Strong commutativity preserving skew derivations on Banach algebras

Abstract

A map $\delta :\mathcal{S}\subseteq \mathcal{A}\to \mathcal{A}$ is called a strong commutativity preserving (SCP) on $\mathcal{S}$ if $\left[\delta \right(x),\delta (y\left)\right]=[x,y]$ for all $x,y\in \mathcal{S}$, where $\mathcal{A}$ is a Banach algebra over $\mathbb{C}$ or $\mathbb{R}$ and $\mathcal{S}$ is a nonempty subset of $\mathcal{A}$. The main goal of this manuscript is to investigate the SCP skew derivations on Banach algebra $\mathcal{A}$ and we obtain action of skew derivations on rings and the structure of prime Banach algebras.

Keywords:

• Prime Banach algebra
• strong commutativity preserving (SCP) map
• skew derivation

2010 Mathematics Subject Classifications:

• 46J10
• 16N60

1. Introduction

In this manuscript, we investigate strong commutativity preserving (SCP) skew derivations on prime Banach algebra. The concept of SCP derivations is first studied by Bell and Daif [1]. From then onward many researchers have discussed the behaviour of SCP maps on different subsets of rings and established the structure of rings/algebras. Our main motive is to study the SCP maps on Banach algebras, instead of using ring theoretic computations, our approach is somewhat different and based on the theory of topology and analysis.

Throughout the paper $\mathcal{A}$ represents a prime Banach algebra. A linear map d of $\mathcal{A}$ into itself is called a linear derivation if $d\left(xy\right)=d\left(x\right)y+xd\left(y\right)$ for all $x,y\in \mathcal{A}$. Let ζ be an automorphism of $\mathcal{A}$. A linear ζ-derivation $d:\mathcal{A}\to \mathcal{A}$ is a linear map satisfying $d\left(xy\right)=d\left(x\right)y+\zeta \left(x\right)d\left(y\right)\text{ for all }x,y\in \mathcal{A}.$ It is generally called a linear skew derivation. When $\zeta =I_{id}$ on $\mathcal{A},$ linear ζ-derivation is simply an ordinary linear derivation. For $\zeta \ne I_{\mathcal{A}}$, the example of linear ζ-derivation is the map $I_{\mathcal{A}}-\zeta$, where $I_{\mathcal{A}}$ denotes the identity automorphism of $\mathcal{A}$. Thus the results on linear skew derivations are the generalizations of both linear derivations and automorphisms. Moreover, linear skew derivation is a non-identity endomorphism on $\mathcal{A}$.

Numerous results in the literature specify how the general structure of rings and algebras are mostly connected to the behaviour of linear maps on rings and algebras [2–9]. In [1], Bell and Daif explored the commutativity of a ring $\mathcal{R}$ if it satisfies SCP condition on a right ideal of $\mathcal{R}$. Later, Brešar and Miers [10] studied additive SCP maps on semiprime rings and characterized them. In particular, they obtained: Let $\mathcal{F}$ be an additive map on a semiprime ring $\mathcal{R}$ satisfying SCP on $\mathcal{R}$, then $\mathcal{F}\left(x\right)=\eta x+\nu \left(x\right)$, where $\eta \in C$, the extended centroid of $\mathcal{R}$, ${\eta }^2=1$, and ν is an additive map of $\mathcal{R}$ to $\mathcal{C}$. Later, Deng and Ashraf [11] established that if there exists a non-identity endomorphism φ on $\mathcal{R}$, where $\mathcal{R}$ is a prime ring of characteristic different from two, such that $\left[\phi \right(x),\phi (y\left)\right]-[x,y]\in Z\left(\mathcal{R}\right)$ for each x,y in some right ideal of $\mathcal{R}$, then $\mathcal{R}$ is commutative. In 2016, De Filippis et al. [12] discussed the strong commutative preserving skew derivation on ideals of prime rings. More exactly, they proved that: Let $\mathcal{R}$ be a prime ring of characteristic different from 2, C be the extended centroid of $\mathcal{R}$, $Z\left(\mathcal{R}\right)$ be the centre of $\mathcal{R}$, I be a nonzero ideal of $\mathcal{R}$, F and G be two nonzero skew derivations of $\mathcal{R}$ with associated automorphism α and m,n be the positive integers such that $\left[F\right(x),G(y){]}_m=[x,y{]}^n$ for all $x,y\in I$. Then $\mathcal{R}$ is commutative.

In the past few years, numerous algebraists has been investigated a lot of ring theoretic results and try to relate it to Banach algebras by using the standard approach and established some relationship between purely ring theoretic results and Banach algebras. Some of them discussed their results on the theory of Singer–Wermer result [13], who proved that every linear derivation on a Banach algebra maps into its Jacobson radical. In this line of investigation, our aim is to explore SCP skew derivations on Banach algebras. Also our approach is somewhat different and based on the theory of rings, topology and analysis. In particular, we prove the following:

Theorem 1.1

Let δ be a linear continuous skew derivation on $\mathcal{A}$ into itself. Suppose $G_1,$ $G_2$ are open subsets of $\mathcal{A}$ and $1<m,n\in {\mathbb{Z}}^{\mathbb{+}},$ depending on x and y such that $\left[\delta \right(x^m),\delta (y^n\left)\right]-[x^m,y^n]\in Z\left(A\right)$ for each $x\in G_1,$ $y\in G_2$. Then $\mathcal{A}$ is commutative.

Corollary 1.1

Let d be a linear continuous derivation on $\mathcal{A}$ into itself. Suppose $G_1,$ $G_2$ are open subsets of $\mathcal{A}$ and $1<m,n\in {\mathbb{Z}}^{\mathbb{+}},$ depending on x and y such that $\left[d\right(x^m),d(y^n\left)\right]-[x^m,y^n]\in Z\left(A\right)$ for each $x\in G_1,$ $y\in G_2$. Then $\mathcal{A}$ is commutative.

Lastly, in favour of our main theorem, we present the following example:

Example 1.1

Let $\mathcal{A}=\left\{\left(\begin{array}{cc}a& b\\ c& d\end{array}\right):a,b,c,d\in \mathbb{C}\right\}$ be a noncommutative unital prime algebra of all $2\times 2$ matrices over $\mathbb{C}$, where $\mathbb{C}$ is a field of complex numbers, with usual matrix addition, and define matrix multiplication as follows: $X{\times }_{K}Y=KXY,\mathrm{for}\mathrm{all}XY\in \mathcal{A},$ where $K=\left(\begin{array}{cc}\lambda & 0\\ 0& \lambda \end{array}\right)$ and $|\lambda |>1$. For $B=\left({\beta }_{ij}\right)\in \mathcal{A}$, set the Frobenius norm $\parallel B{\parallel }_F$ on $\mathcal{A}$ as follows: $\parallel B{\parallel }_F={\left(\sum _{i,j=1}^2|{\beta }_{ij}{|}^2\right)}^{1/2}\mathrm{for}\mathrm{all}B=\left({\beta }_{ij}\right)\in \mathcal{A}.$ Then, $\mathcal{A}$ is a normed linear space under the defined norm. Further, define a map $\delta :\mathcal{A}\to \mathcal{A}$ by $\delta \left(\begin{array}{cc}a& b\\ c& d\end{array}\right)=\left(\begin{array}{cc}0& b\\ c& 0\end{array}\right)$ and $\zeta \left(\begin{array}{cc}a& b\\ c& d\end{array}\right)=\left(\begin{array}{cc}a& -b\\ -c& d\end{array}\right)$ for every $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \mathcal{A}.$ Since, $\mathcal{A}$ is finite-dimensional, it is straightforward to check that δ is a nonzero continuous linear skew-derivation (ζ-derivation) on $\mathcal{A}$. Observe that $G_1=\left\{\left(\begin{array}{cc}{\mathrm{e}}^{\mathrm{i}t}& 0\\ 0& {\mathrm{e}}^{-\mathrm{i}t}\end{array}\right):t\in \mathbb{R}\right\}$ and $G_2=\left\{\left(\begin{array}{cc}{\mathrm{e}}^{-\mathrm{i}t}& 0\\ 0& {\mathrm{e}}^{\mathrm{i}t}\end{array}\right):t\in \mathbb{R}\right\}$ are open subsets of $\mathcal{A}$ such that $\begin{array}{rl}& \left[\delta \right(X^m),\delta (Y^n\left)\right]-[X^m,Y^n]\in Z\left(\mathcal{A}\right)\\ & \mathrm{for}\mathrm{all}X\in G_1\mathrm{and}Y\in G_2.\end{array}$ Therefore, it follows from Theorem 1.1 that $\mathcal{A}$ is not a Banach algebra under the Frobenius norm $\parallel B{\parallel }_F$.

2. The main result

Before starting the proof of our main theorem, we need the following crucial proposition which is used in the sequel.

Proposition 2.1

Let $\mathcal{R}$ be a prime ring with characteristic not two. If there exists a linear skew derivation δ on $\mathcal{R}$ into itself satisfying $\left[\delta \right(x^m),\delta (y^n\left)\right]-[x^m,y^n]\in Z\left(\mathcal{R}\right)$ for all $x,y\in \mathcal{R},$ where $m,n\in {\mathbb{Z}}^{\mathbb{+}}$. Then $\mathcal{R}$ is commutative unless $char\left(\mathcal{R}\right)=2$ and satisfies $s_4,$ the standard identity of degree 4.

Proof.

Let $S_1=\{r^m\mid r\in \mathcal{R}\}$ and $S_2=\{r^n\mid r\in \mathcal{R}\}$ be additive subgroups. It implies that $\left[\delta \right(x),\delta (y\left)\right]-[x,y]\in Z\left(\mathcal{R}\right)$ for all $x\in S_1$ and $y\in S_2$. By [14, Main Theorem] either $S_1$ have a Lie ideal (non-central) $L_1$ or $r^m\in Z\left(\mathcal{R}\right)$. The latter case concludes $\mathcal{R}$ is commutative. Similarly, assume that there exists a Lie ideal $L_2(⊈Z(\mathcal{R}\left)\right)$ of $\mathcal{R}$ such that $L_2\subseteq S_2$. Moreover, by [15] (page 4–5), there exist $I_1$ and $I_2$ ideals of $\mathcal{R}$, such that $[I_1,\mathcal{R}]\subseteq L_1$ and $[I_2,\mathcal{R}]\subseteq L_2$. Thus we have $\left[\delta \right(x),\delta (y\left)\right]-[x,y]\in Z\left(\mathcal{R}\right)$ for each $x\in [I_1,I_1]$ and $y\in [I_2,I_2]$. Since $I_1$, $I_2$ and $\mathcal{R}$ satisfy the same GPIs by [16, 17], so we conclude that (1) $\left[\delta \right(x),\delta (y\left)\right]-[x,y]\in Z\left(\mathcal{R}\right)$(1) for each $x,y\in [\mathcal{R},\mathcal{R}]$. Application of [18, Corollary 1.4] yields $[\mathcal{R},\mathcal{R}]\subseteq Z\left(\mathcal{R}\right)$, i.e. $\mathcal{R}$ is commutative unless $char\left(\mathcal{R}\right)=2$ and satisfies $s_4,$ the standard identity of degree 4.

Proof

Proof of Theorem 1.1

Fix $x\in G_1$ and define $U_{k,l}=${$y\in \mathcal{A}\mid \left[\delta \right(x^k),\delta (y^l\left)\right]-[x^k,y^l]\notin Z\left(\mathcal{A}\right)\}$ for each k,l>1. From here, it is easy to see that $U_{k,l}$ is open. If every $U_{k,l}$ is dense, we know that their intersection is also dense by Baire category theorem. This contradicts the existence of $G_1$ and $G_2$. Hence for $r,s\in {\mathbb{Z}}^{\mathbb{+}}$, $U_{r,s}$ is not a dense set and there exists an nonempty open set $G_3$ in the complement of $U_{r,s}$ such that $\left[\delta \right(x^r),\delta (y^s\left)\right]-[x^r,y^s]\in Z\left(A\right)$ for all $y\in G_3$. Take $v_0\in G_3$, $w\in \mathcal{A}$ and $t\in \mathbb{R}$, $v_0+tw\in G_3,$ we have (2) $\left[\delta \right(x^r),\delta ((v_0+tw{)}^s)]-[x^r,(v_0+tw{)}^s]\in Z\left(\mathcal{A}\right).$(2) The expression $\left[\delta \right(x^r),\delta ((v_0+tw{)}^s)]-[x^r,(v_0+tw{)}^s]$ can be written as $\begin{array}{rl}& \left[\delta \right(x^r),A_{s,0}(v_0,w\left)\right]-[x^r,B_{s,0}(v_0,w\left)\right]\\ & +\left(\right[\delta \left(x^r\right),A_{s-1,1}(v_0,w)]-[x^r,B_{s-1,1}(v_0,w)\left]\right)t\\ & +\cdots \\ & +\left(\right[\delta \left(x^r\right),A_{1,s-1}(v_0,w)]-[x^r,B_{1,s-1}(v_0,w)\left]\right)t^{s-1}\\ & +\left(\right[\delta \left(x^r\right),A_{0,s}(v_0,w)]-[x^r,B_{0,s}(v_0,w)\left]\right)t^s.\end{array}$ Let $\alpha ,\beta \in {\mathbb{Z}}^{\mathbb{+}}$. If $\alpha +\beta =s$, then $A_{\alpha ,\beta }(v_0,w)$ represents the summation of all those terms in which $v_0$ appears α times, and w appears β times in the expansion of $\delta \left(\right(v_0+tw{)}^s)$. Similarly, we can define $B_{\alpha ,\beta }(v_0,w)$ for $(v_0+tw{)}^s$. The coefficient of $t^s$ in above polynomial is just $\left[\delta \right(x^r),\delta (w^s\left)\right]-[x^r,w^s]$. Hence $\left[\delta \right(x^r),\delta (w^s\left)\right]-[x^r,w^s]\in Z\left(\mathcal{A}\right)$. Therefore for $x\in G_1$ there are $r,s\in {\mathbb{Z}}^{\mathbb{+}}$ depending on x and y such that for each $w\in \mathcal{A}$, $\left[\delta \right(x^r),\delta (w^s\left)\right]-[x^r,w^s]\in Z\left(\mathcal{A}\right).$

Now, fix $y\in \mathcal{A}$ and for $p,q\in {\mathbb{Z}}^{\mathbb{+}}$, set $V_{p,q}=${$v\in \mathcal{A}\mid \left[\delta \right(v^p),\delta (y^q\left)\right]-[v^p,y^q]\notin Z\left(\mathcal{A}\right)$}. Obviously, each $V_{p,q}$ is open. If each $V_{p,q}$ is dense, then the intersection is also. However, this is contrary to what was shown earlier concerning the open set $G_1$. Hence there are integers $p_1,q_1>1$ and a nonempty open subset $G_4$ in $V_{p_1,q_1}^c$. If $x_0\in G_4$ and $u\in \mathcal{A}$, $x_0+tu\in G_4$ for all small $t\in \mathbb{R}$. Hence, for positive integers $p_1,q_1>1$ $\left[\delta \right((x_0+tu{)}^{p_1}),\delta \left(y^{q_1}\right)]-[(x_0+tu{)}^{p_1},y^{q_1}]\in Z\left(\mathcal{A}\right)$ for each $u\in \mathcal{A}$ and $x_0\in G_4$. Arguing as above, we see that $\left[\delta \right(u^{p_1}),\delta (y^{q_1}\left)\right]-[u^{p_1},y^{q_1}]\in Z\left(\mathcal{A}\right)$ for each $u\in A.$

Now consider $S_{k,l}=${$y\in \mathcal{A}\mid \left[\delta \right(w^{k_1}),\delta (y^{l_1}\left)\right]-[w^{k_1},y^{l_1}]\in Z\left(A\right)$ for each $w\in \mathcal{A}$, then the union of $S_{k_1,l_1}$ will be $\mathcal{A}.$ It can be easily proven that each $S_{k_1,l_1}$ is closed. Hence some $S_{r_1,s_1}$, $r_1,s_1>1$, must have a nonempty open subset $G_5$, by Baire category theorem. Let $y_0\in G_5$, for all small $t\in \mathbb{R}$ and each $z\in \mathcal{A}$ $\left[\delta \right(w^{r_1}),\delta ((y_0+tz{)}^{s_1})]-[w^{r_1},(y_0+tz{)}^{s_1}]\in Z\left(\mathcal{A}\right).$ Hence by earlier arguments, for each $w,z\in \mathcal{A}$ we have $\left[\delta \right(w^r),\delta (z^s\left)\right]-[w^r,z^s]\in Z\left(\mathcal{A}\right)$. Thus by Proposition 2.1, we get the desire conclusion. This completes the proof.

In conclusion, it is tempting to conjecture as follows:

Conjecture 2.1

Let δ be a continuous skew derivation (linear) on $\mathcal{A}$ into itself, where $\mathcal{A}$ is a semisimple Banach algebra. Suppose $G_1$, $G_2$ are open subsets of $\mathcal{A}$ and $1<m,n\in {\mathbb{Z}}^{\mathbb{+}}$ depending on x and y such that $\left[\delta \right(x^m),\delta (y^n\left)\right]-[x^m,y^n]\in Z\left(\mathcal{A}\right)$ for each $x\in G_1$, $y\in G_2$. Then $\mathcal{A}$ is commutative.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Abdul Nadim Khan http://orcid.org/0000-0002-5861-6137

Mohd Arif Raza http://orcid.org/0000-0001-6799-8969

Husain Alhazmi http://orcid.org/0000-0001-7190-5884