Maps on $C^\ast$-algebras are skew Lie triple derivations or homomorphisms at one point

1.   Introduction and basic definitions

Homomorphisms and derivations are the most intensively studied classes of operators on Banach algebras or other algebras. Let $A$ be an algebra and $M$ be an $A$-bimodule, $d:A\to M$ be a linear map. If for any $x, y\in A$,

then $d$ is called a derivation. If for any $x\in A$, $d(x^2) = d(x)x+xd(x)$, then $d$ is called a Jordan derivation. Additionally, if for any $x, y\in A$, $d([x, y]) = [d(x), y]+[x, d(y)]$, where $[x, y] = xy-yx$ is the Lie product of $x$ and $y$, then $d$ is called a Lie derivation. Clearly, a derivation is a Jordan derivation and is a Lie derivation. However, a Jordan or Lie derivation need not certainly be a derivation. The standard problem is to find conditions implying that a Jordan or a Lie derivation is actually a derivation. Along this line, there are fruitful results. See for example ^{[1,2,3,4,5,6,7]}.

Recently, many studies are concerned with finding the standard form of linear maps satisfying the derivation type equation for special pairs of $x$ and $y$. For example, ^{[8,9,10,11,12,13]} studied linear maps which satisfy the derivation type equation for any $x, y$ with $xy = 0$ and ^[14,15,16] studied linear maps which satisfy the derivation type equation for any $x, y$ with $xy = 1$.

A map $\varphi$ from a $\ast$-algebra $A$ into a bimodule $M$ over $A$ is called a skew Lie derivation if

for all $a, b\in A$, is called skew Lie derivable at the point $z\in A$ if (1.2) holds for all $a, b\in A$ with $ab = z$, where $[a, b]_\ast = ab-ba^\ast$. A linear map $\varphi$ from $A$ into another $C^\ast$-algebra $B$ is called a skew Lie homomorphism at $z$ if $\varphi([a, b]_\ast) = [\varphi(a), \varphi(b)]_\ast$ for all $a, b\in A$ with $ab = z$. If $\varphi$ is a skew Lie homomorphism at all elements of $A$, then it is called a skew Lie homomorphism on $A$. For their special importance, these maps attracted many authors' attention in the past decades (see ^{[17,18,19,20]}).

In ^[21], Li, Zhao and Chen introduced the concept of nonlinear skew Lie triple derivations and showed every nonlinear skew Lie triple derivation between factors is an additive $\ast$-derivation.

Difinition 1.1. An additive map $\varphi:A\to M$ is called a nonlinear skew Lie triple derivation if

for all $a, b, c$ in $A$, and is called skew Lie triple derivable at the point $z\in A$ if Eq (1.3) holds for all $a, b, c\in A$ with $ab = z$ and $c = a$.

For unital $C^\ast$-algebras $A$ and $B$, in the present paper, we study two types of continuous maps. One of them is the type of continuous linear maps from $A$ into $B$, which are skew Lie triple derivable at the identity. Another is the type of continuous linear maps from $A$ into $B$, which are skew Lie triple homomorphisms at the identity (see Section 3 for more details).

2.   Skew Lie triple derivations at the identity

In this section, we will study linear maps between unital $C^\ast$-algebras which are skew Lie triple derivable at the identity. Throughout this section, $B$ will be a unital $C^\ast$-algebra, $A$ will be a $C^\ast$-subalgebra of $B$ and $1_A = 1_B$, $A_{sa}$ will be the set of all self-adjoint elements in $A$, $\varphi:A\to B$ will be a linear map. First, let us explore the behavior of the identity $1$ under $\varphi$ when $\varphi$ is skew Lie triple derivable at the identity.

Lemma 2.1. If $\varphi:A\to B$ is skew Lie triple derivable at the identity, then $\varphi(1) = \varphi(1)^\ast$.

Proof. Since $1\cdot 1 = 1$ and $[[1, 1]_\ast, 1]_\ast = 0$, we have

from which it follows that $\varphi(1) = \varphi(1)^\ast$.

Theorem 2.1. Let $\varphi: A\to B$ be a continuous linear map which is skew Lie triple derivable at the identity, then it is a $\ast$-derivation.

Proof. (1) First, we show $\varphi$ is selfadjoint, i.e., $\varphi(x^\ast) = \varphi(x)^\ast$ for all $x\in A$. Put any $a\in A_{sa}$, $\mathrm{e}^{ita}$ is a unitary for each $t\in\mathbb{R}$ and $[[\mathrm{e}^{ita}, \mathrm{e}^{-ita}]_\ast, \mathrm{e}^{ita}]_\ast = \mathrm{e}^{3ita}-\mathrm{e}^{-ita}$. Thus we deduce that

By taking derivative of Eq (2.1) at $t$, we obtain that

Put $t = 0$ and $a = 1$ in Eq (2.2), then we get $\varphi(1) = 0$. Again put $t = 0$ in Eq (2.2), noted that $\varphi(1) = 0$, we get

For each $x\in A$, there are $a, b\in A_{sa}$ such that $x = a+ib$. Hence, $\varphi(x^\ast) = \varphi(a)-i\varphi(b) = \varphi(x)^\ast$.

(2) Now we show $\varphi$ is a $\ast$-derivation. Taking derivative of Eq (2.2) in $t = 0$ yields that

Put any $a, b\in A_{sa}$, then

So

For any $x\in A$, there are $a, b\in A$ such that $x = a+ib$. Hence,

Therefore, $\varphi$ is a Jordan derivation. By [4, Theorem 6.3], $\varphi$ is a $\ast$-derivation.

3.   Skew Lie triple homomorphisms at the identity

Let $A, B$ be unital $C^\ast$-algebras, $\varphi:A\to B$ a linear map. If

for all $x, y\in A$ with $xy = 1$, then $\varphi$ is called a skew Lie triple homomorphism at the identity. Recall that a Jordan $\ast$-homomorphism between $C^\ast$-algebras is a linear map $\varphi$ such that $\varphi(x^2) = \varphi(x)^2$ and $\varphi(x^\ast) = \varphi(x)^\ast$. In this section we prove that every linear continuous skew Lie triple homomorphism at the identity is a Jordan $\ast$-homomorphism. Throughout this section $A$ and $B$ will be unital $C^\ast$-algebras.

Lemma 3.1. Let $\varphi:A\to B$ be a linear continuous skew Lie triple homomorphism at the identity. Then $\varphi(1)$ is a partial isometry.

Proof. Since the product of $1$ and itself is $1$ and $[[1, 1]_\ast, 1]_\ast = 0$, then

Hence,

For any $a\in A_{sa}$, $\mathrm{e}^{ita}$ is a unitary for each real number $t$ and $[[\mathrm{e}^{ita}, \mathrm{e}^{-ita}]_\ast, \mathrm{e}^{ita}]_\ast = \mathrm{e}^{3ita}-\mathrm{e}^{-ita}$. Thus we deduce that

Take derivative at $t$, then

Put $t = 0$ and $a = 1$, then

By taking derivative of Eq (3.2) at $t = 0$, we get

By putting $a = 1$ in Eq (3.4), we get

Multiplying Eq (3.3) by $2$ and subtracting Eq (3.5) yield that

which implies

It follows from Eqs (3.1) and (3.6) that

Now combine the above two equations and Eq (3.3), then we get $\varphi(1) = \varphi(1)\varphi(1)^\ast\varphi(1)$. Hence, $\varphi(1)$ is a partial isometry.

If furthermore $\varphi(1) = 1$, then we can show the following main theorem.

Theorem 3.2. Let $\varphi:A\to B$ be a linear continuous skew Lie triple homomorphism at the identity. If $\varphi(1) = 1$, then $\varphi$ is a Jordan $\ast$-homomorphism.

Proof. (1) Since $\varphi(1) = 1$, by putting $t = 0$ in Eq (3.2), we obtain that

i.e., $\varphi(a) = \varphi(a)^\ast$. As in the proof of Theorem 2.2, we can see that $\varphi(x^\ast) = \varphi(x)^\ast$ for all $x\in A$.

(2) Since $\varphi(1) = 1$ and $\varphi(a) = \varphi(a)^\ast$, it follows from Eq (3.4) that

Replacing $a$ by $a+b$ for $a, b\in A_{sa}$, we get

Now for each $x\in A$, there are $a, b\in A_{sa}$ such that $x = a+ib$. So

and so $\varphi$ is a Jordan $\ast$-homomorphism.

For any partial isometry $e$ in a $C^\ast$-algebra $A$, $e^\ast e$ and $ee^\ast$ are projections. $A$ can be decomposed as a direct sum of the form

From Lemma 3.1, it follows that $\varphi(1)$ is a partial isometry. Let $\varphi(1)\varphi(1)^\ast = p$ and $\varphi(1)^\ast\varphi(1) = q$, then we can decompose $B$ as

where $p^\perp = 1-p$ and $q^\perp = 1-q$. Let $B_0(\varphi(1)) = pBq, B_2(\varphi(1)) = p^\perp Bq^\perp$, we can show the following corollary.

Corollary 3.1. Let $\varphi:A\to B$ be a linear continuous skew Lie triple homomorphism at the identity. If $\varphi(1)^\ast = \varphi(1)$, then $\varphi(a) = \varphi(a)^\ast$ for all $a\in A_{sa}$ and $\varphi(A)\subset B_0(\varphi(1))$.

Proof. Since $\varphi(1)^\ast = \varphi(1)$, $p = q = \varphi(1)^2$. By putting $t = 0$ in Eq (3.2), we obtain that

Hence, $\varphi(a) = \varphi(a)^\ast$ for all $a\in A_{sa}$. It follows from Eq (3.7) that

So $p^\perp\varphi(a)p^\perp = p\varphi(a)p^\perp = p^\perp\varphi(a)p = 0$ and so $\varphi(a) = p\varphi(a)p\in B_0(\varphi(1))$. Therefore, $\varphi(A)\subset B_0(\varphi(1))$.

Corollary 3.2. Let $\varphi:A\to B$ be a linear continuous skew Lie triple homomorphism at the identity. Then $\varphi$ is a Jordan $\ast$-homomorphism if and only if $\varphi(1)$ is a projection.

Proof. If $\varphi$ is a Jordan $\ast$-homomorphism, then $\varphi(1) = \varphi(1^2) = \varphi(1)^2$ and $\varphi(1)^\ast = \varphi(1)$. So $\varphi(1)$ is a projection.

Conversely, if $\varphi(1)$ is a projection, then $\varphi(1) = p$ is the identity of the subalgebra $B_0(\varphi(1))$. By Corollary 3.1, we can regard $\varphi$ as a map from $A$ into $B_0(\varphi(1))$. Hence, by Theorem 3.2, $\varphi$ is a Jordan $\ast$-homomorphism.

4.   Conclusions

It is not hard to see that the continuity of the linear map $\varphi$ is very important in this paper. The automatical continuity of some maps on operator algebra is an important problem (see for example ^[22]). Let $\varphi$ be a linear map which is skew Lie triple derivable at the identity or is a skew Lie triple homomorphism at the identity. It is natural to ask whether $\varphi$ is automatically continuous.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (No.11901451), Talent Project Foundation of Yunnan Provincial Science and Technology Department (No.202105AC160089), and Natural Science Foundation of Yunnan Province (No.202101BA070001198)

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.