Nonlinear η-∗-Jordan n-Derivation on ∗-Algebras

Abstract

Let $\mathcal{A}$ be a unital ∗-algebra with the unit I over the complex field $\mathbb{C}$ and let $\eta \ne 0,\pm 1$ be a complex number. For any $A,B\in \mathcal{A}$, $A{\diamond }_{\eta }B=AB+\eta B{A}^{*}$ is referred to as the $\eta$-Jordan ∗-product. Suppose that $n\ge 3$ is a fixed positive integer. In this study, it is shown that if a map $\phi :\mathcal{A}\to \mathcal{A}$ satisfies $\phi (A_1{\diamond }_{\eta }{A}_2{\diamond }_{\eta }\cdots {\diamond }_{\eta }{A}_n)={\sum }_{k=1}^n{A}_1{\diamond }_{\eta }\cdots {\diamond }_{\eta }{A}_{k-1}{\diamond }_{\eta }\phi \left(A_k\right){\diamond }_{\eta }{A}_{k+1}{\diamond }_{\eta }\cdots {\diamond }_{\eta }{A}_n$ for all $A_1,A_2\cdots A_{n-3}\in \{I,iI\}$ and $A_{n-2},A_{n-1},A_n\in \mathcal{A}$, then $\phi$ is an additive ∗-derivation and $\phi \left(\eta A\right)=\eta \phi \left(A\right)$ for all $A\in \mathcal{A}$, where i is the imaginary unit. In application, characterizations of prime ∗-algebras, von Neumann algebras with no central summands of type $I_1$ and factor von Neumann algebras are obtained.

1. Introduction

Let $\mathcal{A}$ be an algebra. For $A,B\in \mathcal{A}$, $A\circ B=AB+BA$ is called Jordan product. Recall that a map $\phi :\mathcal{A}\to \mathcal{A}$ is said to be a derivation if $\phi \left(AB\right)=\phi \left(A\right)B+A\phi \left(B\right)$ for all $A,B\in \mathcal{A}$. A map $\phi :\mathcal{A}\to \mathcal{A}$ is called a Jordan derivation which satisfies $\phi (A\circ B)=\phi \left(A\right)\circ B+A\circ \phi \left(B\right)$ for $A,B\in \mathcal{A}$. An involution is a map from $\mathcal{A}$ into itself, usually denoted by ∗, which satisfies ${(A+B)}^{*}=A^{*}+B^{*}$, ${\left(A^{*}\right)}^{*}=A$ and ${\left(AB\right)}^{*}=B^{*}{A}^{*}$, for any $A,B\in \mathcal{A}$. Moreover, if $\mathcal{A}$ is an algebra over the ground field $\mathbb{C}$, ∗ also satisfies ${\left(\lambda A\right)}^{*}=\overline{\lambda }{A}^{*}$, for all $\lambda \in \mathbb{C}$ and $A\in \mathcal{A}$, where $\overline{\lambda }$ denotes the complex conjugation of $\lambda$. A ∗-algebra is an algebra equipped with an involution ∗.

Let $\mathcal{A}$ be a ∗-algebra over the complex field $\mathbb{C}$ and $\eta$ be a nonzero complex number. For any elements $A,B\in \mathcal{A}$, the $\eta$-Jordan ∗-product is defined by $A{\diamond }_{\eta }B=AB+\eta B{A}^{*}$. The $(-1)$-Jordan ∗-product and 1-Jordan ∗-product, which are usually written as ${[A,B]}_{*}=AB-B{A}^{*}$ and $A•B=AB+B{A}^{*}$, are considerably significant and very important for specific research topics, for example, see [1,2,3]. A mapping $\phi :\mathcal{A}\to \mathcal{B}$ is said to preserve the product if $\phi \left(AB\right)=\phi \left(A\right)\phi \left(B\right)$ for all elements $A,B\in \mathcal{A}$. The $\eta$-Jordan ∗-product is said to be preserved if $\phi \left(A{\diamond }_{\eta }B\right)=\phi \left(A\right){\diamond }_{\eta }\phi \left(B\right)$, a feature that has attracted the attention of many researchers (for example, see [4,5,6]).

An additive derivation is said to be an additive ∗-derivation if $\phi \left(A^{*}\right)=\phi {\left(A\right)}^{*}$ for all $A\in \mathcal{A}$. A map $\phi :\mathcal{A}\to \mathcal{A}$ that is not necessarily linear is said to be a nonlinear $\eta$-∗-Jordan derivation if $\phi \left(A{\diamond }_{\eta }B\right)=\phi \left(A\right){\diamond }_{\eta }B+A{\diamond }_{\eta }\phi \left(B\right)$ for all $A,B\in \mathcal{A}$. Clearly, for $\eta =-1$ and $\eta =1$, the $\eta$-∗-Jordan derivation map is a ∗-Lie derivation and a ∗-Jordan derivation, respectively. Similarly, a map $\phi :\mathcal{A}\to \mathcal{A}$ is said to be $\eta$-∗-Jordan triple derivation if it satisfies the condition

$$\phi \left(A{\diamond }_{\eta }B{\diamond }_{\eta }C\right)=\phi \left(A\right){\diamond }_{\eta }B{\diamond }_{\eta }C+A{\diamond }_{\eta }\phi \left(B\right){\diamond }_{\eta }C+A{\diamond }_{\eta }B{\diamond }_{\eta }\phi \left(C\right)$$

(1)

for all $A,B,C\in \mathcal{A}$.

Given the consideration of $\eta$-∗-Jordan derivations and $\eta$-∗-Jordan triple derivations, we can further develop them in a natural way. Suppose that $n\ge 2$ is a fixed positive integer. A map $\phi :\mathcal{A}\to \mathcal{A}$ is called a nonlinear $\eta$-∗-Jordan n-derivation if

$$\phi (A_1{\diamond }_{\eta }{A}_2{\diamond }_{\eta }\cdots {\diamond }_{\eta }{A}_n)=\sum _{k=1}^n{A}_1{\diamond }_{\eta }\cdots {\diamond }_{\eta }{A}_{k-1}{\diamond }_{\eta }\phi \left(A_k\right){\diamond }_{\eta }{A}_{k+1}{\diamond }_{\eta }\cdots {\diamond }_{\eta }{A}_n$$

(2)

for all $A_1,A_2,\cdots ,A_n\in \mathcal{A}$. Given this definition, it is clear that each ∗-Jordan derivation is a 1-∗-Jordan 2-derivation and every ∗-Jordan n-derivation is a 1-∗-Jordan n-derivation. These derivations in different backgrounds have been studied extensively by several authors.

Recently, Zhang [7] proved that every nonlinear 1-∗-Jordan 2-derivation between factor von Neumann algebras is an additive ∗-derivation. On the other hand, Yu and Zhang [8] showed that every nonlinear (-1)-∗-Jordan 2-derivation on a factor von Neumann algebra is an additive ∗-derivation. In [9], it is proved that the map $\phi$ is a nonlinear $\eta$-∗-Jordan 2-derivation on von Neumann algebras without central abelian projections for all nonzero numbers $\eta$ if and only if $\phi$ is an additive ∗-derivation and $\phi \left(\eta A\right)=\eta \phi \left(A\right)$ for all $A\in \mathcal{A}.$ In addition, Zhao and Li [10] found that every nonlinear 1-∗-Jordan 3-derivation between von Neumann algebras with no central abelian projections of type $I_1$ is an additive ∗-derivation. Darvish et al. [11] showed that under the mild condition, nonlinear 1-∗-Jordan 3-derivations on prime ∗-algebras are an additive ∗-derivation. In [12], Li et al. introduced the concept of (-1)-∗-Jordan 3-derivations on factor von Neumann algebras and obtained the same result. In [13], the authors considered that a map $\phi$ on prime ∗-algebras satisfies Equation (1) for a complex scalar $\eta$ with $\left|\eta \right|\ne 0,1$; then, $\phi$ is an additive. Moreover, if $\phi \left(I\right)$ is self-adjoint, then $\phi$ is ∗-derivation. It was shown in [14] that if the map $\phi$ on prime ∗-algebras satisfies Equation (1) for a complex scalar $\eta \in \mathbb{C}\setminus \{0,\pm 1\}$, then $\phi$ is an additive ∗-derivation and $\phi \left(\eta A\right)=\eta \phi \left(A\right)$ for all $A\in \mathcal{A}$.

Li et al. [15] proved that if a ∗-algebra $\mathcal{A}$ with the unit I contains a nontrivial projection P that satisfies $X\mathcal{A}P=0\Rightarrow X=0$ and $X\mathcal{A}(I-P)=0\Rightarrow X=0$, then a nonlinear 1-∗-Jordan n-derivation on ∗-algebra $\mathcal{A}$ is an additive ∗-derivation. Lin [16] showed that if $A\subseteq B\left(H\right)$ is a von Neumann algebra without nonzero central abelian projections, then a map $\phi :\mathcal{A}\to B\left(H\right)$ satisfying Equation (2) for $\eta =-1$ is an additive ∗-derivation. In [17], Arshad Madni et al. proved that under some mild conditions, every nonlinear $(-1)$-∗-Jordan n-derivation ($n\ge 3$) on a unital ∗-algebra is an additive ∗-derivation as well.

Motivated by the above-cited works, we will investigate the nonlinear $\eta$-∗-Jordan n-derivations on some unital ∗-algebras with a nontrivial projection P satisfying $X\mathcal{A}P=0\Rightarrow X=0$ and $X\mathcal{A}(I-P)=0\Rightarrow X=0$ and prove that such maps must be additive ∗-derivations. For $\eta =1$ and $\eta =-1$, the conclusion can be found in [15,17], respectively. Therefore, in this paper, we will prove that the nonlinear $\eta$-∗-Jordan n-derivations are additive ∗-derivations, where $\eta \ne 0,\pm 1$ and $n\ge 3$ is a fixed positive integer. Moreover, we apply the main result to some special classes of unital ∗-algebras. We found that $\eta$-∗-Jordan n-derivations on prime ∗-algebras, von Neumann algebras with no central summands of type $I_1$, or factor von Neumann algebras are additive derivations, which generalized the existing results. Finally, some potential topics for future research are presented.

It is obvious that every nonlinear $\eta$-∗-Jordan derivation on any ∗-algebra is a nonlinear $\eta$-∗-Jordan n-derivation ($n\ge 3$), but we do not know whether the converse is true. Furthermore, we extend the results for $\eta$-∗-Jordan n-derivations with $\eta =\pm 1$ to all nonzero complex numbers.

2. Preliminaries

Before presenting our obtained primary results, it is essential that we lay out some notations and preliminary knowledge. Suppose that $n\ge 2$ is a fixed positive integer and $\eta$ is a nonzero complex number. Let us see a sequence of polynomials with a scalar $\eta$.

$$\begin{array}{cc}\hfill g_{1}I& =I,\hfill \\ \hfill g_{2}I=I{\diamond }_{\eta }I& =(1+\eta {\overline{g}}_1)I,\hfill \\ \hfill g_{3}I=I{\diamond }_{\eta }I{\diamond }_{\eta }I& =g_{2}I{\diamond }_{\eta }I=(g_2+\eta {\overline{g}}_2)I,\hfill \\ \hfill g_{4}I=I{\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }I& =g_{3}I{\diamond }_{\eta }I=(g_3+\eta {\overline{g}}_3)I,\hfill \\ & \cdots ,\hfill \\ \hfill g_{k}I=\underset{k-1}{\underbrace{I{\diamond }_{\eta }I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I}}{\diamond }_{\eta }I& =g_{k-1}I{\diamond }_{\eta }I=(g_{k-1}+\eta {\overline{g}}_{k-1})I,\hfill \\ & \cdots ,\hfill \\ \hfill g_{n}I=\underset{n-1}{\underbrace{I{\diamond }_{\eta }I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I}}{\diamond }_{\eta }I& =g_{n-1}I{\diamond }_{\eta }I=(g_{n-1}+\eta {\overline{g}}_{n-1})I,\hfill \end{array}$$

where ${\overline{g}}_k$ is the conjugate number of the complex number $g_k$ ($k=1,2,\cdots$).

It is not difficult to see that

$$\begin{array}{ll}{g}_n& =\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{0}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{1}}\right)\eta +\cdots +\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{k}}\right){\eta }^{{\displaystyle \frac{1-{(-1)}^k}{2}}}{\left|\eta \right|}^{k-{\displaystyle \frac{1-{(-1)}^k}{2}}}\\ & +\cdots +\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{n-1}}\right){\eta }^{{\displaystyle \frac{1-{(-1)}^{n-1}}{2}}}{\left|\eta \right|}^{n-1-{\displaystyle \frac{1-{(-1)}^{n-1}}{2}}}.\end{array}$$

Obviously, $g_n$ depends on $\eta$. We also have the following result:

Lemma 1.

If$\eta \ne -1$, then $g_k\ne 0 \mathit{for} 1\le k\le n$.

Proof. 

We only prove the case where k is an odd number because the proof for the even case is similar. It is clear that $g_1\ne 0$ if $\eta \ne -1$. Let $\eta ={\eta }_1+{\eta }_{2}i$ where ${\eta }_1={\displaystyle \frac{\eta +\overline{\eta }}{2}}$ and ${\eta }_2={\displaystyle \frac{\eta -\overline{\eta }}{2i}}$. For $1<k\le n$, we have

$$\begin{array}{ll}{g}_k& =\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{0}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{1}}\right)\eta +\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{2}}\right){\left|\eta \right|}^2+\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{3}}\right)\eta {\left|\eta \right|}^2\\ & +\cdots \cdots +\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{k-2}}\right){\eta \left|\eta \right|}^{k-3}+\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{k-1}}\right){\left|\eta \right|}^{k-1}\\ & =(\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{0}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{1}}\right){\eta }_1+\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{2}}\right)|\eta {|}^2+\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{3}}\right){\eta }_1|\eta {|}^2\\ & +\cdots \cdots +\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{k-2}}\right){\eta }_1{\left|\eta \right|}^{k-3}+\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{k-1}}\right){\left|\eta \right|}^{k-1})\\ & +(\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{1}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{3}}\right){\left|\eta \right|}^2+\cdots \cdots +\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{k-2}}\right){\left|\eta \right|}^{k-3}){\eta }_{2}i.\end{array}$$

Assume that $g_k=0$ for $\eta \ne -1$, it follows that

$$\begin{array}{ll}\mathrm{Re}\left(g_k\right)& =(\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{0}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{1}}\right){\eta }_1+\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{2}}\right)|\eta {|}^2\\ & +\cdots \cdots +\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{k-2}}\right){\eta }_1{\left|\eta \right|}^{k-3}+\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{k-1}}\right){\left|\eta \right|}^{k-1})=0\end{array}$$

and

$$\mathrm{Im}\left(g_k\right)=(\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{1}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{3}}\right){\left|\eta \right|}^2+\cdots \cdots +\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{k-2}}\right){\left|\eta \right|}^{k-3}){\eta }_2=0.$$

Because $\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{1}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{3}}\right){\left|\eta \right|}^2+\cdots \cdots +\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{k-2}}\right){\left|\eta \right|}^{k-3}>0$, we obtain ${\eta }_2=0$. Hence

$$\begin{array}{ll}\mathrm{Re}\left(g_k\right)& =(\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{0}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{1}}\right){\eta }_1+\cdots \cdots +\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{k-2}}\right){\eta }_1|{\eta }_1{|}^{k-3}+\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{k-1}}\right)|{\eta }_1{|}^{k-1})\\ & ={(1+{\eta }_1)}^{k-1}=0,\end{array}$$

which indicates $\eta ={\eta }_1=-1$. This contradicts the known condition. Therefore, $g_k\ne 0$. □

Since the first $n-3$ variables are restricted to the set $\{I,iI\}$, we can obtain some expressions involving the combination of $g_k$. Lemma 1 plays a foundational role throughout the entire proof.

Let $P_1$ and $P_2=I-P_1$ be nonzero idempotent elements of $\mathcal{A}$. It is well known that $\mathcal{A}$ has a Peirce decomposition $\mathcal{A}={\mathcal{A}}_{11}\oplus {\mathcal{A}}_{12}\oplus {\mathcal{A}}_{21}\oplus {\mathcal{A}}_{22}$, where ${\mathcal{A}}_{ij}=P_i\mathcal{A}{P}_j$ ($i,j=1,2$), satisfying the following multiplicative relations:

1.

${\mathcal{A}}_{ij}{\mathcal{A}}_{jl}\subseteq {\mathcal{A}}_{il}$$(i,j,l\in \{1,2\left\}\right)$,

2.

${\mathcal{A}}_{ij}{\mathcal{A}}_{kl}=0$ if $j\ne k$$(i,j,k,l\in \{1,2\left\}\right)$.

3. The Main Theorem and Its Proof

Theorem 1.

Let $\mathcal{A}$ be a unital ∗-algebra with the unit I and $\eta \in \mathbb{C}\setminus \{0,\pm 1\}$. Assume that $\mathcal{A}$ contains a nontrivial projection P that satisfies

(♠)

$X\mathcal{A}P=0\Rightarrow X=0$;

(♣)

$X\mathcal{A}(I-P)=0\Rightarrow X=0$.

If the map $\phi :\mathcal{A}\to \mathcal{A}$ satisfies

$$\phi (A_1{\diamond }_{\eta }{A}_2{\diamond }_{\eta }\cdots {\diamond }_{\eta }{A}_n)=\sum _{k=1}^n{A}_1{\diamond }_{\eta }\cdots {\diamond }_{\eta }{A}_{k-1}{\diamond }_{\eta }\phi \left(A_k\right){\diamond }_{\eta }{A}_{k+1}{\diamond }_{\eta }\cdots {\diamond }_{\eta }{A}_n$$

for all $A_1,A_2\cdots A_{n-3}\in \{I,iI\}$ and $A_{n-2},A_{n-1},A_n\in \mathcal{A}$, where $n\ge 3$ is a fixed positive integer, then φ is an additive ∗-derivation and $\phi \left(\eta A\right)=\eta \phi \left(A\right)$ for all $A\in \mathcal{A}$.

Proof. 

We will complete the proof by the following lemmas. □

Lemma 2.

$\phi \left(0\right)=0$.

Proof. 

Note that

$$\phi \left(0\right)=\phi (I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }0{\diamond }_{\eta }0)=0.$$

It follows that $\phi \left(0\right)=0$. □

Lemma 3.

For all $B_{12}\in {\mathcal{A}}_{12}$ and $C_{21}\in {\mathcal{A}}_{21}$, we have $\phi (B_{12}+C_{21})=\phi \left(B_{12}\right)+\phi \left(C_{21}\right)$.

Proof. 

We only need to show that $T=\phi (B_{12}+C_{21})-\phi \left(B_{12}\right)-\phi \left(C_{21}\right)=0$.

Since $I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_1-\frac{1}{\overline{\eta }}{P}_2}{g_{n-1}}}{\diamond }_{\eta }{B}_{12}=0$, we have

$$\begin{array}{l}\phi \left(I\right){\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_1-\frac{1}{\overline{\eta }}{P}_2}{g_{n-1}}}{\diamond }_{\eta }(B_{12}+C_{21})\\ +\cdots +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }\phi \left({\displaystyle \frac{P_1-\frac{1}{\overline{\eta }}{P}_2}{g_{n-1}}}\right){\diamond }_{\eta }(B_{12}+C_{21})\\ +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_1-\frac{1}{\overline{\eta }}{P}_2}{g_{n-1}}}{\diamond }_{\eta }\phi (B_{12}+C_{21})\\ =\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_1-\frac{1}{\overline{\eta }}{P}_2}{g_{n-1}}}{\diamond }_{\eta }(B_{12}+C_{21})\right)\\ =\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_1-\frac{1}{\overline{\eta }}{P}_2}{g_{n-1}}}{\diamond }_{\eta }{B}_{12}\right)+\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_1-\frac{1}{\overline{\eta }}{P}_2}{g_{n-1}}}{\diamond }_{\eta }{C}_{21}\right)\\ =\phi \left(I\right){\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_1-\frac{1}{\overline{\eta }}{P}_2}{g_{n-1}}}{\diamond }_{\eta }(B_{12}+C_{21})\\ +\cdots +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }\phi \left({\displaystyle \frac{P_1-\frac{1}{\overline{\eta }}{P}_2}{g_{n-1}}}\right){\diamond }_{\eta }(B_{12}+C_{21})\\ +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_1-\frac{1}{\overline{\eta }}{P}_2}{g_{n-1}}}{\diamond }_{\eta }(\phi \left(B_{12}\right)+\phi \left(C_{21}\right)).\end{array}$$

Hence, $I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_1-\frac{1}{\overline{\eta }}{P}_2}{g_{n-1}}}{\diamond }_{\eta }T=0$, from which we find that

$$(1+\eta )T_{11}+\left(\eta -{\displaystyle \frac{1}{\overline{\eta }}}\right)T_{21}-\left(1+{\displaystyle \frac{1}{\overline{\eta }}}\right)T_{22}=0.$$

Since $\eta \ne 0,-1$, we find that $T_{11}=T_{22}=0$.

On the other hand, since $I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{B}_{12}{\diamond }_{\eta }{\displaystyle \frac{P_1}{g_{n-1}}}=0$, it follows that

$$\begin{array}{ll}& \phi \left(I\right){\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }(B_{12}+C_{21}){\diamond }_{\eta }{\displaystyle \frac{P_1}{g_{n-1}}}+\cdots +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }\phi (B_{12}+C_{21}){\diamond }_{\eta }{\displaystyle \frac{P_1}{g_{n-1}}}\\ & +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }(B_{12}+C_{21}){\diamond }_{\eta }\phi \left({\displaystyle \frac{P_1}{g_{n-1}}}\right)\\ & =\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }(B_{12}+C_{21}){\diamond }_{\eta }{\displaystyle \frac{P_1}{g_{n-1}}}\right)\\ & =\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{B}_{12}{\diamond }_{\eta }{\displaystyle \frac{P_1}{g_{n-1}}}\right)+\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{C}_{21}{\diamond }_{\eta }{\displaystyle \frac{P_1}{g_{n-1}}}\right)\\ & =\phi \left(I\right){\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }(B_{12}+C_{21}){\diamond }_{\eta }{\displaystyle \frac{P_1}{g_{n-1}}}+\cdots +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }(\phi \left(B_{12}\right)+\phi \left(C_{21}\right)){\diamond }_{\eta }{\displaystyle \frac{P_1}{g_{n-1}}}\\ & +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }(B_{12}+C_{21}){\diamond }_{\eta }\phi \left({\displaystyle \frac{P_1}{g_{n-1}}}\right).\end{array}$$

Hence, $I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }T{\diamond }_{\eta }{\displaystyle \frac{P_1}{g_{n-1}}}=0$. We find that $T_{21}+\eta {\displaystyle \frac{{\overline{g}}_{n-1}}{g_{n-1}}}{T}_{21}^{*}=0$, so $T_{21}=0$. Similarly, by applying $I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{C}_{21}{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}=0$, we find that $T_{12}=0$. □

Lemma 4.

For all $A_{11}\in {\mathcal{A}}_{11}$, $B_{12}\in {\mathcal{A}}_{12}$, $C_{21}\in {\mathcal{A}}_{21}$ and $D_{22}\in {\mathcal{A}}_{22}$, we have

(1) 

$\phi (A_{11}+B_{12})=\phi \left(A_{11}\right)+\phi \left(B_{12}\right)$;

(2) 

$\phi (C_{21}+D_{22})=\phi \left(C_{21}\right)+\phi \left(D_{22}\right)$.

Proof. 

We only need to show (1) because the proof of (2) is similar. Let $T=\phi (A_{11}+B_{12})-\phi \left(A_{11}\right)-\phi \left(B_{12}\right)$.

Since $I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }{A}_{11}=0$, we have

$$\begin{array}{ll}& \phi \left(I\right){\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }(A_{11}+B_{12})+\cdots +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }\phi \left({\displaystyle \frac{P_2}{g_{n-1}}}\right){\diamond }_{\eta }(A_{11}+B_{12})\\ & +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }\phi (A_{11}+B_{12})\\ & =\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }(A_{11}+B_{12})\right)\\ & =\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }{A}_{11}\right)+\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }{B}_{12}\right)\\ & =\phi \left(I\right){\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }(A_{11}+B_{12})+\cdots +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }\phi \left({\displaystyle \frac{P_2}{g_{n-1}}}\right){\diamond }_{\eta }(A_{11}+B_{12})\\ & +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }(\phi \left(A_{11}\right)+\phi \left(B_{12}\right)).\end{array}$$

Thus, $I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }T=0$, which implies that $T_{12}=T_{21}=T_{22}=0$.

To complete the proof, we need to show that $T_{11}=0$. Considering $I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_1-\frac{1}{\overline{\eta }}{P}_2}{g_{n-1}}}{\diamond }_{\eta }{B}_{12}=0$, we find that

$$\begin{array}{ll}& \phi \left(I\right){\diamond }_{\eta }\cdots {\diamond }_{\eta }{I}_{{\diamond }_{\eta }}{\displaystyle \frac{P_1-\frac{1}{\overline{\eta }}{P}_2}{g_{n-1}}}{\diamond }_{\eta }(A_{11}+B_{12})+\cdots +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }\phi \left({\displaystyle \frac{P_1-\frac{1}{\overline{\eta }}{P}_2}{g_{n-1}}}\right){\diamond }_{\eta }(A_{11}+B_{12})\\ & +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_1-\frac{1}{\overline{\eta }}{P}_2}{g_{n-1}}}{\diamond }_{\eta }\phi (A_{11}+B_{12})\\ & =\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_1-\frac{1}{\overline{\eta }}{P}_2}{g_{n-1}}}{\diamond }_{\eta }(A_{11}+B_{12})\right)\\ & =\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_1-\frac{1}{\overline{\eta }}{P}_2}{g_{n-1}}}{\diamond }_{\eta }{A}_{11}\right)+\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }{I}_{{\diamond }_{\eta }}{\displaystyle \frac{P_1-\frac{1}{\overline{\eta }}{P}_2}{g_{n-1}}}{\diamond }_{\eta }{B}_{12}\right)\\ & =\phi \left(I\right){\diamond }_{\eta }\cdots {\diamond }_{\eta }{I}_{{\diamond }_{\eta }}{\displaystyle \frac{P_1-\frac{1}{\overline{\eta }}{P}_2}{g_{n-1}}}{\diamond }_{\eta }(A_{11}+B_{12})\\ & +\cdots +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }\phi \left({\displaystyle \frac{P_1-\frac{1}{\overline{\eta }}{P}_2}{g_{n-1}}}\right){\diamond }_{\eta }(A_{11}+B_{12})\\ & +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_1-\frac{1}{\overline{\eta }}{P}_2}{g_{n-1}}}{\diamond }_{\eta }(\phi \left(A_{11}\right)+\phi \left(B_{12}\right)).\end{array}$$

It follows that $I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_1-\frac{1}{\overline{\eta }}{P}_2}{g_{n-1}}}{\diamond }_{\eta }T=0$. So we derive that $T_{11}=0$. □

Lemma 5.

For all $A_{11}\in {\mathcal{A}}_{11}$, $B_{12}\in {\mathcal{A}}_{12}$, $C_{21}\in {\mathcal{A}}_{21}$ and $D_{22}\in {\mathcal{A}}_{22}$, we have

(1) 

$\phi (A_{11}+B_{12}+C_{21})=\phi \left(A_{11}\right)+\phi \left(B_{12}\right)+\phi \left(C_{21}\right)$;

(2) 

$\phi (B_{12}+C_{21}+D_{22})=\phi \left(B_{12}\right)+\phi \left(C_{21}\right)+\phi \left(D_{22}\right)$.

Proof. 

We only need to show that $T=\phi (A_{11}+B_{12}+C_{21})-\phi \left(A_{11}\right)-\phi \left(B_{12}\right)-\phi \left(C_{21}\right)=0$.

Using Lemma 3, we find that

$$\begin{array}{ll}& \phi \left(I\right){\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }(A_{11}+B_{12}+C_{21})\\ & +\cdots +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }\phi \left({\displaystyle \frac{P_2}{g_{n-1}}}\right){\diamond }_{\eta }(A_{11}+B_{12}+C_{21})\\ & +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }\phi (A_{11}+B_{12}+C_{21})\\ & =\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }(A_{11}+B_{12}+C_{21})\right)\\ & =\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }{A}_{11}\right)+\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }{B}_{12}\right)\\ & +\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }{C}_{21}\right)\\ & =\phi \left(I\right){\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }(A_{11}+B_{12}+C_{21})\\ & +\cdots +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }\phi \left({\displaystyle \frac{P_2}{g_{n-1}}}\right){\diamond }_{\eta }(A_{11}+B_{12}+C_{21})\\ & +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }(\phi \left(A_{11}\right)+\phi \left(B_{12}\right)+\phi \left(C_{21}\right)).\end{array}$$

Hence, $I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }T=0$, which implies that $T_{12}=T_{21}=T_{22}=0$.

Since $I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{-\frac{1}{\overline{\eta }}{P}_1+P_2}{g_{n-1}}}{\diamond }_{\eta }{C}_{21}=0$, using Lemma 4, it follows that

$$\begin{array}{ll}& \phi \left(I\right){\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{-\frac{1}{\overline{\eta }}{P}_1+P_2}{g_{n-1}}}{\diamond }_{\eta }(A_{11}+B_{12}+C_{21})\\ & +\cdots +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }\phi \left({\displaystyle \frac{-\frac{1}{\overline{\eta }}{P}_1+P_2}{g_{n-1}}}\right){\diamond }_{\eta }(A_{11}+B_{12}+C_{21})\\ & +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{-\frac{1}{\overline{\eta }}{P}_1+P_2}{g_{n-1}}}{\diamond }_{\eta }\phi (A_{11}+B_{12}+C_{21})\\ & =\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{-\frac{1}{\overline{\eta }}{P}_1+P_2}{g_{n-1}}}{\diamond }_{\eta }(A_{11}+B_{12}+C_{21})\right)\\ & =\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{-\frac{1}{\overline{\eta }}{P}_1+P_2}{g_{n-1}}}{\diamond }_{\eta }{A}_{11}\right)+\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{-\frac{1}{\overline{\eta }}{P}_1+P_2}{g_{n-1}}}{\diamond }_{\eta }{B}_{12}\right)\\ & +\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{-\frac{1}{\overline{\eta }}{P}_1+P_2}{g_{n-1}}}{\diamond }_{\eta }{C}_{21}\right)\\ & =\phi \left(I\right){\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{-\frac{1}{\overline{\eta }}{P}_1+P_2}{g_{n-1}}}{\diamond }_{\eta }(A_{11}+B_{12}+C_{21})\\ & +\cdots +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }\phi \left({\displaystyle \frac{-\frac{1}{\overline{\eta }}{P}_1+P_2}{g_{n-1}}}\right){\diamond }_{\eta }(A_{11}+B_{12}+C_{21})\\ & +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{-\frac{1}{\overline{\eta }}{P}_1+P_2}{g_{n-1}}}{\diamond }_{\eta }(\phi \left(A_{11}\right)+\phi \left(B_{12}\right)+\phi \left(C_{21}\right)).\end{array}$$

Thus, $I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{-\frac{1}{\overline{\eta }}{P}_1+P_2}{g_{n-1}}}{\diamond }_{\eta }T=0$, which implies that $T_{11}=0$. Using a similar method, we obtain the second case. □

Lemma 6.

For all $A_{11}\in {\mathcal{A}}_{11}$, $B_{12}\in {\mathcal{A}}_{12}$, $C_{21}\in {\mathcal{A}}_{21}$, $D_{22}\in {\mathcal{A}}_{22}$, we have

$$\phi (A_{11}+B_{12}+C_{21}+D_{22})=\phi \left(A_{11}\right)+\phi \left(B_{12}\right)+\phi \left(C_{21}\right)+\phi \left(D_{22}\right).$$

Proof. 

We only need to show that

$$T=\phi (A_{11}+B_{12}+C_{21}+D_{22})-\phi \left(A_{11}\right)-\phi \left(B_{12}\right)-\phi \left(C_{21}\right)-\phi \left(D_{22}\right)=0.$$

Using Lemma 5, it follows that

$$\begin{array}{ll}& \phi \left(I\right){\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }(A_{11}+B_{12}+C_{21}+D_{22})\\ & +\cdots +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }\phi \left({\displaystyle \frac{P_2}{g_{n-1}}}\right){\diamond }_{\eta }(A_{11}+B_{12}+C_{21}+D_{22})\\ & +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }\phi (A_{11}+B_{12}+C_{21}+D_{22})\\ & =\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }(A_{11}+B_{12}+C_{21}+D_{22})\right)\\ & =\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }{A}_{11}\right)+\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }{B}_{12}\right)\\ & +\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }{C}_{21}\right)+\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }{D}_{22}\right)\\ & =\phi \left(I\right){\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }(A_{11}+B_{12}+C_{21}+D_{22})\\ & +\cdots +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }\phi \left({\displaystyle \frac{P_2}{g_{n-1}}}\right){\diamond }_{\eta }(A_{11}+B_{12}+C_{21}+D_{22})\\ & +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }(\phi \left(A_{11}\right)+\phi \left(B_{12}\right)+\phi \left(C_{21}\right)+\phi \left(D_{22}\right)).\end{array}$$

Hence, $I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_2}{g_{n-1}}}{\diamond }_{\eta }T=0$, which implies that $T_{12}=T_{21}=T_{22}=0$. Analogously, using $I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_1}{g_{n-1}}}{\diamond }_{\eta }T=0$, we find that $T_{11}=0$. □

Lemma 7.

For all $A_{ij},B_{ij}\in {\mathcal{A}}_{ij}$ with $i\ne j$, we have $\phi (A_{ij}+B_{ij})=\phi \left(A_{ij}\right)+\phi \left(B_{ij}\right)$.

Proof. 

Since $I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_i+A_{ij}}{g_{n-1}}}{\diamond }_{\eta }(P_j+B_{ij})=A_{ij}+B_{ij}+\eta A_{ij}^{*}+\eta B_{ij}{A}_{ij}^{*}$, using Lemma 6, we have

$$\begin{array}{ll}& \phi (A_{ij}+B_{ij})+\phi \left(\eta A_{ij}^{*}\right)+\phi \left(\eta B_{ij}{A}_{ij}^{*}\right)\\ & =\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_i+A_{ij}}{g_{n-1}}}{\diamond }_{\eta }(P_j+B_{ij})\right)\\ & =\phi \left(I\right){\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_i+A_{ij}}{g_{n-1}}}{\diamond }_{\eta }(P_j+B_{ij})\\ & +\cdots +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }\phi \left({\displaystyle \frac{P_i+A_{ij}}{g_{n-1}}}\right){\diamond }_{\eta }(P_j+B_{ij})\\ & +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_i+A_{ij}}{g_{n-1}}}{\diamond }_{\eta }\phi (P_j+B_{ij})\\ & =\phi \left(I\right){\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_i+A_{ij}}{g_{n-1}}}{\diamond }_{\eta }(P_j+B_{ij})\\ & +\cdots +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }\left(\phi \left({\displaystyle \frac{P_i}{g_{n-1}}}\right)+\phi \left({\displaystyle \frac{A_{ij}}{g_{n-1}}}\right)\right){\diamond }_{\eta }(P_j+B_{ij})\\ & +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_i+A_{ij}}{g_{n-1}}}{\diamond }_{\eta }(\phi \left(P_j\right)+\phi \left(B_{ij}\right))\\ & =\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_i}{g_{n-1}}}{\diamond }_{\eta }{P}_j\right)+\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{A_{ij}}{g_{n-1}}}{\diamond }_{\eta }{P}_j\right)\\ & +\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_i}{g_{n-1}}}{\diamond }_{\eta }{B}_{ij}\right)+\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{A_{ij}}{g_{n-1}}}{\diamond }_{\eta }{B}_{ij}\right)\\ & =\phi \left(A_{ij}\right)+\phi \left(B_{ij}\right)+\phi \left(\eta A_{ij}^{*}\right)+\phi \left(\eta B_{ij}{A}_{ij}^{*}\right).\end{array}$$

So, $\phi (A_{ij}+B_{ij})=\phi \left(A_{ij}\right)+\phi \left(B_{ij}\right)$. □

Lemma 8.

For all $A_{ii},B_{ii}\in {\mathcal{A}}_{ii}$, we have $\phi (A_{ii}+B_{ii})=\phi \left(A_{ii}\right)+\phi \left(B_{ii}\right)$.

Proof. 

Let $T=\phi (A_{ii}+B_{ii})-\phi \left(A_{ii}\right)-\phi \left(B_{ii}\right)$. For $i,j\in \{1,2\}:i\ne j$, we have

$$\begin{array}{l}\phi \left(I\right){\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_j}{g_{n-1}}}{\diamond }_{\eta }(A_{ii}+B_{ii})+\cdots +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }\phi \left({\displaystyle \frac{P_j}{g_{n-1}}}\right){\diamond }_{\eta }(A_{ii}+B_{ii})\\ +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_j}{g_{n-1}}}{\diamond }_{\eta }\phi (A_{ii}+B_{ii})\\ =\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_j}{g_{n-1}}}{\diamond }_{\eta }(A_{ii}+B_{ii})\right)\\ =\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_j}{g_{n-1}}}{\diamond }_{\eta }{A}_{ii}\right)+\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_j}{g_{n-1}}}{\diamond }_{\eta }{B}_{ii}\right)\\ =\phi \left(I\right){\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_j}{g_{n-1}}}{\diamond }_{\eta }(A_{ii}+B_{ii})+\cdots +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }\phi \left({\displaystyle \frac{P_j}{g_{n-1}}}\right){\diamond }_{\eta }(A_{ii}+B_{ii})\\ +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_j}{g_{n-1}}}{\diamond }_{\eta }(\phi \left(A_{ii}\right)+\phi \left(B_{ii}\right)).\end{array}$$

Hence, $I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{P_j}{g_{n-1}}}{\diamond }_{\eta }T=0$, which implies that $T_{ij}=T_{ji}=T_{jj}=0$.

On the other hand, for any $F_{ij}\in {\mathcal{A}}_{ij}$ with $i\ne j$, using Lemmas 3 and 7, it follows that

$$\begin{array}{l}\phi \left(I\right){\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{F_{ij}^{*}}{g_{n-1}}}{\diamond }_{\eta }(A_{ii}+B_{ii})+\cdots +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }\phi \left({\displaystyle \frac{F_{ij}^{*}}{g_{n-1}}}\right){\diamond }_{\eta }(A_{ii}+B_{ii})\\ +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{F_{ij}^{*}}{g_{n-1}}}{\diamond }_{\eta }\phi (A_{ii}+B_{ii})\\ =\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{F_{ij}^{*}}{g_{n-1}}}{\diamond }_{\eta }(A_{ii}+B_{ii})\right)\\ =\phi (F_{ij}^{*}{A}_{ii}+F_{ij}^{*}{B}_{ii}+\eta A_{ii}{F}_{ij}+\eta B_{ii}{F}_{ij})\\ =\phi (F_{ij}^{*}{A}_{ii}+\eta A_{ii}{F}_{ij})+\phi (F_{ij}^{*}{B}_{ii}+\eta B_{ii}{F}_{ij})\\ =\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{F_{ij}^{*}}{g_{n-1}}}{\diamond }_{\eta }{A}_{ii}\right)+\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{F_{ij}^{*}}{g_{n-1}}}{\diamond }_{\eta }{B}_{ii}\right)\\ =\phi \left(I\right){\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{F_{ij}^{*}}{g_{n-1}}}{\diamond }_{\eta }(A_{ii}+B_{ii})+\cdots +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }\phi \left({\displaystyle \frac{F_{ij}^{*}}{g_{n-1}}}\right){\diamond }_{\eta }(A_{ii}+B_{ii})\\ +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{F_{ij}^{*}}{g_{n-1}}}{\diamond }_{\eta }(\phi \left(A_{ii}\right)+\phi \left(B_{ii}\right)).\end{array}$$

Then, we find that $I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }{\displaystyle \frac{F_{ij}^{*}}{g_{n-1}}}{\diamond }_{\eta }T=0$, which implies that $F_{ij}^{*}{T}_{ii}+\eta T_{ii}{F}_{ij}=0$. Therefore, we obtain $T_{ii}{F}_{ij}=0$, and it follows from (♠) and (♣) that $T_{ii}=0$. □

Lemma 9.

φ is an additive.

Proof. 

For every $A,B\in \mathcal{A}$, we have $A={\sum }_{i,j=1}^2{A}_{ij}$, $B={\sum }_{i,j=1}^2{B}_{ij}$. Using Lemmas 6–8, we can show that $\phi (A+B)=\phi \left(A\right)+\phi \left(B\right)$. □

Lemma 10.

$\phi {\left(I\right)}^{*}=\phi \left(I\right)$.

Proof. 

Case 1: $\left|\eta \right|=1$ and $\eta \ne \pm 1$.

Since $I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }iI{\diamond }_{\eta }I=0$, we obtain

$$\begin{array}{ll}0& =\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }iI{\diamond }_{\eta }I\right)\\ & =\phi \left(I\right){\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }iI{\diamond }_{\eta }I+\cdots +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }\phi \left(iI\right){\diamond }_{\eta }I\\ & +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }iI{\diamond }_{\eta }\phi \left(I\right)\\ & =2^{n-3}(1+\eta )\left(\phi \left(iI\right)+\phi {\left(iI\right)}^{*}\right).\end{array}$$

Thus,

$$\phi {\left(iI\right)}^{*}=-\phi \left(iI\right).$$

(3)

Then, we have the following:

$$\begin{array}{ll}{2}^{n-2}\phi \left((1+\eta )I\right)& =\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }I\right)\\ & =2^{n-3}(n+2+n\eta )\phi \left(I\right)+2^{n-3}(n-2+n\eta )\phi {\left(I\right)}^{*}\end{array}$$

and

$$\begin{array}{ll}{2}^{n-2}\phi (-(1-\eta )I)& =\phi \left(iI{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }iI\right)\\ & =2^{n-2}i(2-2\eta )\phi \left(iI\right)+2^{n-3}(n-2)(\eta -1)\phi \left(I\right)\\ & +2^{n-3}(n-2)(\eta -1)\phi {\left(I\right)}^{*}.\end{array}$$

Using the above two equations, we find that

$$4\phi \left(I\right)=-4i(1-\eta )\phi \left(iI\right)+2(n+\eta )\phi \left(I\right)+2(n+\eta -2)\phi {\left(I\right)}^{*}.$$

(4)

On the other hand, we find that

$$\begin{array}{ll}{2}^{n-2}\phi \left(i(1+\eta )I\right)& =\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }iI\right)\\ & =2^{n-3}i(n+(n-2)\eta )\phi \left(I\right)+2^{n-3}i(n-2+n\eta )\phi {\left(I\right)}^{*}\\ & +2^{n-2}(1+\eta )\phi \left(iI\right)\end{array}$$

(5)

and

$$\begin{array}{ll}{2}^{n-2}\phi \left(i(1-\eta )I\right)& =\phi \left(iI{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }I\right)\\ & =2^{n-2}(1-\eta )\phi \left(iI\right)+2^{n-3}in(1-\eta )\phi \left(I\right)\\ & +2^{n-3}i(n-2)(1-\eta )\phi {\left(I\right)}^{*}.\end{array}$$

(6)

Then, by adding (5) and (6), we obtain

$$(n+\eta -2)\phi {\left(I\right)}^{*}=(\eta -n)\phi \left(I\right).$$

(7)

Hence, using (4) and (7), it follows that

$$(1-\eta )\phi \left(iI\right)=i(1-\eta )\phi \left(I\right),$$

which gives

$$\phi \left(iI\right)=i\phi \left(I\right).$$

(8)

Using (3) and (8), we find that $\phi {\left(I\right)}^{*}=\phi \left(I\right).$

• Case 2: $\left|\eta \right|\ne 1$.

Since $\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }iI{\diamond }_{\eta }iI\right)=\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }iI{\diamond }_{\eta }iI{\diamond }_{\eta }I\right)$ and $\phi \left(I\right){\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }iI{\diamond }_{\eta }iI=\phi \left(I\right){\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }iI{\diamond }_{\eta }iI{\diamond }_{\eta }I,\cdots \cdots ,I{\diamond }_{\eta }\cdots {\diamond }_{\eta }\phi \left(I\right){\diamond }_{\eta }I{\diamond }_{\eta }iI{\diamond }_{\eta }iI=I{\diamond }_{\eta }\cdots {\diamond }_{\eta }\phi \left(I\right){\diamond }_{\eta }iI{\diamond }_{\eta }iI{\diamond }_{\eta }I$, we have

$$\begin{array}{l}I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }\phi \left(I\right){\diamond }_{\eta }iI{\diamond }_{\eta }iI+I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }\phi \left(iI\right){\diamond }_{\eta }iI\\ +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }iI{\diamond }_{\eta }\phi \left(iI\right)\\ =I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }\phi \left(iI\right){\diamond }_{\eta }iI{\diamond }_{\eta }I+I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }iI{\diamond }_{\eta }\phi \left(iI\right){\diamond }_{\eta }I\\ +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }iI{\diamond }_{\eta }iI{\diamond }_{\eta }\phi \left(I\right).\end{array}$$

Thus,

$$\begin{array}{ll}-& g_{n-2}\phi \left(I\right)+{\left|\eta \right|}^2{g}_{n-2}\phi \left(I\right)+2g_{n-2}i\phi \left(iI\right)+\eta {\overline{g}}_{n-2}i\phi \left(iI\right)\\ & +\eta {\overline{g}}_{n-2}i\phi {\left(iI\right)}^{*}+{\left|\eta \right|}^2{g}_{n-2}i\phi {\left(iI\right)}^{*}-{\left|\eta \right|}^2{g}_{n-2}i\phi \left(iI\right)\\ =& -g_{n-2}\phi \left(I\right)+{\left|\eta \right|}^2{g}_{n-2}\phi \left(I\right)+2g_{n-2}i\phi \left(iI\right)-\eta {\overline{g}}_{n-2}i\phi \left(iI\right)\\ & -\eta {\overline{g}}_{n-2}i\phi {\left(iI\right)}^{*}+{\left|\eta \right|}^2{g}_{n-2}i\phi {\left(iI\right)}^{*}-{\left|\eta \right|}^2{g}_{n-2}i\phi \left(iI\right).\end{array}$$

Then, $2i\eta {\overline{g}}_{n-2}\phi \left(iI\right)+2i\eta {\overline{g}}_{n-2}\phi {\left(iI\right)}^{*}=0$, which indicates that

$$\phi {\left(iI\right)}^{*}=-\phi \left(iI\right).$$

(9)

On the other hand, with $\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }iI{\diamond }_{\eta }iI{\diamond }_{\eta }iI\right)=-\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }iI{\diamond }_{\eta }I\right)$ and $\phi \left(I\right){\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }iI{\diamond }_{\eta }iI{\diamond }_{\eta }iI=-\phi \left(I\right){\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }iI{\diamond }_{\eta }I,\cdots \cdots ,I{\diamond }_{\eta }\cdots {\diamond }_{\eta }\phi \left(I\right){\diamond }_{\eta }iI{\diamond }_{\eta }iI{\diamond }_{\eta }iI=-I{\diamond }_{\eta }\cdots {\diamond }_{\eta }\phi \left(I\right){\diamond }_{\eta }I{\diamond }_{\eta }iI{\diamond }_{\eta }I$, we find that

$$\begin{array}{l}I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }\phi \left(iI\right){\diamond }_{\eta }iI{\diamond }_{\eta }iI+I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }iI{\diamond }_{\eta }\phi \left(iI\right){\diamond }_{\eta }iI\\ +I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }iI{\diamond }_{\eta }iI{\diamond }_{\eta }\phi \left(iI\right)\\ =-I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }\phi \left(I\right){\diamond }_{\eta }iI{\diamond }_{\eta }I-I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }\phi \left(iI\right){\diamond }_{\eta }I\\ -I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }iI{\diamond }_{\eta }\phi \left(I\right).\end{array}$$

It follows that

$$\begin{array}{ll}-& 3g_{n-2}\phi \left(iI\right)+2\eta {\overline{g}}_{n-2}\phi {\left(iI\right)}^{*}+{2\left|\eta \right|}^2{g}_{n-2}\phi \left(iI\right)-{\left|\eta \right|}^2{g}_{n-2}i\phi {\left(iI\right)}^{*}\\ =& -2g_{n-2}i\phi \left(I\right)-g_{n-2}\phi \left(iI\right)-2\eta {\overline{g}}_{n-2}\phi \left(iI\right)\\ & +{2\left|\eta \right|}^2{g}_{n-2}i\phi \left(I\right)-{\left|\eta \right|}^2{g}_{n-2}i\phi {\left(iI\right)}^{*}.\end{array}$$

Using (9), we obtain $2g_{n-2}{(1-|\eta |}^2)\phi \left(iI\right)=2g_{n-2}{(1-|\eta |}^2)i\phi \left(I\right)$, which indicates that

$$\phi \left(iI\right)=i\phi \left(I\right).$$

(10)

Combining (9) and (10), we find that $\phi {\left(I\right)}^{*}=\phi \left(I\right)$. □

Lemma 11.

$\phi \left(I\right)=0$.

Proof. 

Firstly, we need to prove that $\phi \left(g_{k}I\right)=n{g}_k\phi \left(I\right)$ for $1\le k\le n$.

For $k=n$, it is obvious that

$$\phi \left(g_{n}I\right)=\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }I\right)=n{g}_n\phi \left(I\right),$$

that is,

$$\phi \left(\left({\displaystyle \genfrac{}{}{0pt}{}{1}{0}}\right)g_{n-1}I+\left({\displaystyle \genfrac{}{}{0pt}{}{1}{1}}\right)\eta {\overline{g}}_{n-1}I\right)=n\left(\left({\displaystyle \genfrac{}{}{0pt}{}{1}{0}}\right)g_{n-1}+\left({\displaystyle \genfrac{}{}{0pt}{}{1}{1}}\right)\eta {\overline{g}}_{n-1}\right)\phi \left(I\right)$$

(11)

and

$$\begin{array}{l}\phi \left(\left({\displaystyle \genfrac{}{}{0pt}{}{2}{0}}\right)g_{n-2}I+\left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right)\eta {\overline{g}}_{n-2}I+\left({\displaystyle \genfrac{}{}{0pt}{}{2}{2}}\right){\left|\eta \right|}^2{g}_{n-2}I\right)\\ =n\left(\left({\displaystyle \genfrac{}{}{0pt}{}{2}{0}}\right)g_{n-2}+\left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right)\eta {\overline{g}}_{n-2}+\left({\displaystyle \genfrac{}{}{0pt}{}{2}{2}}\right){\left|\eta \right|}^2{g}_{n-2}\right)\phi \left(I\right).\end{array}$$

(12)

For $k=n-1$, since it follows that

$$\begin{array}{ll}\phi & \left(-\left({\displaystyle \genfrac{}{}{0pt}{}{1}{0}}\right)g_{n-1}I+\left({\displaystyle \genfrac{}{}{0pt}{}{1}{1}}\right)\eta {\overline{g}}_{n-1}I\right)\\ & =\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }iI{\diamond }_{\eta }iI\right)\\ & =n\left(-\left({\displaystyle \genfrac{}{}{0pt}{}{1}{0}}\right)g_{n-1}+\left({\displaystyle \genfrac{}{}{0pt}{}{1}{1}}\right)\eta {\overline{g}}_{n-1}\right)\phi \left(I\right),\end{array}$$

(13)

when subtracting (13) by (11), we find that

$$\phi \left(f_1{g}_{n-1}I\right)=n{f}_1{g}_{n-1}\phi \left(I\right)\hspace{1em}(\mathrm{where} f_1=\left|\begin{array}{cc}-\left({\displaystyle \genfrac{}{}{0pt}{}{1}{0}}\right)& \left({\displaystyle \genfrac{}{}{0pt}{}{1}{0}}\right)\\ \left({\displaystyle \genfrac{}{}{0pt}{}{1}{1}}\right)& \left({\displaystyle \genfrac{}{}{0pt}{}{1}{1}}\right)\end{array}\right|=-2),$$

which gives

$$\phi \left(g_{n-1}I\right)=n{g}_{n-1}\phi \left(I\right).$$

Then, we find that

$$\phi \left(\left({\displaystyle \genfrac{}{}{0pt}{}{1}{0}}\right)g_{n-2}I+\left({\displaystyle \genfrac{}{}{0pt}{}{1}{1}}\right)\eta {\overline{g}}_{n-2}I\right)=n\left(\left({\displaystyle \genfrac{}{}{0pt}{}{1}{0}}\right)g_{n-2}+\left({\displaystyle \genfrac{}{}{0pt}{}{1}{1}}\right)\eta {\overline{g}}_{n-2}\right)\phi \left(I\right).$$

(14)

By repeating this construction, we obtain the case for $k=n-2$. Since

$$\begin{array}{ll}& \phi \left(-\left({\displaystyle \genfrac{}{}{0pt}{}{2}{0}}\right)g_{n-2}I+\left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right)\eta {\overline{g}}_{n-2}I-\left({\displaystyle \genfrac{}{}{0pt}{}{2}{2}}\right){\left|\eta \right|}^2{g}_{n-2}I\right)\\ & =\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }iI{\diamond }_{\eta }I{\diamond }_{\eta }iI\right)\\ & =n\left(-\left({\displaystyle \genfrac{}{}{0pt}{}{2}{0}}\right)g_{n-2}+\left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right)\eta {\overline{g}}_{n-2}-\left({\displaystyle \genfrac{}{}{0pt}{}{2}{2}}\right){\left|\eta \right|}^2{g}_{n-2}\right)\phi \left(I\right),\end{array}$$

(15)

using (12) and (15), we find that

$$\phi \left(4\eta {\overline{g}}_{n-2}I\right)=n\left(4\eta {\overline{g}}_{n-2}\right)\phi \left(I\right).$$

(16)

Hence, using (14) and (16), we have

$$\phi \left(f_2{g}_{n-2}I\right)=n\left(f_2{g}_{n-2}\right)\phi \left(I\right)\hspace{1em}(\mathrm{where} f_2=\left|\begin{array}{ccc}-\left({\displaystyle \genfrac{}{}{0pt}{}{2}{0}}\right)& \left({\displaystyle \genfrac{}{}{0pt}{}{1}{0}}\right)& \left({\displaystyle \genfrac{}{}{0pt}{}{2}{0}}\right)\\ \left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right)& \left({\displaystyle \genfrac{}{}{0pt}{}{1}{1}}\right)& \left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right)\\ -\left({\displaystyle \genfrac{}{}{0pt}{}{2}{2}}\right)& 0& \left({\displaystyle \genfrac{}{}{0pt}{}{2}{2}}\right)\end{array}\right|=-4).$$

Then, we obtain $\phi \left(g_{n-2}I\right)=n\left(g_{n-2}\right)\phi \left(I\right)$.

We will prove the following assertion: if $\phi \left(g_{m}I\right)=n{g}_m\phi \left(I\right)$ holds for all integers m with $k\le m\le n$, then $\phi \left(g_{k-1}I\right)=n{g}_{k-1}\phi \left(I\right)$ for a fixed integer k: $2\le k\le n-1$.

Let $t=k-1$ and $s=n-t$. Then, we use $h_j$ to define some expressions related to $\eta$:

$$h_j=\left\{\begin{array}{ll}{\eta \left|\eta \right|}^{j-1}{\overline{g}}_t& j\mathrm{is}\mathrm{an}\mathrm{odd}\mathrm{number}\mathrm{in}\{0,1,2,\cdots ,s-1,s\},\\ {\left|\eta \right|}^j{g}_t& j\mathrm{is}\mathrm{an}\mathrm{even}\mathrm{number}\mathrm{in}\{0,1,2,\cdots ,s-1,s\}.\end{array}\right.$$

Since $\phi \left(g_m\left(\eta \right)I\right)=n{g}_m\left(\eta \right)\phi \left(I\right)$ for $k\le m\le n$, we obtain the following:

For $m=n$, $\phi \left(g_{n}I\right)=n{g}_n\phi \left(I\right)$, that is,

$$\begin{array}{ll}& \phi \left(\left(\left({\displaystyle \genfrac{}{}{0pt}{}{s}{0}}\right)h_0+\left({\displaystyle \genfrac{}{}{0pt}{}{s}{1}}\right)h_1+\cdots +\left({\displaystyle \genfrac{}{}{0pt}{}{s}{j}}\right)h_j+\cdots +\left({\displaystyle \genfrac{}{}{0pt}{}{s}{s-1}}\right)h_{s-1}+\left({\displaystyle \genfrac{}{}{0pt}{}{s}{s}}\right)h_s\right)I\right)\\ & =n\left(\left({\displaystyle \genfrac{}{}{0pt}{}{s}{0}}\right)h_0+\left({\displaystyle \genfrac{}{}{0pt}{}{s}{1}}\right)h_1+\cdots +\left({\displaystyle \genfrac{}{}{0pt}{}{s}{j}}\right)h_j+\cdots +\left({\displaystyle \genfrac{}{}{0pt}{}{s}{s-1}}\right)h_{s-1}+\left({\displaystyle \genfrac{}{}{0pt}{}{s}{s}}\right)h_s\right)\phi \left(I\right).\end{array}$$

(17)

For $m=n-1$, $\phi \left(g_{n-1}I\right)=n{g}_{n-1}\phi \left(I\right)$, that is,

$$\begin{array}{cc}& \phi \left(\left(\left({\displaystyle \genfrac{}{}{0pt}{}{s-1}{0}}\right)h_0+\left({\displaystyle \genfrac{}{}{0pt}{}{s-1}{1}}\right)h_1+\cdots +\left({\displaystyle \genfrac{}{}{0pt}{}{s-1}{j}}\right)h_j+\cdots +\left({\displaystyle \genfrac{}{}{0pt}{}{s-1}{s-1}}\right)h_{s-1}\right)I\right)\hfill \\ & =n\left(\left({\displaystyle \genfrac{}{}{0pt}{}{s-1}{0}}\right)h_0+\left({\displaystyle \genfrac{}{}{0pt}{}{s-1}{1}}\right)h_1+\cdots +\left({\displaystyle \genfrac{}{}{0pt}{}{s-1}{j}}\right)h_j+\cdots +\left({\displaystyle \genfrac{}{}{0pt}{}{s-1}{s-1}}\right)h_{s-1}\right)\phi \left(I\right).\hfill \\ & \hfill \cdots \cdots ,\hfill \end{array}$$

(18)

For $m=k+1$, $\phi \left(g_{k+1}I\right)=n{g}_{k+1}\phi \left(I\right)$, that is,

$$\begin{array}{ll}& \phi \left(\left(\left({\displaystyle \genfrac{}{}{0pt}{}{2}{0}}\right)h_0+\left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right)h_1+\left({\displaystyle \genfrac{}{}{0pt}{}{2}{2}}\right)h_2\right)I\right)\\ & =n\left(\left({\displaystyle \genfrac{}{}{0pt}{}{2}{0}}\right)h_0+\left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right)h_1+\left({\displaystyle \genfrac{}{}{0pt}{}{2}{2}}\right)h_2\right)\phi \left(I\right).\end{array}$$

(19)

For $m=k$, $\phi \left(g_{k}I\right)=n{g}_k\phi \left(I\right)$, that is,

$$\phi \left(\left(\left({\displaystyle \genfrac{}{}{0pt}{}{1}{0}}\right)h_0+\left({\displaystyle \genfrac{}{}{0pt}{}{1}{1}}\right)h_1\right)I\right)=n\left(\left({\displaystyle \genfrac{}{}{0pt}{}{1}{0}}\right)h_0+\left({\displaystyle \genfrac{}{}{0pt}{}{1}{1}}\right)h_1\right)\phi \left(I\right).$$

(20)

On the other hand, since

$$\begin{array}{ll}& \phi \left(\left(-\left({\displaystyle \genfrac{}{}{0pt}{}{s}{0}}\right)h_0+\left({\displaystyle \genfrac{}{}{0pt}{}{s}{1}}\right)h_1-\left({\displaystyle \genfrac{}{}{0pt}{}{s}{2}}\right)h_2+\cdots +{(-1)}^{j+1}\left({\displaystyle \genfrac{}{}{0pt}{}{s}{j}}\right)h_j+\cdots +{(-1)}^{s+1}\left({\displaystyle \genfrac{}{}{0pt}{}{s}{s}}\right)h_s\right)I\right)\\ & =\phi \left(\underset{k-1}{\underbrace{I{\diamond }_{\eta }\cdots {\diamond }_{\eta }iI}}{\diamond }_{\eta }\underset{s}{\underbrace{I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }iI}}\right)\\ & =n\left(-\left({\displaystyle \genfrac{}{}{0pt}{}{s}{0}}\right)h_0+\left({\displaystyle \genfrac{}{}{0pt}{}{s}{1}}\right)h_1-\left({\displaystyle \genfrac{}{}{0pt}{}{s}{2}}\right)h_2+\cdots +{(-1)}^{j+1}\left({\displaystyle \genfrac{}{}{0pt}{}{s}{j}}\right)h_j+\cdots +{(-1)}^{s+1}\left({\displaystyle \genfrac{}{}{0pt}{}{s}{s}}\right)h_s\right)\phi \left(I\right),\end{array}$$

using (17), if s is even, then

$$\begin{array}{ll}& 2\phi \left(\left(\left({\displaystyle \genfrac{}{}{0pt}{}{s}{1}}\right)h_1+\left({\displaystyle \genfrac{}{}{0pt}{}{s}{3}}\right)h_3+\cdots +\left({\displaystyle \genfrac{}{}{0pt}{}{s}{s-1}}\right)h_{s-1}\right)I\right)\\ & =2n\left(\left({\displaystyle \genfrac{}{}{0pt}{}{s}{1}}\right)h_1+\left({\displaystyle \genfrac{}{}{0pt}{}{s}{3}}\right)h_3+\cdots +\left({\displaystyle \genfrac{}{}{0pt}{}{s}{s-1}}\right)h_{s-1}\right)\phi \left(I\right),\end{array}$$

and if s is odd, then

$$\begin{array}{ll}& 2\phi \left(\left(-\left({\displaystyle \genfrac{}{}{0pt}{}{s}{0}}\right)h_0-\left({\displaystyle \genfrac{}{}{0pt}{}{s}{2}}\right)h_2-\cdots -\left({\displaystyle \genfrac{}{}{0pt}{}{s}{s-1}}\right)h_{s-1}\right)I\right)\\ & =2n\left(-\left({\displaystyle \genfrac{}{}{0pt}{}{s}{0}}\right)h_0-\left({\displaystyle \genfrac{}{}{0pt}{}{s}{2}}\right)h_2-\cdots -\left({\displaystyle \genfrac{}{}{0pt}{}{s}{s-1}}\right)h_{s-1}\right)\phi \left(I\right).\end{array}$$

Next, using (18), we find that if s is even, then

$$\begin{array}{ll}& 2\phi \left(\left(-\left({\displaystyle \genfrac{}{}{0pt}{}{s}{s-1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{s-1}{0}}\right)h_0+\left(\left({\displaystyle \genfrac{}{}{0pt}{}{s}{1}}\right)-\left({\displaystyle \genfrac{}{}{0pt}{}{s}{s-1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{s-1}{1}}\right)\right)h_1+\cdots -\left({\displaystyle \genfrac{}{}{0pt}{}{s}{s-1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{s-1}{s-2}}\right)h_{s-2}\right)I\right)\\ & =2n\left(-\left({\displaystyle \genfrac{}{}{0pt}{}{s}{s-1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{s-1}{0}}\right)h_0+\left(\left({\displaystyle \genfrac{}{}{0pt}{}{s}{1}}\right)-\left({\displaystyle \genfrac{}{}{0pt}{}{s}{s-1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{s-1}{1}}\right)\right)h_1+\cdots -\left({\displaystyle \genfrac{}{}{0pt}{}{s}{s-1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{s-1}{s-2}}\right)h_{s-2}\right)\phi \left(I\right).\end{array}$$

and if s is odd, then

$$\begin{array}{ll}& 2\phi \left(\left((-\left({\displaystyle \genfrac{}{}{0pt}{}{s}{0}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{s}{s-1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{s-1}{0}}\right))h_0+\left({\displaystyle \genfrac{}{}{0pt}{}{s}{s-1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{s-1}{1}}\right)h_1+\cdots +\left({\displaystyle \genfrac{}{}{0pt}{}{s}{s-1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{s-1}{s-2}}\right)h_{s-2}\right)I\right)\\ & =2n\left((-\left({\displaystyle \genfrac{}{}{0pt}{}{s}{0}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{s}{s-1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{s-1}{0}}\right))h_0+\left({\displaystyle \genfrac{}{}{0pt}{}{s}{s-1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{s-1}{1}}\right)h_1+\cdots +\left({\displaystyle \genfrac{}{}{0pt}{}{s}{s-1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{s-1}{s-2}}\right)h_{s-2}\right)\phi \left(I\right).\end{array}$$

Continuing the computation step by step, we eventually obtain

$$\phi \left(f_s{g}_{t}I\right)=n{f}_s{g}_t\phi \left(I\right),$$

where

$$f_s=\left|\begin{array}{cccccc}-\left({\displaystyle \genfrac{}{}{0pt}{}{s}{0}}\right)& \left({\displaystyle \genfrac{}{}{0pt}{}{1}{0}}\right)& \left({\displaystyle \genfrac{}{}{0pt}{}{2}{0}}\right)& \left({\displaystyle \genfrac{}{}{0pt}{}{3}{0}}\right)& \cdots & \left({\displaystyle \genfrac{}{}{0pt}{}{s}{0}}\right)\\ \left({\displaystyle \genfrac{}{}{0pt}{}{s}{1}}\right)& \left({\displaystyle \genfrac{}{}{0pt}{}{1}{1}}\right)& \left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right)& \left({\displaystyle \genfrac{}{}{0pt}{}{3}{1}}\right)& \cdots & \left({\displaystyle \genfrac{}{}{0pt}{}{s}{1}}\right)\\ -\left({\displaystyle \genfrac{}{}{0pt}{}{s}{2}}\right)& 0& \left({\displaystyle \genfrac{}{}{0pt}{}{2}{2}}\right)& \left({\displaystyle \genfrac{}{}{0pt}{}{3}{2}}\right)& \cdots & \left({\displaystyle \genfrac{}{}{0pt}{}{s}{2}}\right)\\ \left({\displaystyle \genfrac{}{}{0pt}{}{s}{3}}\right)& 0& 0& \left({\displaystyle \genfrac{}{}{0pt}{}{3}{3}}\right)& \cdots & \left({\displaystyle \genfrac{}{}{0pt}{}{s}{3}}\right)\\ ⋮& ⋮& ⋮& ⋮& & ⋮\\ {(-1)}^{s+1}\left({\displaystyle \genfrac{}{}{0pt}{}{s}{s}}\right)& 0& 0& 0& \cdots & \left({\displaystyle \genfrac{}{}{0pt}{}{s}{s}}\right)\end{array}\right|=-2^s,$$

which indicates that $\phi \left(g_{k-1}I\right)=n{g}_{k-1}\phi \left(I\right)$.

Then, we find that $\phi \left(g_{k}I\right)=n{g}_k\phi \left(I\right)$ for $1\le k\le n$. Taking $k=1$, we find that $\phi \left(I\right)=n\phi \left(I\right)$. Then, $\phi \left(I\right)=0$. □

Remark 1.

We infer that $\phi \left(I\right)=0$ by virtue of the recursive property, proceeding from higher to lower orders.

Lemma 12.

$\phi \left(A^{*}\right)=\phi {\left(A\right)}^{*}$ for all $A\in \mathcal{A}$.

Proof. 

For all $A\in \mathcal{A}$, we have $\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }A{\diamond }_{\eta }iI{\diamond }_{\eta }iI\right)=I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }\phi \left(A\right){\diamond }_{\eta }iI{\diamond }_{\eta }iI$; that is,

$$\phi (g_{n-2}{\left(\right|\eta |}^2-1\left)A\right)=g_{n-2}{\left(\right|\eta |}^2-1)\phi \left(A\right).$$

(21)

Since $\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }A{\diamond }_{\eta }I{\diamond }_{\eta }I\right)=I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }\phi \left(A\right){\diamond }_{\eta }I{\diamond }_{\eta }I$, it follows that

$$\phi (g_{n-2}A+2\eta {\overline{g}}_{n-2}{A}^{*}+{\left|\eta \right|}^2{g}_{n-2}A)=g_{n-2}\phi \left(A\right)+2\eta {\overline{g}}_{n-2}\phi {\left(A\right)}^{*}+{\left|\eta \right|}^2{g}_{n-2}\phi \left(A\right).$$

This, along with (21), indicates that

$$\phi (g_{n-2}A+\eta {\overline{g}}_{n-2}{A}^{*})=g_{n-2}\phi \left(A\right)+\eta {\overline{g}}_{n-2}\phi {\left(A\right)}^{*}.$$

(22)

Furthermore, for all $A\in \mathcal{A}$, we have $\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }A\right)=I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }\phi \left(A\right),$ which gives

$$\phi (g_{n-2}A+2\eta {\overline{g}}_{n-2}A+{\left|\eta \right|}^2{g}_{n-2}A)=g_{n-2}\phi \left(A\right)+2\eta {\overline{g}}_{n-2}\phi \left(A\right)+{\left|\eta \right|}^2{g}_{n-2}\phi \left(A\right).$$

(23)

Using (21) and (23), we find that

$$\phi (g_{n-2}A+\eta {\overline{g}}_{n-2}A)=g_{n-2}\phi \left(A\right)+\eta {\overline{g}}_{n-2}\phi \left(A\right),$$

(24)

On the other hand, since for $A\in \mathcal{A}$,

$$\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }iI{\diamond }_{\eta }A{\diamond }_{\eta }iI\right)=I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }iI{\diamond }_{\eta }\phi \left(A\right){\diamond }_{\eta }iI$$

and

$$\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }A{\diamond }_{\eta }I\right)=I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }\phi \left(A\right){\diamond }_{\eta }I,$$

it follows that

$$\begin{array}{ll}& \phi (-g_{n-2}A+\eta {\overline{g}}_{n-2}A+\eta {\overline{g}}_{n-2}{A}^{*}-|\eta {|}^2{g}_{n-2}{A}^{*})\\ & =-g_{n-2}\phi \left(A\right)+\eta {\overline{g}}_{n-2}\phi \left(A\right)+\eta {\overline{g}}_{n-2}\phi {\left(A\right)}^{*}-{\left|\eta \right|}^2{g}_{n-2}\phi {\left(A\right)}^{*}\end{array}$$

(25)

and

$$\begin{array}{ll}& \phi (g_{n-2}A+\eta {\overline{g}}_{n-2}A+\eta {\overline{g}}_{n-2}{A}^{*}+|\eta {|}^2{g}_{n-2}{A}^{*})\\ & =g_{n-2}\phi \left(A\right)+\eta {\overline{g}}_{n-2}\phi \left(A\right)+\eta {\overline{g}}_{n-2}\phi {\left(A\right)}^{*}+{\left|\eta \right|}^2{g}_{n-2}\phi {\left(A\right)}^{*}.\end{array}$$

(26)

By applying (25) and (26), we obtain

$$\phi \left(\eta {\overline{g}}_{n-2}(A+A^{*})\right)=\eta {\overline{g}}_{n-2}(\phi \left(A\right)+\phi {\left(A\right)}^{*}).$$

Moreover, using (22) and (24), we have

$$\phi \left(\eta {\overline{g}}_{n-2}(A-A^{*})\right)=\eta {\overline{g}}_{n-2}(\phi \left(A\right)-\phi {\left(A\right)}^{*}).$$

Hence,

$$\phi \left(\eta {\overline{g}}_{n-2}A\right)=\eta {\overline{g}}_{n-2}\phi \left(A\right).$$

(27)

In addition, using (24) and (27), we find that

$$\phi \left(g_{n-2}A\right)=g_{n-2}\phi \left(A\right).$$

(28)

Finally, using (22), (27) and (28), we obtain $\phi \left(A^{*}\right)=\phi {\left(A\right)}^{*}$. □

Lemma 13.

$\phi \left(\eta A\right)=\eta \phi \left(A\right)$ for all $A\in \mathcal{A}$.

Proof. 

For all $A\in \mathcal{A}$, using (24), we find that

$$\phi \left(g_{n-1}A\right)=g_{n-1}\phi \left(A\right).$$

(29)

Since

$$\phi (g_{n-1}A+\eta {\overline{g}}_{n-1}A)=\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }A\right)=g_{n-1}\phi \left(A\right)+\eta {\overline{g}}_{n-1}\phi \left(A\right),$$

it follows that

$$\phi \left(\eta {\overline{g}}_{n-1}A\right)=\eta {\overline{g}}_{n-1}\phi \left(A\right).$$

Then,

$$\begin{array}{ll}& \eta {\overline{g}}_{n-1}\phi \left(A\right)=\phi \left(\eta {\overline{g}}_{n-1}A\right)=\phi \left({\left(g_{n-1}\overline{\eta }{A}^{*}\right)}^{*}\right)\\ & =\phi {\left(g_{n-1}\overline{\eta }{A}^{*}\right)}^{*}={\left(g_{n-1}\phi \left(\overline{\eta }{A}^{*}\right)\right)}^{*}={\overline{g}}_{n-1}\phi {\left(\overline{\eta }{A}^{*}\right)}^{*}={\overline{g}}_{n-1}\phi \left(\eta A\right).\end{array}$$

Thus, $\phi \left(\eta A\right)=\eta \phi \left(A\right)$. □

Lemma 14.

$\phi \left(iA\right)=i\phi \left(A\right)$ for all $A\in \mathcal{A}$.

Proof. 

For every $A=A^{*}\in \mathcal{A}$, we have

$$\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }iI{\diamond }_{\eta }A{\diamond }_{\eta }I\right)=I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }iI{\diamond }_{\eta }\phi \left(A\right){\diamond }_{\eta }I$$

and

$$\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }A{\diamond }_{\eta }iI\right)=I{\diamond }_{\eta }\cdots {\diamond }_{\eta }I{\diamond }_{\eta }I{\diamond }_{\eta }\phi \left(A\right){\diamond }_{\eta }iI,$$

and it follows that

$$\phi (g_{n-2}iA-2\eta {\overline{g}}_{n-2}iA+{\left|\eta \right|}^2{g}_{n-2}iA)=g_{n-2}i\phi \left(A\right)-2\eta {\overline{g}}_{n-2}i\phi \left(A\right)+{\left|\eta \right|}^2{g}_{n-2}i\phi \left(A\right)$$

and

$$\phi (g_{n-2}iA+2\eta {\overline{g}}_{n-2}iA+{\left|\eta \right|}^2{g}_{n-2}iA)=g_{n-2}i\phi \left(A\right)+2\eta {\overline{g}}_{n-2}i\phi \left(A\right)+{\left|\eta \right|}^2{g}_{n-2}i\phi \left(A\right).$$

Then,

$$\phi \left(\eta {\overline{g}}_{n-2}\left(iA\right)\right)=\eta {\overline{g}}_{n-2}i\phi \left(A\right).$$

Using (27), we find that

$$\phi \left(\eta {\overline{g}}_{n-2}\left(iA\right)\right)=\eta {\overline{g}}_{n-2}\phi \left(iA\right)=\eta {\overline{g}}_{n-2}i\phi \left(A\right),$$

which indicates that $\phi \left(iA\right)=i\phi \left(A\right)$ for every $A=A^{*}\in \mathcal{A}$.

Therefore, for all $A\in \mathcal{A}$, since $A=A_1+A_{2}i$, where $A_1={\displaystyle \frac{A+A^{*}}{2}}$ and $A_2={\displaystyle \frac{A-A^{*}}{2i}}$, we have

$$\begin{array}{ll}\phi \left(iA\right)& =\phi \left(i(A_1+A_{2}i)\right)=\phi (i{A}_1-A_2)=\phi \left(i{A}_1\right)-\phi \left(A_2\right)\\ & =i\phi \left(A_1\right)+i^2\phi \left(A_2\right)=i(\phi \left(A_1\right)+i\phi \left(A_2\right))=i\phi \left((A_1+i{A}_2)\right)=i\phi \left(A\right).\end{array}$$

Then, we can prove that $\phi \left(iA\right)=i\phi \left(A\right).$ □

Lemma 15.

$\phi \left(AB\right)=\phi \left(A\right)B+A\phi \left(B\right)$ for all $A,B\in \mathcal{A}$.

Proof. 

For every $A,B\in \mathcal{A}$, we have

$$\begin{array}{ll}& \phi (g_{n-1}AB+\eta {\overline{g}}_{n-1}B{A}^{*})\\ & =\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }A{\diamond }_{\eta }B\right)\\ & =I{\diamond }_{\eta }\cdots {\diamond }_{\eta }\phi \left(A\right){\diamond }_{\eta }B+I{\diamond }_{\eta }\cdots {\diamond }_{\eta }A{\diamond }_{\eta }\phi \left(B\right)\\ & =g_{n-1}\phi \left(A\right)B+\eta {\overline{g}}_{n-1}B\phi {\left(A\right)}^{*}+g_{n-1}A\phi \left(B\right)+\eta {\overline{g}}_{n-1}\phi \left(B\right)A^{*}.\end{array}$$

(30)

On the other hand, we have

$$\begin{array}{ll}& \phi (-g_{n-1}AB+\eta {\overline{g}}_{n-1}B{A}^{*})\\ & =\phi \left(I{\diamond }_{\eta }\cdots {\diamond }_{\eta }iA{\diamond }_{\eta }iB\right)\\ & =I{\diamond }_{\eta }\cdots {\diamond }_{\eta }\phi \left(iA\right){\diamond }_{\eta }iB+I{\diamond }_{\eta }\cdots {\diamond }_{\eta }iA{\diamond }_{\eta }\phi \left(iB\right)\\ & =-g_{n-1}\phi \left(A\right)B+\eta {\overline{g}}_{n-1}B\phi {\left(A\right)}^{*}-g_{n-1}A\phi \left(B\right)+\eta {\overline{g}}_{n-1}\phi \left(B\right)A^{*}.\end{array}$$

(31)

Combining (30) and (31), we find that

$$\phi \left(g_{n-1}AB\right)=g_{n-1}(\phi \left(A\right)B+A\phi \left(B\right)).$$

Then, using (29), we obtain

$$g_{n-1}\phi \left(AB\right)=g_{n-1}(\phi \left(A\right)B+A\phi \left(B\right)).$$

It follows that $\phi \left(AB\right)=\phi \left(A\right)B+A\phi \left(B\right)$. □

Remark 2.

Firstly, we find that φ is an additive by virtue of the Peirce decomposition. Then, we derive that $\phi \left(I\right)=0$ by virtue of the recursive property, proceeding from higher to lower orders. Finally, we verify that φ is a ∗-derivation.

4. Corollaries

Taking into account certain special classes of ∗-algebras and our main result, which have great significance in functional analysis and operator theory, such as in the Classification of Operator Algebras, we can present some consequences. The first one is as follows:

Corollary 1.

Let $\mathcal{A}$ be a unital ∗-algebra with the unit I and let $\eta \ne 0,\pm 1$ be a complex number. Assume that $\mathcal{A}$ contains a nontrivial projection P, which satisfies

(♠)

$X\mathcal{A}P=0\Rightarrow X=0$;

(♣)

$X\mathcal{A}(I-P)=0\Rightarrow X=0$.

If the map $\phi :\mathcal{A}\to \mathcal{A}$ satisfies

$$\phi (A_1{\diamond }_{\eta }{A}_2{\diamond }_{\eta }\cdots {\diamond }_{\eta }{A}_n)=\sum _{k=1}^n{A}_1{\diamond }_{\eta }\cdots {\diamond }_{\eta }{A}_{k-1}{\diamond }_{\eta }\phi \left(A_k\right){\diamond }_{\eta }{A}_{k+1}{\diamond }_{\eta }\cdots {\diamond }_{\eta }{A}_n$$

for all $A_1,A_2\cdots A_n\in \mathcal{A}$, where $n\ge 3$ is a fixed positive integer, then φ is an additive ∗-derivation and $\phi \left(\eta A\right)=\eta \phi \left(A\right)$ for all $A\in \mathcal{A}$.

An algebra $\mathcal{A}$ is prime if for $A,B\in \mathcal{A}$, $A\mathcal{A}B=0$ implies that $A=0\mathrm{or}B=0$. Obviously, the prime ∗-algebras satisfy (♠) and (♣), then we have the following result:

Corollary 2.

Suppose $\mathcal{A}$ is a unital prime ∗-algebra with a nontrivial projection P and let $\eta \ne 0,\pm 1$ be a complex number. If a map $\phi :\mathcal{A}\to \mathcal{A}$ satisfies

$$\phi (A_1{\diamond }_{\eta }{A}_2{\diamond }_{\eta }\cdots {\diamond }_{\eta }{A}_n)=\sum _{k=1}^n{A}_1{\diamond }_{\eta }\cdots {\diamond }_{\eta }{A}_{k-1}{\diamond }_{\eta }\phi \left(A_k\right){\diamond }_{\eta }{A}_{k+1}{\diamond }_{\eta }\cdots {\diamond }_{\eta }{A}_n$$

for all $A_1,A_2\cdots A_n\in \mathcal{A}$, where $n\ge 3$ is a fixed positive integer, then φ is an additive ∗-derivation and $\phi \left(\eta A\right)=\eta \phi \left(A\right)$ for all $A\in \mathcal{A}$.

Remark 3.

We can see that a unital ∗-algebra with the unit I contains a nontrivial projection P satisfying (♠) and (♣) has a broader scope than the unital prime ∗-algebras.

Example 1

([18]). Let $\mathcal{R}$ be an alternative ring with an idempotent $e\ne 0,1$. Consider the multiplication table given by

$$\begin{array}{|cccccc|}\hline ·& e& a_{11}& b_{12}& c_{21}& d_{22}\\ e& e& a_{11}& b_{12}& 0& 0\\ a_{11}& a_{11}& a_{11}& b_{12}& 0& 0\\ b_{12}& 0& 0& 0& a_{11}& b_{12}\\ c_{21}& c_{21}& 0& d_{22}& 0& 0\\ d_{22}& 0& 0& 0& c_{21}& d_{22}\\ \hline\end{array}$$

By a direct computation it can be verified that $\mathfrak{R}$ is not prime and satisfies the conditions of Theorem 1.

A von Neumann algebra $\mathcal{A}$ is a weakly closed, self-adjoint algebra of operators on a Hilbert space H containing the identity operator I. It is shown in [9] that every von Neumann algebra with no central summands of type $I_1$ satisfies (♠) and (♣). Therefore, we have the following result:

Corollary 3.

Let $\mathcal{A}$ be a von Neumann algebra with no central summands of type $I_1$ and let $\eta \ne 0,\pm 1$ be a complex number. Consider the map $\phi :\mathcal{A}\to \mathcal{A}$ satisfying

$$\phi (A_1{\diamond }_{\eta }{A}_2{\diamond }_{\eta }\cdots {\diamond }_{\eta }{A}_n)=\sum _{k=1}^n{A}_1{\diamond }_{\eta }\cdots {\diamond }_{\eta }{A}_{k-1}{\diamond }_{\eta }\phi \left(A_k\right){\diamond }_{\eta }{A}_{k+1}{\diamond }_{\eta }\cdots {\diamond }_{\eta }{A}_n$$

for all $A_1,A_2\cdots A_n\in \mathcal{A}$, where $n\ge 3$ is a fixed positive integer, then φ is an additive ∗-derivation and $\phi \left(\eta A\right)=\eta \phi \left(A\right)$ for all $A\in \mathcal{A}$.

$\mathcal{A}$ is a factor von Neumann algebra if its center only contains the scalar operators. It is well known that a factor von Neumann algebra is prime, which indicates that it always satisfies (♠) and (♣). Hence, we obtain the following:

Corollary 4.

Let $\mathcal{A}$ be a factor von Neumann algebra with $dim\left(\mathcal{A}\right)\ge 2$ and let $\eta \ne 0,\pm 1$ be a complex number. Then, if $\phi :\mathcal{A}\to \mathcal{A}$, satisfying

$$\phi (A_1{\diamond }_{\eta }{A}_2{\diamond }_{\eta }\cdots {\diamond }_{\eta }{A}_n)=\sum _{k=1}^n{A}_1{\diamond }_{\eta }\cdots {\diamond }_{\eta }{A}_{k-1}{\diamond }_{\eta }\phi \left(A_k\right){\diamond }_{\eta }{A}_{k+1}{\diamond }_{\eta }\cdots {\diamond }_{\eta }{A}_n$$

for all $A_1,A_2\cdots A_n\in \mathcal{A}$, where $n\ge 3$ is a fixed positive integer, then φ is an additive ∗-derivation and $\phi \left(\eta A\right)=\eta \phi \left(A\right)$ for all $A\in \mathcal{A}$.

5. Further Discussion

Let $\mathcal{A}$ be a ∗-algebra over the complex field $\mathbb{C}$ and $\eta$ be a nonzero scalar. For any $A,B\in \mathcal{A}$, define a new product of A and B by $A{♢}_{\eta }B=A^{*}B+\eta B^{*}A$. This new product is said to be the bi-skew $\eta$-Jordan product. Clearly, the bi-skew 1-Jordan product is the so-called bi-skew Jordan product ($A^{*}B+B^{*}A$ product), and the bi-skew $(-1)$-Jordan product is the so-called bi-skew Lie product ($A^{*}B-B^{*}A$ product). The products have attracted much authors’ attention [19,20,21,22,23,24,25,26,27,28]. Let $\phi :\mathcal{A}⟶\mathcal{A}$ be a mapping (without the additivity assumption). The map $\phi$ is called a nonlinear bi-skew $\eta$-Jordan derivation if

$$\phi \left(A{♢}_{\eta }B\right)=\phi \left(A\right){♢}_{\eta }B+A{♢}_{\eta }\phi \left(B\right),$$

holds ture for all $A,B\in \mathcal{A}$. Similarly, a mapping $\phi :\mathcal{A}⟶\mathcal{A}$ is called a nonlinear bi-skew $\eta$-Jordan triple derivation if it satisfies the condition

$$\phi \left(A{♢}_{\eta }B{♢}_{\eta }C\right)=\phi \left(A\right){♢}_{\eta }B{♢}_{\eta }C+A{♢}_{\eta }\phi \left(B\right){♢}_{\eta }C+A{♢}_{\eta }B{♢}_{\eta }\phi \left(C\right)$$

for all $A,B,C\in \mathcal{A}$, where $A{♢}_{\eta }B{♢}_{\eta }C=\left(A{♢}_{\eta }B\right){♢}_{\eta }C$. We should note that ${♢}_{\eta }$ is not necessarily associative.

Taking into account the definitions of bi-skew $\eta$-Jordan derivations and bi-skew $\eta$-Jordan triple derivations, we can further develop them in one natural way. Suppose that $n\ge 2$ is a fixed positive integer. Let us see a sequence of polynomials with scalar $\eta$ and *:

$$\begin{array}{cc}\hfill q_1\left(x_1\right)& =x_1,\hfill \\ \hfill q_2(x_1,x_2)& =x_1{♢}_{\eta }{x}_2=x_1^{*}{x}_2+\eta x_2^{*}{x}_1,\hfill \\ \hfill q_3(x_1,x_2,x_3)& =q_2(x_1,x_2){♢}_{\eta }{x}_3=\left(x_1{♢}_{\eta }{x}_2\right){♢}_{\eta }{x}_3,\hfill \\ \hfill q_4(x_1,x_2,x_3,x_4)& =q_3(x_1,x_2,x_3){♢}_{\eta }{x}_4=\left(\left(x_1{♢}_{\eta }{x}_2\right){♢}_{\eta }{x}_3\right){♢}_{\eta }{x}_4,\hfill \\ & \cdots \cdots ,\hfill \\ \hfill q_n(x_1,x_2,\cdots ,x_n)& =q_{n-1}(x_1,x_2,\cdots ,x_{n-1}){♢}_{\eta }{x}_n\hfill \\ & =\underset{n-1}{\underbrace{(\cdots \left(\left(x_1{♢}_{\eta }{x}_2\right){♢}_{\eta }{x}_3\right){♢}_{\eta }\cdots {♢}_{\eta }{x}_{n-1})}}{♢}_{\eta }{x}_n.\hfill \end{array}$$

Accordingly, a nonlinear bi-skew $\eta$-Jordan n-derivation is a mapping $\phi :\mathcal{A}⟶\mathcal{A}$ satisfying the condition

$$\phi \left(q_n(A_1,A_2,\cdots ,A_n)\right)=\sum _{k=1}^n{q}_n(A_1,\cdots ,A_{k-1},\phi \left(A_k\right),A_{k+1},\cdots ,A_n)$$

for all $A_1,A_2,\cdots ,A_n\in \mathcal{A}$. From the definition, it is clear that each bi-skew Jordan derivation is a bi-skew 1-Jordan 2-derivation, and every bi-skew Jordan n-derivation is a bi-skew 1-Jordan n-derivation. Zhang and Zhu [24] studied bi-skew 1-Jordan n-derivations on ∗-algebra. Shavandi et al. [27] and Zhang [28] studied bi-skew (-1)-Jordan n-derivations on factor von Neumann algebras and on ∗-algebra, respectively. A fundamental advancement for this line of research is to investigate whether each nonlinear bi-skew $\eta$-Jordan n-derivation on ∗-algebras is an additive ∗-derivation. In view of the current work and existing results in this direction, we propose the following open question:

Question. Let $\mathcal{A}$ be a unital ∗-algebra with the unit I. Assume that $\mathcal{A}$ contains a nontrivial projection P which satisfies

$$X\mathcal{A}P=0\Rightarrow X=0$$

and

$$X\mathcal{A}(I-P)=0\Rightarrow X=0.$$

Then, $\phi$ is a nonlinear bi-skew $\eta$-Jordan n-derivation on $\mathcal{A}$ if and only if $\mathsf{\Phi }$ is an additive ∗-derivation and $\mathsf{\Phi }\left(\eta A\right)=\eta \mathsf{\Phi }\left(A\right)$ for all $A\in \mathcal{A}$?

6. Conclusions

The purpose of this article was to prove that a nonlinear $\eta$-∗-Jordan n-derivation($n\ge 3$) on ∗-algebras is an additive ∗-derivation under mild conditions. In application, we applied the result to prime ∗-algebras, von Neumann algebras with no central summands of type $I_1$, and factor von Neumann algebras.