorem 4. Let $T^{*}$ be the triangular bimodule $\left(\begin{array}{cc} A^{*} & X^{*} \\ 0 & B^{*}\\\end{array}\right)$ associated to the triangular Banach algebra $T$. Assume that $G:T\times T\rightarrow T^{*}$ is a bounded generalized biderivation. Then, there exist bounded generalized biderivations $\delta_{A}:A\times A\rightarrow A^{*}$ and $\delta_{B}:B\times B\rightarrow B^{*}$, and $h_{1}, h_{2}\in X^{*}$ such that

for every $a, a^{\prime}\in A$, $b, b^{\prime}\in B$ and $x, x^{\prime}\in X$.

Proof. Suppose $G$ is a generalized biderivation on $T$. Write $G$ as

where $\delta_{A}:A\times A\rightarrow A^{*}$, $\delta_{B}:B\times B\rightarrow B^{*}$, $d_{A}:A\times A\rightarrow B^{*}$, $d_{B}:B\times B\rightarrow A^{*}$, $k_{1}:X\times X\rightarrow A^{*}$, $k_{2}:X\times X\rightarrow B^{*}$, $r_{1}:A\times A\rightarrow X^{*}$, $r_{2}:B\times B\rightarrow X^{*}$ and $r_{3}:X\times X\rightarrow X^{*}$ are all bilinear maps. Let

and

for some $f_{1}, f_{2}\in A^{*}$, $h_{1}, h_{2}\in X^{*}$ and $g_{1}, g_{2}\in B^{*}$. Then, we have

On the other hand,

Thus, for all $a, a^{\prime}\in A$, we have $f_{2}\cdot a = 0$, $r_{1}(a, a^{\prime}) = h_{1}\cdot a$ and $d_{A}(a, a^{\prime}) = 0$. So $f_{2} = 0$. Similarly, we have

Thus, we have $d_{B}(b, b^{\prime}) = 0$, $b\cdot g_{1} = 0$, and $r_{2}(b, b^{\prime}) = b\cdot h_{2}$ for all $b, b^{\prime}\in B$. So $g_{1} = 0$. For $x\in X$, we have

Thus, $r_{3}(x, x^{\prime}) = 0$ and $k_{2}(x, x^{\prime}) = h_{1}\cdot x$ for all $x, x^{\prime}\in X$. Also, we have

Thus, $k_{1}(x, x^{\prime}) = x\cdot h_{2}$ and $r_{3}(x, x^{\prime}) = 0$ for all $x, x^{\prime}\in X$. Hence, we have

for all $a, a^{\prime}\in A$, $b, b^{\prime}\in B$, $x, x^{\prime}\in X$. So, for each $a, a^{\prime}\in A$, $b^{\prime}\in B$, and $x^{\prime}\in X$, we have

Moreover,

for all $a_{1}, a_{2}\in A$. So, we have $\delta_{A}(a_{1}a_{2}, a^{\prime}) = \delta_{A}(a_{1}, a^{\prime})\cdot a_{2}+a_{1}\cdot\delta_{A}(a_{2}, a^{\prime})-a_{1}\cdot f_{1}\cdot a_{2}$ for each $a_{1}, a_{2}\in A$. In particular, taking $a_{1} = a_{2} = 1_{A}$, we get $\delta_{A}(1_{A}, a^{\prime}) = \delta_{A}(1_{A}, a^{\prime})+\delta_{A}(1_{A}, a^{\prime})-f_{1}$, that is, $f_{1} = \delta_{A}(1_{A}, a^{\prime})$. Similarly, we can show

Therefore, $\delta_{A}$ is a generalized biderivation. Further, since $G$ is bounded, so

It follows that $\delta_{A}$ is bounded. Also, for each $a^{\prime}\in A$, $b, b^{\prime}\in B$ and $x^{\prime}\in X$, we have

Moreover,

for all $b_{1}, b_{2}\in B$. So, we have

for each $b_{1}, b_{2}\in B$. Taking $b_{1} = b_{2} = 1_{B}$, we have $g_{2} = \delta_{B}(1_{B}, b^{\prime})$. Similarly, we can show

Therefore, $\delta_{B}$ is a generalized biderivation. Further, since $G$ is bounded, it is clear that $\delta_{B}$ is bounded. □

Theorem 5. Let $A$ and $B$ be unital Banach algebras, and let $T = Tri(A, 0, B)$ be the corresponding triangular Banach algebra. Then,

Proof. Define $f:GZ^{1}(A, A^{*})\oplus GZ^{1}(B, B^{*})\rightarrow GH^{1}(T, T^{*})$ by $f((\delta_{A}, \delta_{B})): = [G^{\prime}]$, where $[G^{\prime}]$ is the equivalent class of $G^{\prime}: = \left(\begin{array}{cc} \delta_{A} & 0 \\ 0 & \delta_{B}\\\end{array}\right)$ in $GH^{1}(T, T^{*})$. Clearly, $f$ is linear. By Theorems 3 and 4, $f$ is surjective. Also, by Theorem 3, we have

Therefore, we have

Then, we have the desired result. □

Corollary 2. $T = Tri(A, 0, B)$ is weakly generalized biamenable if and only if $A$ and $B$ are weakly generalized biamenable.

Now, we give a result about generalized biamenability for triangular Banach algebras.

Theorem 6. If $T = Tri(A, 0, B)$ is generalized biamenable, then $A$ and $B$ are both generalized biamenable Banach algebras.

Proof. Suppose that $T: = \left(\begin{array}{cc} A & 0 \\ 0 & B \end{array}\right)$ is generalized biamenable. Since $\left(\begin{array}{cc} A & 0 \\ 0 & 0 \end{array}\right)$ is a closed ideal of $T$, the quotient algebra $\left(\begin{array}{cc} A & 0 \\ 0 & B \end{array}\right)/\left(\begin{array}{cc} A & 0 \\ 0 & 0 \end{array}\right)$ is generalized biamenable by Corollary 1. On the other hand, $\left(\begin{array}{cc} A & 0 \\ 0 & B \end{array}\right)/\left(\begin{array}{cc} A & 0 \\ 0 & 0 \end{array}\right)\cong B$, thus, $B$ is generalized biamenable. Similarly, since $\left(\begin{array}{cc} 0 & 0 \\ 0 & B \end{array}\right)$ is a closed ideal of $T$, we have $\left(\begin{array}{cc} A & 0 \\ 0 & B \end{array}\right)/\left(\begin{array}{cc} 0 & 0 \\ 0 & B \end{array}\right)\cong A$. Thus, it follows from Corollary 1 that $A$ is generalized biamenable. □

4.   Conclusions

As we mentioned in the introduction, in this paper, we studied the generalized biderivations of unital Banach algebras. Some results of this new concept have been obtained, and the author thinks that using various methods the results on generalized biamenability can be extended to areas related to Banach algebras in the future.

Use of Generative-AI tools declaration

The author declares that she has not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The author declares no conflicts of interest.