Weak 2-Local Inner Derivations of Semiprime ∗-Banach Algebras

Abstract

This paper introduces and examines the concept of weak 2-local inner derivations and their relationship to $\mathcal{P}$-2-local mappings. It is established that every such mapping on a semiprime ∗-Banach algebra with a faithful trace is a derivation, which also provides a complete characterization on finite von Neumann algebras. Additionally, it is shown that weak 2-local inner derivations coincide with the 2-local reflection closure of the inner derivations.

1. Introduction

Research on derivations and their generalizations constitutes a central focus within the fields of operator algebras and functional analysis. A linear operator D from an algebra A into an A-bimodule X is called a derivation if it satisfies the Leibniz identity $D\left(ab\right)=D\left(a\right)b+aD\left(b\right)$ for all $a,b\in A.$ A canonical example is provided by inner derivations: for any fixed $a\in A$, the map $D_a:A\to A$ defined by $D_a\left(x\right)=[a,x]=ax-xa$ for $x\in A,$ readily verifies the derivation property. A central theme in this field is the identification of conditions under which linear operators on various classes of algebras are necessarily derivations. Kadison made significant progress on this question by introducing the concept of local derivations and demonstrating that every norm-continuous local derivation from a von Neumann algebra into its dual bimodule is a derivation [1].

Šemrl introduced a significant nonlinear generalization known as 2-local derivations [2]. An operator $\Delta :A\to A$ (not necessarily linear) is defined as a 2-local derivation if, for every pair $a,b\in A$, there exists a derivation $D_{a,b}$, which depends on a and b, such that $\Delta \left(a\right)=D_{a,b}\left(a\right)$ and $\Delta \left(b\right)=D_{a,b}\left(b\right).$ Šemrl demonstrated that every 2-local derivation on $B\left(H\right)$, the algebra of all bounded linear operators on an infinite-dimensional separable Hilbert space H, constitutes a derivation.

The framework was later generalized by examining the local action of derivations through continuous linear functionals. In a significant advancement, Essaleh, Peralta, and Ramírez introduced the concept of weak local derivations and demonstrated that every such map on a $C*$-algebra is a derivation [3]. These developments have spurred substantial research, leading to numerous characterizations of local and 2-local derivations for a range of associative algebras [4,5,6]. Expanding upon these results, Niazi and Peralta introduced the notion of weak 2-local derivations [7]. A map $\Delta :A\to A$ is a weak 2-local derivation if for every $x,y\in A$ and every $\phi \in A*$, there exists a derivation $D_{x,y,\phi }:A\to A$ (depending on x, y, and $\phi$) such that $\phi (\Delta \left(x\right))=\phi \left(D_{x,y,\phi }\left(x\right)\right)$ and $\phi (\Delta \left(y\right))=\phi \left(D_{x,y,\phi }\left(y\right)\right).$ This shift in perspective is significant. The “weak” formulation requires that local implementation by derivations holds only when tested by continuous linear functionals. This requirement makes the concept particularly natural in the context of infinite-dimensional algebras equipped with weak topologies. This line of research has yielded significant results, demonstrating that weak 2-local derivations are derivations on matrix algebras $M_n$[8], on $B\left(H\right)$ and $K\left(H\right)$ (the compact operators) for any Hilbert space H [9], and on finite von Neumann algebras [10]. For further developments and applications, see Refs. [11,12,13,14] and the references therein.

This paper refines the aforementioned concepts by restricting the implementing derivations to inner derivations, thereby introducing the notion of a weak 2-local inner derivation. Let X be a Banach algebra. A mapping $\Delta :X\to X$ is a weak 2-local inner derivation if for every $x,y\in X$ and every $\phi \in X*$, there exists an inner derivation $D_{x,y,\phi }$ (depending on x, y, and $\phi$) such that

$$\phi (\Delta \left(x\right))=\phi \left(D_{x,y,\phi }\left(x\right)\right)\mathrm{and}\phi (\Delta \left(y\right))=\phi \left(D_{x,y,\phi }\left(y\right)\right).$$

The motivation for this study arises from the fundamental role of inner derivations in key operator algebras, such as von Neumann and $C*$-algebras, where their structure is closely related to non-commutativity. Given this definition, a natural question emerges: must mappings that are locally indistinguishable from inner derivations by every continuous linear functional necessarily be derivations, or even inner derivations themselves? The concept of a weak 2-local inner derivation is introduced to address this question.

The principal result shows that every weak 2-local inner derivation on a semiprime ∗-Banach algebra with a faithful trace is a derivation. This result extends previous research by including non-self-adjoint operator algebras and provides a complete characterization on finite von Neumann algebras. To unify the study of such local mappings, a general framework of $(2,\mathcal{P})$-reflection and 2-reflexivity is introduced. Here, $\mathcal{P}$ denotes a point-separating family of seminorms and $\mathcal{S}$ a set of bounded linear operators. The $(2,\mathcal{P})$-reflection ${ref}_{(2,\mathcal{P})}\left(\mathcal{S}\right)$ comprises all maps that are $\mathcal{P}$-locally approximated at every pair of points by some operator in $\mathcal{S}$. If $\mathcal{S}$ coincides with its reflection, it is termed 2-reflexive. Weak 2-local inner derivations are connected to this framework, and it is established that they constitute a 2-reflexive set with respect to the natural seminorm structure. These findings extend and unify existing research. Earlier work established 2-reflexivity for all derivations on various algebras [8,9,10]. This work shows the same rigidity for the stricter class of weak 2-local inner derivations.

2. 2-Reflexivity

Let X be a Banach space, $X*$ the dual space of X (i.e., the space of all continuous linear functionals on X), and $\mathcal{P}$ a family of seminorms on X that separates points in X (i.e., for any nonzero $x\in X$, there exists $p\in \mathcal{P}$ such that $p\left(x\right)\ne 0$). Let $Map\left(X\right)=\{T:X\to X\}$ denote the set of all maps on X, and $B\left(X\right)=\{T\in Map(X)\mid T\mathrm{is}\mathrm{a}\mathrm{bounded}\mathrm{linear}\mathrm{operator}\}$.

Suppose $\mathcal{S}$ is a subset of $B\left(X\right)$. We define the $(2,\mathcal{P})$-reflection ${ref}_{(2,\mathcal{P})}\left(\mathcal{S}\right)$ of $\mathcal{S}$ as the set of all maps $T\in Map\left(X\right)$ satisfying the following condition: for every $x,y\in X$ and every $p\in \mathcal{P}$, there exists an operator $T_{x,y,p}\in \mathcal{S}$ (depending on x, y, and p) such that

$$p(T\left(x\right)-T_{x,y,p}\left(x\right))=0\mathrm{and}p(T\left(y\right)-T_{x,y,p}\left(y\right))=0.$$

One can verify that for any $T\in \mathcal{S}$, by choosing $T_{x,y,p}=T$, the above conditions are satisfied. Therefore, $\mathcal{S}\subseteq {ref}_{(2,\mathcal{P})}\left(\mathcal{S}\right)$.

Definition 1.

A mapping in ${ref}_{(2,\mathcal{P})}\left(\mathcal{S}\right)$ is called a $\mathcal{P}$-2-local mapping. If ${ref}_{(2,\mathcal{P})}\left(\mathcal{S}\right)=\mathcal{S}$, then $\mathcal{S}$ is said to be 2-reflexive with respect to $\mathcal{P}$.

A key question is whether the set $\mathcal{S}$ is always 2-reflexive for a given seminorm family $\mathcal{P}$. The following examples demonstrate how this property does not always hold.

Example 1.

Let $\mathcal{S}=B\left(X\right)$ and $\mathcal{P}=\{\parallel ·\parallel \}$ consists of a single element. Then $\mathcal{S}$ is not 2-reflexive with respect to $\mathcal{P}$.

Proof. 

Let $0\ne x_0\in X$ and $f_0\in X*$. Define the rank-one operator $x_0\otimes f_0$ by $(x_0\otimes f_0)\left(z\right)=f_0\left(z\right)x_0$ for all $z\in X$. Now, choose any homogeneous but non-additive mapping $T:X\to X$. Since T is not additive, it is not linear, and hence $T\notin \mathcal{S}$. We will show that $T\in {ref}_{(2,\mathcal{P})}\left(\mathcal{S}\right)$ by constructing, for any $x,y\in X$, an operator $T_{x,y,\parallel ·\parallel }\in \mathcal{S}$ such that $\parallel T\left(x\right)-T_{x,y,\parallel ·\parallel }\left(x\right)\parallel =0$ and $\parallel T\left(y\right)-T_{x,y,\parallel ·\parallel }\left(y\right)\parallel =0$. To illustrate this, we consider the following three cases.

Case 1: $x=0$ and $y\ne 0$. Take $f\in X*$ such that $f\left(x\right)=0$ and $f\left(y\right)=1$. Define $T_{x,y,\parallel ·\parallel }=T\left(y\right)\otimes f\in \mathcal{S}$. Then

$$\begin{array}{cc}\hfill T_{x,y,\parallel ·\parallel }\left(x\right)& =\left(T\right(y)\otimes f)\left(0\right)=0=T\left(0\right)=T\left(x\right),\hfill \\ \hfill T_{x,y,\parallel ·\parallel }\left(y\right)& =\left(T\right(y)\otimes f)\left(y\right)=f\left(y\right)T\left(y\right)=T\left(y\right).\hfill \end{array}$$

Case 2: x and y are linearly dependent and both nonzero. Then $x=\lambda y$ for some scalar λ. Take $f\in X*$ with $f\left(y\right)=1$, and define $T_{x,y,\parallel ·\parallel }=T\left(y\right)\otimes f\in \mathcal{S}$. Since T is homogeneous, $T\left(\lambda y\right)=\lambda T\left(y\right)$. Thus,

$$\begin{array}{cc}\hfill T_{x,y,\parallel ·\parallel }\left(x\right)& =\left(T\right(y)\otimes f)\left(\lambda y\right)=\lambda f\left(y\right)T\left(y\right)=\lambda T\left(y\right)=T\left(\lambda y\right)=T\left(x\right),\hfill \\ \hfill T_{x,y,\parallel ·\parallel }\left(y\right)& =\left(T\right(y)\otimes f)\left(y\right)=f\left(y\right)T\left(y\right)=T\left(y\right).\hfill \end{array}$$

Case 3: x and y are linearly independent. By the Hahn–Banach theorem, there exist $f_1,f_2\in X*$ such that $f_1\left(x\right)=1,f_1\left(y\right)=0$ and $f_2\left(x\right)=0,f_2\left(y\right)=1$. Define $T_{x,y,\parallel ·\parallel }=T\left(x\right)\otimes f_1+T\left(y\right)\otimes f_2\in \mathcal{S}$. Then

$$\begin{array}{cc}\hfill T_{x,y,\parallel ·\parallel }\left(x\right)& =(T\left(x\right)\otimes f_1)\left(x\right)+(T\left(y\right)\otimes f_2)\left(x\right)=f_1\left(x\right)T\left(x\right)+f_2\left(x\right)T\left(y\right)=T\left(x\right),\hfill \\ \hfill T_{x,y,\parallel ·\parallel }\left(y\right)& =(T\left(x\right)\otimes f_1)\left(y\right)+(T\left(y\right)\otimes f_2)\left(y\right)=f_1\left(y\right)T\left(x\right)+f_2\left(y\right)T\left(y\right)=T\left(y\right).\hfill \end{array}$$

In all cases, for any $x,y\in X$, we found an operator $T_{x,y,\parallel ·\parallel }\in \mathcal{S}$ such that the required conditions hold. This means $T\in {ref}_{(2,\mathcal{P})}\left(\mathcal{S}\right)$. However, since T is not linear, $T\notin \mathcal{S}$. We conclude that ${ref}_{(2,\mathcal{P})}\left(\mathcal{S}\right)\ne \mathcal{S}$, so $\mathcal{S}$ is not 2-reflexive with respect to $\mathcal{P}$.   □

Example 2.

Let $\mathcal{S}=B\left(X\right)$, where $X={\mathbb{R}}^2$ equipped with the Euclidean norm ${\parallel ·\parallel }_2$, and $\mathcal{P}=\left\{p\right\}$ with $p\left(x\right)={\parallel x\parallel }_2$. Then $\mathcal{S}$ is not 2-reflexive with respect to $\mathcal{P}$.

Proof. 

Define a mapping $T:{\mathbb{R}}^2\to {\mathbb{R}}^2$ by $T(x,y)=\left\{\begin{array}{cc}(x,y)\hfill & \mathrm{if}x\ne 0,\hfill \\ (0,0)\hfill & \mathrm{if}x=0.\hfill \end{array}\right.$ Take $x=(1,0)$ and $y=(0,1)$. Then

$$T\left(x\right)+T\left(y\right)=(1,0)+(0,0)=(1,0),\mathrm{but}T(x+y)=T(1,1)=(1,1)\ne (1,0).$$

Hence, T is nonlinear. We now show that for any $x,y\in {\mathbb{R}}^2$, there exists a linear operator $L\in B\left(X\right)$ such that $L\left(x\right)=T\left(x\right)$ and $L\left(y\right)=T\left(y\right)$. Consider the following cases:

Case 1: If both x and y are not on the y-axis (i.e., $x_1\ne 0$ and $y_1\ne 0$), take L to be the identity operator.

Case 2: If x is on the y-axis ($x_1=0$) and y is not, then $T\left(x\right)=(0,0)$ and $T\left(y\right)=(y_1,y_2)$. Define the linear operator $L(u,v)=\left(u,{\displaystyle \frac{y_2}{y_1}}u\right).$ Then $L\left(x\right)=(0,0)=T\left(x\right)$ and $L\left(y\right)=(y_1,y_2)=T\left(y\right)$.

Case 3: If x is not on the y-axis and y is, set $L(u,v)=\left(u,{\displaystyle \frac{x_2}{x_1}}u\right)$. Then $L\left(x\right)=(x_1,x_2)=T\left(x\right)$ and $L\left(y\right)=(0,0)=T\left(y\right)$.

Case 4: If both x and y are on the y-axis, take L to be the zero operator.

Thus, $T\in {ref}_{(2,\mathcal{P})}\left(\mathcal{S}\right)$. Since T is nonlinear, we conclude that ${ref}_{(2,\mathcal{P})}\left(\mathcal{S}\right)⊋\mathcal{S}$, and therefore $\mathcal{S}$ is not 2-reflexive with respect to $\mathcal{P}$.   □

Proposition 1.

Let $\mathcal{S}=B\left(X\right)$. Each $\mathcal{P}$-2-local mapping $T\in {ref}_{(2,\mathcal{P})}\left(\mathcal{S}\right)$ is homogeneous; namely, for any $x\in X$ and $\lambda \in \mathbb{C}$, we have $T\left(\lambda x\right)=\lambda T\left(x\right)$.

Proof. 

Fix an arbitrary $x\in X$ and $\lambda \in \mathbb{C}$, and define $y=\lambda x$. By definition, for every $p\in \mathcal{P}$, there exists an operator $T_{x,y,p}\in \mathcal{S}=B\left(X\right)$ (depending on x, y, and p) such that

$$p(T\left(x\right)-T_{x,y,p}\left(x\right))=0$$

(1)

and

$$p(T\left(y\right)-T_{x,y,p}\left(y\right))=0$$

(2)

Since $T_{x,y,p}$ is a bounded linear operator, it is homogeneous. Noting that $y=\lambda x$, we have $T_{x,y,p}\left(\lambda x\right)=T_{x,y,p}\left(y\right)=\lambda T_{x,y,p}\left(x\right).$ Substituting this into Equation (2) yields

$$p(T\left(\lambda x\right)-\lambda T_{x,y,p}\left(x\right))=0.$$

(3)

Using the triangle inequality, we have

$$\begin{array}{cc}\hfill p\left(T\right(\lambda x)-\lambda T(x\left)\right)& =p\left([T\left(\lambda x\right)-\lambda T_{x,y,p}\left(x\right)]+[\lambda T_{x,y,p}\left(x\right)-\lambda T\left(x\right)]\right)\hfill \\ & \le p(T\left(\lambda x\right)-\lambda T_{x,y,p}\left(x\right))+p(\lambda T_{x,y,p}\left(x\right)-\lambda T\left(x\right)).\hfill \end{array}$$

By Equation (3), the first term satisfies $p(T\left(\lambda x\right)-\lambda T_{x,y,p}\left(x\right))=0$. By Equation (1) and the homogeneity of the seminorm, the second term satisfies $p(\lambda T_{x,y,p}\left(x\right)-\lambda T\left(x\right))=\left|\lambda \right|p(T_{x,y,p}\left(x\right)-T\left(x\right))=0.$ Therefore, $p\left(T\right(\lambda x)-\lambda T(x\left)\right)\le 0$ for every $p\in \mathcal{P}$. As $\mathcal{P}$ separates points in X, we conclude that $T\left(\lambda x\right)-\lambda T\left(x\right)=0$, i.e., $T\left(\lambda x\right)=\lambda T\left(x\right)$. Since x and $\lambda$ were arbitrary, T is homogeneous.   □

When $X=\mathcal{A}$ is a Banach algebra, a $C*$-algebra, or a von Neumann algebra, it is often appropriate to consider $\mathcal{S}$ as a set of derivations or automorphisms (or isomorphisms) on $\mathcal{A}$. For each operator T in $\mathcal{S}$, a corresponding seminorm on $\mathcal{A}$ can be defined. Consequently, the following families of seminorms on $\mathcal{A}$ may be constructed:

$$\begin{array}{cc}\hfill {\mathcal{P}}_1& =\{p_{\varphi }:\varphi \in \mathcal{A}*\},\mathrm{where}{p}_{\varphi }\left(x\right)=\left|\varphi \left(x\right)\right|,\hfill \\ \hfill {\mathcal{P}}_2& =\{p_{\varphi }:\varphi \in \mathcal{A}*\mathrm{is}\mathrm{a}\mathrm{state}\},\hfill \\ \hfill {\mathcal{P}}_3& =\{p_{\varphi }:\varphi \in \mathcal{A}*\mathrm{is}\mathrm{a}\mathrm{normal}\mathrm{state}\},\hfill \\ \hfill {\mathcal{P}}_4& =\{p_{\varphi }:\varphi \in {\mathcal{A}}_{sa}^{*}\},\mathrm{where}{\mathcal{A}}_{sa}^{*}\mathrm{denotes}\mathrm{the}\mathrm{self}\text{-}\mathrm{adjoint}\mathrm{part}\mathrm{of}\mathcal{A}*.\hfill \end{array}$$

One can easily verify that ${\mathcal{P}}_3\subseteq {\mathcal{P}}_2\subseteq {\mathcal{P}}_1$. Note that ${\mathcal{P}}_1$, ${\mathcal{P}}_2$, and ${\mathcal{P}}_3$ separate points in $\mathcal{A}$, but ${\mathcal{P}}_4$ does not.

Proposition 2.

Let ${\mathcal{P}}_1$, ${\mathcal{P}}_2$ be two families of seminorms on $\mathcal{A}$ and suppose that ${\mathcal{P}}_1\subseteq {\mathcal{P}}_2$. If $\mathcal{S}$ is 2-reflexive with respect to ${\mathcal{P}}_1$, then $\mathcal{S}$ is 2-reflexive with respect to ${\mathcal{P}}_2$.

Proof. 

Assume ${\mathcal{P}}_1\subseteq {\mathcal{P}}_2$, one can first show that ${ref}_{(2,{\mathcal{P}}_2)}\left(\mathcal{S}\right)\subseteq {ref}_{(2,{\mathcal{P}}_1)}\left(\mathcal{S}\right)$. Since $\mathcal{S}$ is 2-reflexive with respect to ${\mathcal{P}}_1$, we have ${ref}_{(2,{\mathcal{P}}_1)}\left(\mathcal{S}\right)=\mathcal{S}$. Combining this with the trivial inclusion $\mathcal{S}\subseteq {ref}_{(2,{\mathcal{P}}_2)}\left(\mathcal{S}\right)$, we obtain

$$\mathcal{S}\subseteq {ref}_{(2,{\mathcal{P}}_2)}\left(\mathcal{S}\right)\subseteq {ref}_{(2,{\mathcal{P}}_1)}\left(\mathcal{S}\right)=\mathcal{S}.$$

Therefore, ${ref}_{(2,{\mathcal{P}}_2)}\left(\mathcal{S}\right)=\mathcal{S}$, which means $\mathcal{S}$ is 2-reflexive with respect to ${\mathcal{P}}_2$.   □

The preceding proposition demonstrates that 2-reflexivity is maintained when the family of seminorms is expanded. A natural question is whether the converse holds. To investigate, we ask: if $\mathcal{S}$ is 2-reflexive with respect to the larger family ${\mathcal{P}}_2$, does this imply 2-reflexivity with respect to the smaller family ${\mathcal{P}}_1$? The following example shows that, in general, this implication does not hold.

Example 3.

Let $X={\mathbb{C}}^2$, $\mathcal{S}=B\left(X\right)$. Define two families of seminorms:

$$\begin{array}{cc}\hfill {\mathcal{P}}_1& =\left\{p_1\right\},\mathit{where}{p}_1(x,y)=\left|x\right|,\hfill \\ \hfill {\mathcal{P}}_2& =\{p_1,p_2\},\mathit{where}{p}_2(x,y)=\left|y\right|.\hfill \end{array}$$

Then ${\mathcal{P}}_1\subseteq {\mathcal{P}}_2$. We claim that $\mathcal{S}$ is 2-reflexive with respect to ${\mathcal{P}}_2$ but not with respect to ${\mathcal{P}}_1$.

Proof. 

Let $T\in {ref}_{(2,{\mathcal{P}}_2)}\left(\mathcal{S}\right)$. We show that T is linear. Since $X={\mathbb{C}}^2$ is finite-dimensional and ${\mathcal{P}}_2=\{p_1,p_2\}$ separates points, it suffices to show that for every $x,y\in X$, there exists a linear operator $T_{x,y}\in \mathcal{S}$ such that $T\left(x\right)=T_{x,y}\left(x\right)$ and $T\left(y\right)=T_{x,y}\left(y\right)$.

Fix $x,y\in X$. For each seminorm $p\in {\mathcal{P}}_2$, there exists a linear operator $T_{x,y,p}\in \mathcal{S}$ such that

$$p(T\left(x\right)-T_{x,y,p}\left(x\right))=0\mathrm{and}p(T\left(y\right)-T_{x,y,p}\left(y\right))=0.$$

This means that for $p_1$, the first coordinates of $T\left(x\right)$ and $T\left(y\right)$ match those of $T_{x,y,p_1}\left(x\right)$ and $T_{x,y,p_1}\left(y\right)$, respectively. Similarly, for $p_2$, the second coordinates match those of $T_{x,y,p_2}\left(x\right)$ and $T_{x,y,p_2}\left(y\right)$. Now, define a linear operator $T_{x,y}$ as follows.

Case 1: If x and y are linearly independent, they form a basis for X. Define $T_{x,y}$ by $T_{x,y}\left(x\right)=T\left(x\right)$ and $T_{x,y}\left(y\right)=T\left(y\right)$. This is well-defined and linear.

Case 2: If x and y are linearly dependent, then $y=\alpha x$ for some $\alpha \in \mathbb{C}$. We first prove homogeneity. For any $z\in X$ and $\beta \in \mathbb{C}$, consider the pair $(z,\beta z)$. For each $p\in {\mathcal{P}}_2$, there exists $T_{z,\beta z,p}$ with $p(T\left(z\right)-T_{z,\beta z,p}\left(z\right))=0$ and $p(T\left(\beta z\right)-T_{z,\beta z,p}\left(\beta z\right))=0$. Since $T_{z,\beta z,p}$ is linear, $T_{z,\beta z,p}\left(\beta z\right)=\beta T_{z,\beta z,p}\left(z\right)$, so $p\left(T\right(\beta z)-\beta T(z\left)\right)=0$ for all p, hence $T\left(\beta z\right)=\beta T\left(z\right)$ by point separation. Thus, T is homogeneous. In particular, $T\left(y\right)=T\left(\alpha x\right)=\alpha T\left(x\right)$. Then any linear operator $T^{\prime }$ with $T^{\prime }\left(x\right)=T\left(x\right)$ will satisfy $T^{\prime }\left(y\right)=\alpha T\left(x\right)=T\left(y\right)$.

Therefore, for all $x,y\in X$, there exists a linear operator $T_{x,y}$ with $T\left(x\right)=T_{x,y}\left(x\right)$ and $T\left(y\right)=T_{x,y}\left(y\right)$, it follows that T is linear. Hence, ${ref}_{(2,{\mathcal{P}}_2)}\left(\mathcal{S}\right)=\mathcal{S}$, so $\mathcal{S}$ is 2-reflexive with respect to ${\mathcal{P}}_2$.

Define a map $T:X\to X$ by $T(x,y)=(x,y^2)$. Then T is not linear (e.g., $T(0,2)=(0,4)\ne 2·T(0,1)=(0,2)$), so $T\notin \mathcal{S}$. Now, we show $T\in {ref}_{(2,{\mathcal{P}}_1)}\left(\mathcal{S}\right)$. Let $x=(x_1,x_2),y=(y_1,y_2)\in X$ and $p_1\in {\mathcal{P}}_1$. Define $T_{x,y,p_1}(a,b)=(a,0)$, which is linear. Then

$$\begin{array}{cc}\hfill T\left(x\right)-T_{x,y,p_1}\left(x\right)& =(x_1,x_2^2)-(x_1,0)=(0,x_2^2),\hfill \\ \hfill T\left(y\right)-T_{x,y,p_1}\left(y\right)& =(y_1,y_2^2)-(y_1,0)=(0,y_2^2),\hfill \end{array}$$

so $p_1(T\left(x\right)-T_{x,y,p_1}\left(x\right))=\left|0\right|=0$ and $p_1(T\left(y\right)-T_{x,y,p_1}\left(y\right))=0$. Thus, $T\in {ref}_{(2,{\mathcal{P}}_1)}\left(\mathcal{S}\right)$. Hence, $\mathcal{S}$ is not 2-reflexive with respect to ${\mathcal{P}}_1$. □

3. Weak 2-Local Inner Derivations

Let $\mathcal{A}$ denote a complex ∗-algebra. A linear functional $\tau :\mathcal{A}\to \mathbb{C}$ is defined as a trace if it is positive, meaning $\tau \left(x*x\right)\ge 0$ for all $x\in \mathcal{A}$, and satisfies the tracial property $\tau \left(xy\right)=\tau \left(yx\right)$ for all $x,y\in \mathcal{A}$. A trace $\tau$ is considered faithful if $\tau \left(x*x\right)=0$ implies $x=0$.

Assume that $\mathcal{A}$ is a semiprime unital ∗-Banach algebra endowed with a faithful continuous trace $\tau$, and we discuss some of its properties.

Remark 1.

The family of self-adjoint linear functionals ${\mathcal{A}}_{sa}^{*}$ separates points in $\mathcal{A}$.

In fact, let τ be a faithful trace on $\mathcal{A}$. For an arbitrary element $a\in \mathcal{A}$, define the linear functional ${\phi }_a:\mathcal{A}\to \mathbb{C}$ by ${\phi }_a\left(x\right)=\tau \left(ax\right)$. We show that if a is self-adjoint, then ${\phi }_a$ is self-adjoint, i.e., ${\phi }_a\left(x*\right)=\overline{{\phi }_a\left(x\right)}$ for all $x\in \mathcal{A}$. Indeed,

$${\phi }_a\left(x*\right)=\tau \left(ax*\right)=\overline{\tau \left({\left(ax*\right)}^{*}\right)}=\overline{\tau \left(xa*\right)}=\overline{{\phi }_{a*}\left(x\right)}.$$

If $a^{*}=a$, then ${\phi }_{a*}\left(x\right)={\phi }_a\left(x\right)$, so ${\phi }_a\left(x*\right)=\overline{{\phi }_a\left(x\right)}$, confirming that ${\phi }_a\in {\mathcal{A}}_{sa}^{*}$.

Now, let $x\in \mathcal{A}$ be nonzero. Since $\mathcal{A}$ is semiprime and τ is faithful, we have $x*x\ne 0$ and $\tau \left(x*x\right)>0$. Decompose x into its self-adjoint and skew-adjoint parts as $x=x_1+i{x}_2$, where $x_1={\displaystyle \frac{x+x*}{2}}$ and $x_2={\displaystyle \frac{x-x*}{2}}$ are self-adjoint. Since $x\ne 0$, at least one of $x_1$ or $x_2$ is nonzero. We consider two cases.

If $x_1\ne 0$, take $a=x_1$. Then

$${\phi }_{x_1}\left(x\right)=\tau \left(x_{1}x\right)=\tau \left(x_1(x_1+i{x}_2)\right)=\tau \left(x_1^2\right)+i\tau \left(x_1{x}_2\right).$$

Since $x_1$ is self-adjoint and nonzero, $x_1^2$ is positive and nonzero, so $\tau \left(x_1^2\right)>0$ by faithfulness of τ. Thus, the real part of ${\phi }_{x_1}\left(x\right)$ is positive, implying ${\phi }_{x_1}\left(x\right)\ne 0$.

If $x_1=0$, then $x_2\ne 0$ and $x=i{x}_2$. Take $a=x_2$. Then

$${\phi }_{x_2}\left(x\right)=\tau \left(x_{2}x\right)=\tau \left(x_2\left(i{x}_2\right)\right)=i\tau \left(x_2^2\right).$$

Since $x_2$ is self-adjoint and nonzero, $x_2^2$ is positive and nonzero, so $\tau \left(x_2^2\right)>0$. Thus, ${\phi }_{x_2}\left(x\right)=i\tau \left(x_2^2\right)\ne 0$.

In both cases, we have found a self-adjoint element $a\in \mathcal{A}$ such that ${\phi }_a\in {\mathcal{A}}_{sa}^{*}$ and ${\phi }_a\left(x\right)\ne 0$. Therefore, ${\mathcal{A}}_{sa}^{*}$ separates points in $\mathcal{A}$.

Remark 2.

Every weak 2-local inner derivation Δ on $\mathcal{A}$ vanishes on the center $Z\left(\mathcal{A}\right)$. Specifically, for any central element $z\in Z\left(\mathcal{A}\right)$ and $\phi \in \mathcal{A}*$, the weak 2-local property ensures that $\phi (\Delta (z\left)\right)=\phi \left(\right[a,z\left]\right)$ for some $a\in \mathcal{A}$. Because z is central, $[a,z]=0$, so $\phi (\Delta (z\left)\right)=0$ for all φ. Therefore, $\Delta \left(z\right)=0$.

Lemma 1.

Every weak 2-local inner derivation on $\mathcal{A}$ is linear.

Proof. 

Let $\Delta$ be a weak 2-local inner derivation on $\mathcal{A}$, and let $\tau$ represent a faithful trace on $\mathcal{A}$. Let $x\in \mathcal{A}$ and $\lambda \in \mathbb{C}$ be arbitrary. For any $\phi \in {\mathcal{A}}_{sa}^{*}$, by the definition of weak 2-local derivation, there exists an inner derivation $D_{x,\lambda x,\phi }$ such that

$$\phi (\Delta \left(x\right))=\phi \left(D_{x,\lambda x,\phi }\left(x\right)\right),\phi (\Delta \left(\lambda x\right))=\phi \left(D_{x,\lambda x,\phi }\left(\lambda x\right)\right).$$

Since $D_{x,\lambda x,\phi }$ is linear, we have $D_{x,\lambda x,\phi }\left(\lambda x\right)=\lambda D_{x,\lambda x,\phi }\left(x\right)$. Thus,

$$\phi (\Delta \left(\lambda x\right))=\phi \left(\lambda D_{x,\lambda x,\phi }\left(x\right)\right)=\lambda \phi \left(D_{x,\lambda x,\phi }\left(x\right)\right)=\lambda \phi (\Delta \left(x\right))=\phi (\lambda \Delta \left(x\right)).$$

Therefore, $\phi (\Delta (\lambda x)-\lambda \Delta (x\left)\right)=0$ for every $\phi \in {\mathcal{A}}_{sa}^{*}$. By Remark 1, ${\mathcal{A}}_{sa}^{*}$ separates points, so $\Delta \left(\lambda x\right)=\lambda \Delta \left(x\right)$. Hence, every weak 2-local inner derivation $\Delta$ on $\mathcal{A}$ is homogeneous.

For any $x\in {\mathcal{A}}_{sa}$, define ${\phi }_x\left(a\right)=\tau \left(xa\right)\in {\mathcal{A}}_{sa}^{*}$. By the definition of weak 2-local derivation, for the pair $(x,x)$ and ${\phi }_x$, there exists $T_x\in \mathcal{A}$ such that ${\phi }_x(\Delta \left(x\right))={\phi }_x\left([T_x,x]\right).$ Thus, $\tau (x\Delta \left(x\right))=\tau \left(x[T_x,x]\right)=\tau \left(x(T_{x}x-x{T}_x)\right)=\tau (x{T}_{x}x-x^2{T}_x).$ Using the tracial property, $\tau \left(x{T}_{x}x\right)=\tau \left(T_x{x}^2\right)$ and $\tau \left(x^2{T}_x\right)=\tau \left(T_x{x}^2\right)$, so $\tau (x\Delta \left(x\right))=\tau (T_x{x}^2-T_x{x}^2)=0.$ Hence, $\tau (x\Delta (x\left)\right)=0$ for all $x\in {\mathcal{A}}_{sa}$.

Consider the pair $(y,x+y)$ and the functional ${\phi }_x$. There exists $T_{x,x+y}\in \mathcal{A}$ (depending on x and $x+y$) such that ${\phi }_x(\Delta \left(y\right))={\phi }_x\left([T_{x,x+y},y]\right)$ and ${\phi }_x(\Delta (x+y))={\phi }_x\left([T_{x,x+y},x+y]\right).$ Then, $\tau (x\Delta \left(y\right))=\tau \left(x[T_{x,x+y},y]\right)=\tau \left(x(T_{x,x+y}y-y{T}_{x,x+y})\right)=\tau (x{T}_{x,x+y}y-xy{T}_{x,x+y}),$ and

$$\begin{array}{cc}\hfill \tau (x\Delta (x+y\left)\right)& =\tau \left(x[T_{x,x+y},x+y]\right)\hfill \\ & =\tau \left(x(T_{x,x+y}(x+y)-(x+y)T_{x,x+y})\right)\hfill \\ & =\tau (x{T}_{x,x+y}x+x{T}_{x,x+y}y-x^2{T}_{x,x+y}-xy{T}_{x,x+y}).\hfill \end{array}$$

Note that $\tau (x{T}_{x,x+y}x-x^2{T}_{x,x+y})=0$, so

$$\tau (x\Delta (x+y))=\tau (x{T}_{x,x+y}y-xy{T}_{x,x+y})=\tau (x\Delta \left(y\right)).$$

(4)

Similarly, consider the pair $(x,x+y)$ and the functional ${\phi }_y$. There exists $S\in \mathcal{A}$ (depending on x and $x+y$) such that ${\phi }_y(\Delta \left(x\right))={\phi }_y\left([S,x]\right)$ and ${\phi }_y(\Delta (x+y))={\phi }_y\left([S,x+y]\right).$ Then, $\tau (y\Delta (x\left)\right)=\tau \left(y\right[S,x\left]\right)=\tau \left(y\right(Sx-xS\left)\right)=\tau (ySx-yxS),$ and

$$\begin{array}{cc}\hfill \tau (y\Delta (x+y\left)\right)& =\tau \left(y\right[S,x+y\left]\right)\hfill \\ & =\tau \left(y\right(S(x+y)-(x+y)S\left)\right)\hfill \\ & =\tau (ySx+ySy-yxS-y^{2}S).\hfill \end{array}$$

Note that $\tau (ySy-y^{2}S)=0$, so

$$\tau (y\Delta (x+y\left)\right)=\tau (ySx-yxS)=\tau (y\Delta (x\left)\right).$$

(5)

Now, consider the pair $(x+y,x+y)$ and the functional ${\phi }_{x+y}$. By above, we have $\tau \left(\right(x+y)\Delta (x+y\left)\right)=0.$ That is, $\tau (x\Delta (x+y\left)\right)+\tau (y\Delta (x+y\left)\right)=0.$ Substituting (4) and (5) into this equation, we get $\tau (x\Delta (y\left)\right)+\tau (y\Delta (x\left)\right)=0.$ Since $\tau (y\Delta (x\left)\right)=\tau (\Delta (x\left)y\right)$ by the tracial property, we obtain $\tau (x\Delta (y\left)\right)=-\tau (\Delta (x\left)y\right).$

Now we prove that for any $u,v\in {\mathcal{A}}_{sa}$, $\Delta (u+v)=\Delta \left(u\right)+\Delta \left(v\right)$. For any $x\in {\mathcal{A}}_{sa}$, we have

$$\begin{array}{cc}\hfill \tau (x\Delta (u+v\left)\right)& =-\tau (\Delta (x\left)\right(u+v\left)\right)\hfill \\ & =-\tau (\Delta (x\left)u\right)-\tau (\Delta (x\left)v\right)\hfill \\ & =\tau (x\Delta (u\left)\right)+\tau (x\Delta (v\left)\right)\hfill \\ & =\tau \left(x\right(\Delta \left(u\right)+\Delta \left(v\right)\left)\right).\hfill \end{array}$$

Thus,

$$\tau \left(x(\Delta (u+v)-\Delta \left(u\right)-\Delta \left(v\right))\right)=0\mathrm{for}\mathrm{all}x\in {\mathcal{A}}_{sa}.$$

Since $\tau$ is faithful and the family $\{{\phi }_x:x\in {\mathcal{A}}_{sa}\}$ separates points in $\mathcal{A}$ by Remark 1, it follows that $\Delta (u+v)-\Delta \left(u\right)-\Delta \left(v\right)=0,$ i.e., $\Delta (u+v)=\Delta \left(u\right)+\Delta \left(v\right)$ for all $u,v\in {\mathcal{A}}_{sa}$.

Finally, since $\Delta$ is homogeneous and additive on ${\mathcal{A}}_{sa}$, it follows that $\Delta$ is linear on ${\mathcal{A}}_{sa}$. As every element of $\mathcal{A}$ can be written as a linear combination of self-adjoint elements, $\Delta$ is linear on $\mathcal{A}$.   □

Lemma 2.

Every weak 2-local inner derivation Δ on $\mathcal{A}$ is norm continuous.

Proof. 

Let $\left\{x_n\right\}\subset \mathcal{A}$ be a sequence such that $x_n\stackrel{\parallel ·\parallel }{\to }x$ and $\Delta \left(x_n\right)\stackrel{\parallel ·\parallel }{\to }y$ for some $x,y\in \mathcal{A}$. To show that $\Delta$ is closed, we must prove that $y=\Delta \left(x\right)$. By Lemma 1, we have the identity $\tau (\Delta (z\left)a\right)=-\tau (z\Delta (a\left)\right)$ for all $z,a\in \mathcal{A}.$

Since $\Delta \left(x_n\right)\stackrel{\parallel ·\parallel }{\to }y$ and multiplication is continuous, we have $\Delta \left(x_n\right)a\stackrel{\parallel ·\parallel }{\to }ya$ for any $a\in \mathcal{A}$. By the continuity of $\tau$, it follows that $\tau (\Delta \left(x_n\right)a)\to \tau \left(ya\right).$ Applying the identity with $z=x_n$, we get $\tau (\Delta \left(x_n\right)a)=-\tau (x_n\Delta \left(a\right)).$ As $x_n\stackrel{\parallel ·\parallel }{\to }x$, we obtain $-\tau (x_n\Delta \left(a\right))\to -\tau (x\Delta \left(a\right)).$ But by the same identity, $-\tau (x\Delta (a\left)\right)=\tau (\Delta (x\left)a\right)$. Therefore,

$$\tau \left(ya\right)=\underset{n\to \infty }{lim}\tau (\Delta \left(x_n\right)a)=\underset{n\to \infty }{lim}-\tau (x_n\Delta \left(a\right))=-\tau (x\Delta \left(a\right))=\tau (\Delta \left(x\right)a).$$

Thus, $\tau \left(\right(y-\Delta \left(x\right)\left)a\right)=0$ for all $a\in \mathcal{A}$. In particular, taking $a=(y-\Delta \left(x\right))*$, we get $\tau ((y-\Delta \left(x\right))(y-\Delta \left(x\right))*)=0.$ Since $\tau$ is faithful, this implies $y-\Delta \left(x\right)=0$, i.e., $y=\Delta \left(x\right)$. Therefore, $\Delta$ is a closed operator. Since $\Delta$ is linear by Lemma 1, the closed graph theorem implies that $\Delta$ is bounded, hence norm continuous.   □

Remark 3.

Both linearity and norm continuity are necessary, but not sufficient, conditions for a mapping to qualify as a weak 2-local inner derivation. This is illustrated by a linear, continuous counterexample that does not satisfy the criteria for a weak 2-local inner derivation.

Consider the map $\Delta :\mathcal{A}\to \mathcal{A}$ defined by $\Delta \left(x\right)=\tau \left(x\right)·1_{\mathcal{A}}$ on a finite von Neumann algebra $\mathcal{A}$ with a faithful normal trace τ (normalized such that $\tau \left(1_{\mathcal{A}}\right)=1$). For any $x,y\in \mathcal{A}$ and $\lambda \in \mathbb{C}$,

$$\Delta (x+y)=\tau (x+y)·1_{\mathcal{A}}=(\tau \left(x\right)+\tau \left(y\right))·1_{\mathcal{A}}=\Delta \left(x\right)+\Delta \left(y\right),$$

and

$$\Delta \left(\lambda x\right)=\tau \left(\lambda x\right)·1_{\mathcal{A}}=\lambda \tau \left(x\right)·1_{\mathcal{A}}=\lambda \Delta \left(x\right).$$

Thus, Δ is linear. Since τ is a normal trace on a von Neumann algebra, it is bounded, and hence norm continuous.

Suppose for contradiction that Δ is a weak 2-local inner derivation. Then for $u=1_{\mathcal{A}}$, $v=0$, and for any $\phi \in \mathcal{A}*$, there exists an inner derivation $D_a$ such that $\phi (\Delta \left(1_{\mathcal{A}}\right))=\phi \left(D_a\left(1_{\mathcal{A}}\right)\right)$ and $\phi (\Delta \left(0\right))=\phi \left(D_a\left(0\right)\right).$ By direct computation, we have

$$\Delta \left(1_{\mathcal{A}}\right)=\tau \left(1_{\mathcal{A}}\right)·1_{\mathcal{A}}=1_{\mathcal{A}},\Delta \left(0\right)=0,D_a\left(1_{\mathcal{A}}\right)=0,D_a\left(0\right)=0.$$

Substituting these yields $\phi \left(1_{\mathcal{A}}\right)=\phi \left(0\right)=0$ for all $\phi \in \mathcal{A}*.$ Since $\mathcal{A}*$ separates points, this implies $1_{\mathcal{A}}=0$, a contradiction.

Therefore, Δ is both linear and continuous, but it does not constitute a weak 2-local inner derivation. This result demonstrates that linearity and continuity, whether individually or combined, do not ensure that a mapping is a weak 2-local inner derivation.

Lemma 3.

Every weak 2-local inner derivation Δ on $\mathcal{A}$ satisfies $\Delta (1-x)+\Delta \left(x\right)=0$ for all $x\in \mathcal{A}$.

Proof. 

Let $x\in \mathcal{A}$ and $\phi \in {\mathcal{A}}_{sa}^{*}$ be arbitrary. By the definition of a weak 2-local inner derivation, there exists an inner derivation $D_{x,1-x,\phi }$ (depending on x, $1-x$, and $\phi$) such that

$$\phi (\Delta \left(x\right))=\phi \left(D_{x,1-x,\phi }\left(x\right)\right)\mathrm{and}\phi (\Delta (1-x))=\phi \left(D_{x,1-x,\phi }(1-x)\right).$$

Since $D_{x,1-x,\phi }$ is an inner derivation, it is linear and satisfies $D_{x,1-x,\phi }\left(1\right)=0$ because any derivation annihilates the unit. Therefore,

$$\phi (\Delta (1-x)+\Delta \left(x\right))=\phi (D_{x,1-x,\phi }(1-x)+D_{x,1-x,\phi }\left(x\right))=\phi \left(D_{x,1-x,\phi }\left(1\right)\right)=0.$$

This holds for all $\phi \in {\mathcal{A}}_{sa}^{*}$. Since ${\mathcal{A}}_{sa}^{*}$ separates points in $\mathcal{A}$, we conclude that $\Delta (1-x)+\Delta \left(x\right)=0$.   □

Proposition 3.

Eevery weak 2-local inner derivation Δ on $\mathcal{A}$ satisfies $\tau (\Delta (x\left)\right)=0$ for all $x\in \mathcal{A}$.

Proof. 

Let $x\in \mathcal{A}$ be arbitrary. By the definition of a weak 2-local inner derivation, for the pair $(x,1)$ and for any $\phi \in \mathcal{A}*$, there exists an inner derivation $D_{x,1,\phi }$ (implemented by some $a\in \mathcal{A}$ depending on x, 1, and $\phi$) such that

$$\phi (\Delta \left(x\right))=\phi \left(D_{x,1,\phi }\left(x\right)\right)\mathrm{and}\phi (\Delta \left(1\right))=\phi \left(D_{x,1,\phi }\left(1\right)\right).$$

From Lemma 3, we have $\Delta \left(1\right)=0$. Therefore, the second equality becomes $\phi (\Delta (1\left)\right)=\phi \left(0\right)=0,$ which is consistent. The first equality gives

$$\phi (\Delta (x\left)\right)=\phi \left(\right[a,x\left]\right)=\phi (ax-xa).$$

Since $\tau$ is a continuous trace, it belongs to $\mathcal{A}*$. Applying the above with $\phi =\tau$, we have

$$\tau (\Delta (x\left)\right)=\tau (ax-xa)=\tau \left(ax\right)-\tau \left(xa\right)=0.$$

Hence, $\tau (\Delta (x\left)\right)=0$ for all $x\in \mathcal{A}$.   □

Lemma 4.

Every weak 2-local inner derivation Δ on $\mathcal{A}$ satisfies $\Delta \left(P\right)=\Delta \left(P\right)P+P\Delta \left(P\right)$ for every projection $P\in \mathcal{A}$.

Proof. 

Let $P\in \mathcal{A}$ be a projection. Take any $\phi \in {\mathcal{A}}_{sa}^{*}$ and define $f\in {\mathcal{A}}_{sa}^{*}$ by $f\left(a\right)=\phi \left(\right(I-P\left)a\right(I-P\left)\right)$ for all $a\in \mathcal{A}$. By the weak 2-local property, there exists an inner derivation $D_{P,f}$ such that $f(\Delta \left(P\right))=f\left(D_{P,f}\left(P\right)\right).$ Since $D_{P,f}$ is an inner derivation and P is a projection, we have $D_{P,f}\left(P\right)=D_{P,f}\left(P\right)P+P{D}_{P,f}\left(P\right).$ Then

$$\begin{array}{cc}\hfill f\left(D_{P,f}\left(P\right)\right)& =\phi \left((I-P)(D_{P,f}\left(P\right)P+P{D}_{P,f}\left(P\right))(I-P)\right)\hfill \\ & =\phi ((I-P)D_{P,f}\left(P\right)P(I-P)+(I-P)P{D}_{P,f}\left(P\right)(I-P))\hfill \\ & =0,\hfill \end{array}$$

because $P(I-P)=0$ and $(I-P)P=0$. Thus, $f(\Delta (P\left)\right)=0$, which implies $\phi \left(\right(I-P)\Delta (P\left)\right(I-P\left)\right)=0$ for all $\phi \in {\mathcal{A}}_{sa}^{*}$. By the separation property of ${\mathcal{A}}_{sa}^{*}$, we conclude that $(I-P)\Delta \left(P\right)(I-P)=0$.

Replace P with $I-P$ in the preceding argument (noting that $I-P$ is also a projection), we have $(I-(I-P\left)\right)\Delta (I-P)(I-(I-P\left)\right)=P\Delta (I-P)P=0.$ From Lemma 3, we have $\Delta (I-P)=-\Delta \left(P\right)$, whence $P\Delta (I-P)P=-P\Delta \left(P\right)P=0,$ and therefore $P\Delta \left(P\right)P=0$.

Consider the canonical Peirce decomposition of $\Delta \left(P\right)$ with respect to the projection P, $\Delta \left(P\right)=P\Delta \left(P\right)P+P\Delta \left(P\right)(I-P)+(I-P)\Delta \left(P\right)P+(I-P)\Delta \left(P\right)(I-P).$ Then we have $P\Delta \left(P\right)P=0$ and $(I-P)\Delta \left(P\right)(I-P)=0$, which simplifies the expression $\Delta \left(P\right)=P\Delta \left(P\right)(I-P)+(I-P)\Delta \left(P\right)P.$ Note that $P\Delta \left(P\right)(I-P)=P\Delta \left(P\right)-P\Delta \left(P\right)P=P\Delta \left(P\right),$ and similarly, $(I-P)\Delta \left(P\right)P=\Delta \left(P\right)P-P\Delta \left(P\right)P=\Delta \left(P\right)P.$ Therefore $\Delta \left(P\right)=P\Delta \left(P\right)+\Delta \left(P\right)P.$   □

Lemma 5

([15]). Let $\mathcal{A}$ be a semiprime ∗-Banach algebra. Denote by $D_{\mathcal{A}}$ the set of elements in $\mathcal{A}$ that can be represented as finite real-linear combinations of mutually orthogonal projections. Assume that $D_{\mathcal{A}}$ is dense in ${\mathcal{A}}_{sa}$. Let $\delta :\mathcal{A}\to \mathcal{A}$ be a continuous linear map. If δ satisfies $\delta \left(P\right)=\delta \left(P\right)P+P\delta \left(P\right)$ for every projection $P\in \mathcal{A}$, then δ is a derivation.

Combining above lemmas, we can obtain the following result.

Theorem 1.

Let $\mathcal{A}$ be a semiprime ∗-Banach algebra endowed with a faithful trace τ satisfying $D_{\mathcal{A}}$ is dense in ${\mathcal{A}}_{sa}$. Then every weak 2-local inner derivation on $\mathcal{A}$ is a derivation.

In particular, if $\mathcal{A}$ is a finite von Neumann algebra, then every weak 2-local inner derivation on $\mathcal{A}$ is a derivation.

Remark 4.

Every $C*$-algebra is semiprime. To focus our discussion, consider $C*$-algebras with real rank zero, such as approximately finite-dimensional (AF) algebras, that also have a faithful trace. For these particular algebras, the set $D_{\mathcal{A}}$-defined as the set of finite real-linear combinations of mutually orthogonal projections-is dense in ${\mathcal{A}}_{sa}$. Consequently, by Theorem 1, every weak 2-local inner derivation on such algebras is a derivation.

Note that, by classical results in [16,17], it is readily verified that for any mapping $\Delta :\mathcal{A}\to \mathcal{A}$ on a finite von Neumann algebra $\mathcal{A}$, we have the following equivalences in our setting:

$$\mathrm{Inner}⟺\mathrm{Derivation}⟺\mathrm{Jordan}\mathrm{derivation}⟺\mathrm{Weak}2\text{-}\mathrm{local}\mathrm{inner}\mathrm{derivation}.$$

Corollary 1.

Let $\mathcal{M}$ be a finite von Neumann algebra, and let $\mathcal{I}$ be a σ-weakly closed two-sided ideal of $\mathcal{M}$. If $\Delta :\mathcal{I}\to \mathcal{I}$ is a weak 2-local inner derivation, then there exists an element $a\in \mathcal{I}$ such that the map $\tilde{\Delta }:\mathcal{M}\to \mathcal{M}$ defined by $\tilde{\Delta }\left(x\right)=[a,x]$ is an inner derivation that extends Δ.

Proof. 

Since $\mathcal{I}$ is a σ-weakly closed ideal of the finite von Neumann algebra $\mathcal{M}$, by the structure theorem, there exists a central projection $e\in \mathcal{M}$ such that $\mathcal{I}=e\mathcal{M}$. In particular, I is a finite von Neumann algebra with unit e.

The restriction of the faithful trace to $\mathcal{I}$ remains faithful. By Theorem 1, Δ is a derivation on $\mathcal{I}$. Since $\mathcal{I}$ is a finite von Neumann algebra, every derivation is inner. Thus, there exists $a\in \mathcal{I}$ such that $\Delta \left(x\right)=[a,x]$ for all $x\in \mathcal{I}$.

Define $\tilde{\Delta }\left(x\right)=[a,x]$ for all $x\in \mathcal{M}$. This is clearly an inner derivation on $\mathcal{M}$ that extends Δ.   □

Corollary 2.

Let $\mathcal{A}$ be a finite von Neumann algebra. Then the set $\mathcal{S}$ of weak 2-local inner derivations forms a Lie algebra that is isomorphic to the quotient Lie algebra $\mathcal{A}/\mathcal{Z}\left(\mathcal{A}\right)$, where $\mathcal{Z}\left(\mathcal{A}\right)$ denotes the center of $\mathcal{A}$.

More precisely, the map

$$\mathsf{\Psi }:\mathcal{S}\to \mathcal{A}/\mathcal{Z}\left(\mathcal{A}\right),\Delta \mapsto a+\mathcal{Z}\left(\mathcal{A}\right)$$

where $\Delta \left(x\right)=[a,x]$ for all $x\in \mathcal{A}$, is a well-defined Lie algebra isomorphism.

Proof. 

Let $\mathrm{Inn}\left(\mathcal{A}\right)$ denote the set of inner derivations on $\mathcal{A}$. By Theorem 1, we have $\mathcal{S}=\mathrm{Inn}\left(\mathcal{A}\right)$. An elementary verification shows that $\mathrm{Inn}\left(\mathcal{A}\right)$ forms a Lie algebra under the commutator bracket $[{\Delta }_1,{\Delta }_2]={\Delta }_1{\Delta }_2-{\Delta }_2{\Delta }_1$. Specifically, for ${\Delta }_a\left(x\right)=[a,x]$ and ${\Delta }_b\left(x\right)=[b,x]$, we have ${\Delta }_a+{\Delta }_b={\Delta }_{a+b}$, $\lambda {\Delta }_a={\Delta }_{\lambda a}$ for $\lambda \in \mathbb{C}$ and $[{\Delta }_a,{\Delta }_b]={\Delta }_{[a,b]}$ by the Jacobi identity

Define $\mathsf{\Psi }:\mathrm{Inn}\left(\mathcal{A}\right)\to \mathcal{A}/\mathcal{Z}\left(\mathcal{A}\right)$ by $\mathsf{\Psi }(\Delta )=a+\mathcal{Z}\left(\mathcal{A}\right)$, where $\Delta \left(x\right)=[a,x]$ for all $x\in \mathcal{A}$. If $\Delta \left(x\right)=[a,x]=[b,x]$ for all $x\in \mathcal{A}$, then $a-b\in \mathcal{Z}\left(\mathcal{A}\right)$, so $a+\mathcal{Z}\left(\mathcal{A}\right)=b+\mathcal{Z}\left(\mathcal{A}\right)$. Thus Ψ is well-defined. An easy check shows that Ψ is linear. Note that the Lie bracket on $\mathcal{A}/\mathcal{Z}\left(\mathcal{A}\right)$ is well-defined by $[x+\mathcal{Z}(\mathcal{A}),y+\mathcal{Z}(\mathcal{A}\left)\right]=[x,y]+\mathcal{Z}\left(\mathcal{A}\right).$ This is independent of representatives since for $z_1,z_2\in \mathcal{Z}\left(\mathcal{A}\right)$, we have $[x+z_1,y+z_2]=[x,y]+[x,z_2]+[z_1,y]+[z_1,z_2]=[x,y]$. Then

$$\begin{array}{cc}\hfill \mathsf{\Psi }\left([{\Delta }_a,{\Delta }_b]\right)& =\mathsf{\Psi }\left({\Delta }_{[a,b]}\right)=[a,b]+\mathcal{Z}\left(\mathcal{A}\right)\hfill \\ \hfill & =[a+\mathcal{Z}\left(\mathcal{A}\right),b+\mathcal{Z}\left(\mathcal{A}\right)]=[\mathsf{\Psi }\left({\Delta }_a\right),\mathsf{\Psi }\left({\Delta }_b\right)]\hfill \end{array}$$

Thus, Ψ preserves the Lie bracket.

Consider the map $\mathsf{\Phi }:\mathcal{A}\to \mathrm{Inn}\left(\mathcal{A}\right)$ defined by $\mathsf{\Phi }\left(a\right)={\Delta }_a$. This is a Lie algebra homomorphism with kernel $\mathcal{Z}\left(\mathcal{A}\right)$ and image $\mathrm{Inn}\left(\mathcal{A}\right)$. By the first isomorphism theorem, Φ induces an isomorphism $\mathcal{A}/\mathcal{Z}\left(\mathcal{A}\right)\cong \mathrm{Inn}\left(\mathcal{A}\right)$, and Ψ is the inverse of this isomorphism.

Therefore, Ψ is a well-defined Lie algebra isomorphism, and we have $\mathcal{S}=\mathrm{Inn}\left(\mathcal{A}\right)\cong \mathcal{A}/\mathcal{Z}\left(\mathcal{A}\right).$   □

Proposition 4.

Let $\mathcal{A}$ be a semiprime ∗-Banach algebra endowed with a faithful trace. Then the set $\mathcal{S}$ of all weak 2-local inner derivations on $\mathcal{A}$ is 2-reflexive with respect to the family of seminorms $\mathcal{P}$ induced by $\mathcal{A}*$.

Proof. 

The inclusion $\mathcal{S}\subseteq {ref}_{(2,\mathcal{P})}\left(\mathcal{S}\right)$ is immediate from the definition. For the reverse inclusion, let $T\in {ref}_{(2,\mathcal{P})}\left(\mathcal{S}\right)$. For any $x,y\in \mathcal{A}$ and any $\phi \in \mathcal{A}*$, consider the seminorm $p_{\phi }\in {\mathcal{P}}_1$. Since $T\in ref(2,\mathcal{P})\left(\mathcal{S}\right)$, there exists $T_{x,y,\phi }\in \mathcal{S}$ such that $p_{\phi }(T\left(x\right)-T_{x,y,\phi }\left(x\right))=0$ and $p_{\phi }(T\left(y\right)-T_{x,y,\phi }\left(y\right))=0.$ This implies $\phi \left(T\left(x\right)\right)=\phi \left(T_{x,y,\phi }\left(x\right)\right)$ and $\phi \left(T\left(y\right)\right)=\phi \left(T_{x,y,\phi }\left(y\right)\right).$ Since $T_{x,y,\phi }\in \mathcal{S}$ is a weak 2-local inner derivation, there exists an inner derivation $D_{x,y,\phi }$ such that $\phi \left(T_{x,y,\phi }\left(x\right)\right)=\phi \left(D_{x,y,\phi }\left(x\right)\right)$ and $\phi \left(T_{x,y,\phi }\left(y\right)\right)=\phi \left(D_{x,y,\phi }\left(y\right)\right).$ Combining these, we obtain

$$\phi \left(T\left(x\right)\right)=\phi \left(D_{x,y,\phi }\left(x\right)\right)\mathrm{and}\phi \left(T\left(y\right)\right)=\phi \left(D_{x,y,\phi }\left(y\right)\right).$$

Since this holds for all $\phi \in \mathcal{A}*$, by the Hahn–Banach theorem, we conclude that T is a weak 2-local inner derivation. By Lemmas 1 and 2, every weak 2-local inner derivation is linear and bounded, so $T\in \mathcal{S}$. Hence, ${ref}_{(2,\mathcal{P})}\left(\mathcal{S}\right)\subseteq \mathcal{S}$.

Therefore, ${ref}_{(2,\mathcal{P})}\left(\mathcal{S}\right)=\mathcal{S}$, and $\mathcal{S}$ is 2-reflexive with respect to $\mathcal{P}$.   □

Corollary 3.

Let $\mathcal{A}$ be a semiprime ∗-Banach algebra endowed with a faithful trace. Denote by $\mathcal{ID}$ the set of all inner derivations on $\mathcal{A}$ and by $\mathcal{S}$ the set of all weak 2-local inner derivations. Then ${ref}_{(2,\mathcal{P})}\left(\mathcal{ID}\right)=\mathcal{S}.$

Proof. 

Let $T\in \mathcal{S}$ be a weak 2-local inner derivation. By definition, for any $x,y\in \mathcal{A}$ and $\phi \in \mathcal{A}*$, there exists an inner derivation $D_{x,y,\phi }\in \mathcal{ID}$ such that $\phi \left(T\left(x\right)\right)=\phi \left(D_{x,y,\phi }\left(x\right)\right)$ and $\phi \left(T\left(y\right)\right)=\phi \left(D_{x,y,\phi }\left(y\right)\right)$. This is equivalent to $p_{\phi }(T\left(x\right)-D_{x,y,\phi }\left(x\right))=0$ and $p_{\phi }(T\left(y\right)-D_{x,y,\phi }\left(y\right))=0.$ Therefore, $T\in {ref}_{(2,\mathcal{P})}\left(\mathcal{ID}\right)$.

Since every inner derivation is trivially a weak 2-local inner derivation, we have $\mathcal{ID}\subseteq \mathcal{S}$. It follows from Proposition 2 that ${ref}_{(2,\mathcal{P})}\left(\mathcal{ID}\right)\subseteq {ref}_{(2,\mathcal{P})}\left(\mathcal{S}\right).$ By Proposition 4, ${ref}_{(2,\mathcal{P})}\left(\mathcal{S}\right)=\mathcal{S}$, so ${ref}_{(2,\mathcal{P})}\left(\mathcal{ID}\right)\subseteq \mathcal{S}.$   □

Remark 5.

Corollary 3 provides a profound characterization: the weak 2-local inner derivations are precisely those maps that can be locally approximated by inner derivations in the $(2,\mathcal{P})$-reflection sense. This establishes a hierarchical structure:

$$\mathcal{ID}\subseteq \mathcal{S}={ref}_{(2,\mathcal{P})}\left(\mathcal{S}\right)={ref}_{(2,\mathcal{P})}\left(\mathcal{ID}\right)\subseteq B\left(\mathcal{A}\right),$$

where each inclusion and equality has significant mathematical meaning.

4. Conclusions

This research systematically examines two interrelated classes of nonlinear mappings on Banach algebras: weak 2-local inner derivations and $\mathcal{P}$-2-local mappings. Within the framework of semiprime ∗-Banach algebras endowed with a faithful trace, we have established profound connections between these concepts, leading to several key results:

• Main Theorem: Every weak -local inner derivation on a semiprime ∗-Banach algebra $\mathcal{A}$ with a faithful trace $\tau$ and a dense set $D_{\mathcal{A}}$ in ${\mathcal{A}}_{sa}$ is a derivation. This theorem yields a complete characterization of finite von Neumann algebras and applies to several other important classes of operator algebras satisfying the technical density condition.
• Structural Corollaries: We derive additional structural results, including:
  • Extension properties of ideals: Every weak 2-local inner derivation on a $\sigma$-weakly closed ideal of a finite von Neumann algebra can be extended to an inner derivation on the entire algebra.
  • On a finite von Neumann algebra, the set $\mathcal{S}$ of all weak 2-local inner derivations constitutes a Lie algebra that is isomorphic to the quotient $\mathcal{A}/\mathcal{Z}\left(\mathcal{A}\right)$.
  • The fundamental connection is established: $\mathcal{S}$ is precisely the $(2,\mathcal{P})$-reflection of the inner derivations, that is, ${ref}_{(2,\mathcal{P})}\left(\mathcal{ID}\right)=\mathcal{S}$.

Applicability to Non-classical Settings: The primary results extend to semiprime ∗-Banach algebras with faithful traces and dense projections. These findings indicate significant potential for application to a wider class of operator algebras beyond the classical $C*$ and von Neumann frameworks. Notable examples warranting further investigation include crossed product algebras such as $L^1\left(\mathbb{Z}{⋉}_{\tau }{L}^{\infty }\left([0,1]\right)\right)$, matrix algebras equipped with ${\ell }^1$-norms such as ${⨁}_{{\ell }^1}{M}_{n_k}\left(\mathbb{C}\right)$, and twisted group algebras $L^1(G,\sigma )$. These non-classical structures exhibit key properties analogous to those in the present framework and constitute promising avenues for extending the theory.

The findings presented in this paper suggest several promising and challenging research directions that merit further investigation:

• Given the stated assumptions, every derivation appears to be a weak 2-local inner derivation. However, constructing a counterexample, specifically a derivation that is not weak 2-local inner, within a suitable semiprime ∗-Banach algebra remains a significant challenge.
• A logical extension is to investigate whether analogous results apply to non-unital or non-semiprime Banach algebras and to identify potential counterexamples within these broader contexts.
• $\mathcal{P}$-2-local reflexivity theory extends to additional significant operator classes, including self-adjoint derivations, *-homomorphisms, isometries, and Jordan derivations.