2-local $ (\upvarphi, \uppsi) $-Derivations on Finite von Neumann Algebras

TL;DR

This paper introduces the concept of $(, \u03a8)$-finite von Neumann algebras and proves that under certain conditions, (approximately) 2-local $(, \u03a8)$-derivations on such algebras are actual $(, \u03a8)$-derivations.

Contribution

The paper defines $(, \u03a8)$-finite von Neumann algebras and establishes that (approximately) 2-local $(, \u03a8)$-derivations are genuine $(, \u03a8)$-derivations under specific conditions.

Findings

Introduction of $(, \u03a8)$-finite von Neumann algebras

Characterization of 2-local $(, \u03a8)$-derivations

Conditions ensuring 2-local derivations are actual derivations

Abstract

In this paper, I introduce the concept of $ (\upvarphi, \uppsi) $-finite von Neumann algebras and I show that if $ \mathscr{M} $ is a finite and $ (\upvarphi, \uppsi) $-finite von Neumann algebra togather with condition $ \{ \big( \Delta(u+v)-\Delta(u)-\Delta(v)\big)^{*}\} \subseteq \uppsi(\mathscr{M}) $, then each (approximately) 2-local $ (\upvarphi, \uppsi) $-derivation $ \delta $ on $ \mathscr{M} $, is a $ (\upvarphi, \uppsi) $-derivation.

Equations

$$a.z:=\text{upvarphi}(a)z,z.a:=\text{uppsi}(a)z,(a\in \mathcal{A},z\in \mathbb{K})$$

$$a.x=x.a,(a\in \mathcal{A},x\in E).$$

$$d(ab)=\text{upvarphi}(a)d(b)+\text{upvarphi}(b)d(a),(a,b\in \mathcal{A}).$$

$$a\text{smallblacktriangleright}b:=\text{upvarphi}(a)b,b\text{smallblacktriangleleft}a:=b\text{upvarphi}(a),(a\in \mathcal{A},b\in \mathcal{B})$$

$$\Vert b\cdot x\cdot c{\Vert }_E⩽\Vert b{\Vert }_{\mathcal{B}}\Vert x{\Vert }_E\Vert c{\Vert }_{\mathcal{C}}$$

$$a.x\ne 0⟺x.a\ne 0,(a\in \mathcal{A},x\in E).$$

$$D$$

$$b⟼D(b):=⟨\text{upvarphi}(b_0),{\lambda }_0⟩{\lambda }_0$$

$$\displaystyle\big{\langle}\upvarphi(b),\upvarphi(b_{1})D(b_{2})$$

$$\displaystyle=\big{\langle}\upvarphi(b)\upvarphi(b_{1}),D(b_{2})\big{\rangle}+\big{\langle}\upvarphi(b_{2})\upvarphi(b),D(b_{1})\big{\rangle}$$

$$\displaystyle=\big{\langle}\upvarphi(bb_{1}),\big{\langle}\upvarphi(b_{2}),\lambda_{0}\big{\rangle}\lambda_{0}\big{\rangle}+\big{\langle}\upvarphi(b_{2}b),\big{\langle}\upvarphi(b_{1}),\lambda_{0}\big{\rangle}\lambda_{0}\big{\rangle}$$

$$\displaystyle=\big{\langle}\upvarphi(bb_{1}),\lambda_{0}\big{\rangle}\big{\langle}\upvarphi(b_{2}),\lambda_{0}\big{\rangle}+\big{\langle}\upvarphi(b_{2}b),\lambda_{0}\big{\rangle}\big{\langle}\upvarphi(b_{1}),\lambda_{0}\big{\rangle}=0.$$

$$\displaystyle\langle\upvarphi(b_{0}),D(b_{0})\big{\rangle}$$

$$\displaystyle=\big{\langle}\upvarphi(b_{0}),\lambda_{0}\big{\rangle}\big{\langle}\upvarphi(b_{0}),\lambda_{0}\big{\rangle}=1$$

$$\displaystyle\langle\upvarphi(b_{0}),\delta_{\lambda}(b_{0})\big{\rangle}$$

$$\displaystyle=\big{\langle}\upvarphi(b_{0}),\lambda\upvarphi(b_{0})\big{\rangle}-\big{\langle}\upvarphi(b_{0}),\upvarphi(b_{0})\lambda\big{\rangle}$$

$$\displaystyle=\big{\langle}\upvarphi(b_{0}^{2}),\lambda\big{\rangle}-\big{\langle}\upvarphi(b_{0}^{2}),\lambda\big{\rangle}=0,\hskip 14.22636pt(\lambda\in\mathscr{C}^{*})$$

$$D^{\prime }(b_0^2)=D\circ \text{upvarphi}(b_0^2)\ne 0.$$

$$\displaystyle\big{\langle}a,R_{\lambda}(x.c)\big{\rangle}$$

$$\displaystyle=\big{\langle}x.(ac),\lambda\big{\rangle}\big{\langle}(ac).x,\lambda\big{\rangle}=\big{\langle}a.(c.x),\lambda\big{\rangle}=\big{\langle}a,R_{\lambda}(c.x)\big{\rangle}$$

$$\displaystyle\big{\langle}a,R_{\lambda}(c.x)\big{\rangle}$$

$$\displaystyle=\big{\langle}ac,R_{\lambda}(x)\big{\rangle}=\big{\langle}a,c.R_{\lambda}(x)\big{\rangle}$$

$$\displaystyle\big{\langle}a,R_{\lambda}(x.c)\big{\rangle}$$

$$\displaystyle=\big{\langle}ca,R_{\lambda}(x)\big{\rangle}=\big{\langle}a,R_{\lambda}(x).c\big{\rangle}$$

$$\displaystyle 1=\big{\langle}\upvarphi(b_{0}).D^{\prime}(b_{0}),\lambda\big{\rangle}$$

$$\displaystyle=\big{\langle}\upvarphi(b_{0}),\mu.\upvarphi(b_{0})-\upvarphi(b_{0}).\mu\big{\rangle}$$

$$\displaystyle=\big{\langle}\upvarphi(b_{0}),\mu.\upvarphi(b_{0})\big{\rangle}-\big{\langle}\upvarphi(b_{0}),\upvarphi(b_{0}).\mu\big{\rangle}$$

$$\displaystyle=\big{\langle}\upvarphi(b_{0}^{2}),\mu\big{\rangle}-\big{\langle}\upvarphi(b_{0}^{2}),\mu\big{\rangle}=0.$$

$$a.x=\mathcal{R}(a).x,x.a=x.\mathcal{R}(a)(a\in \mathcal{A},x\in E)$$

$$\delta {|}_{\mathcal{A}}=\delta \circ \mathcal{R}\in {\mathcal{Z}}_{\mathcal{R}}^1(\mathcal{B},E)\Rightarrow \delta \circ \mathcal{R}\in {\mathcal{Z}}_{\mathcal{R}}^1(\mathcal{A},E)={\mathcal{Z}}^1(\mathcal{A},E).$$

$${\mathcal{H}}_{\mathcal{R}}^1(\mathcal{A},{\mathcal{A}}^{\ast })={\mathcal{H}}^1(\mathcal{A},{\mathcal{A}}^{\ast })=\{0\}.$$

$$\sigma (ab)\cup \{0\}=\sigma (ba)\cup \{0\}$$

$$2\tau (x.\delta (x))=\tau (x.\delta (x)+\delta (x).x)\ne 0.$$

$$\big{\langle}a,R_{\tau}(x)\big{\rangle}=\big{\langle}a.x,\tau\big{\rangle},\hskip 14.22636pt(a\in\mathscr{A},x\in\mathscr{M}).$$

$$\displaystyle\big{\langle}a,R_{\tau}(x)\big{\rangle}$$

$$\displaystyle=\lim_{\alpha}\big{\langle}a.x_{\alpha},\tau\big{\rangle}=\lim_{\alpha}\big{\langle}a,R_{\tau}(x_{\alpha})\big{\rangle}=\big{\langle}a,\lim_{\alpha}R_{\tau}(x_{\alpha})\big{\rangle}$$

$$D(ab)=R_{\tau }\circ \delta (ab)$$

Full text

1. Introduction

Let $\mathscr{A}$ be a commutative Banach algebra and $\Phi_{\mathscr{A}}$ its spectrum. Then for each $\upvarphi,\uppsi\in\Phi_{\mathscr{A}}\cup\{0\}$, $\mathbb{K}$ is an $\mathscr{A}$-bimodule with the following actions

[TABLE]

this module is denoted by $\mathbb{K}_{\upvarphi,\uppsi}$ and we write $\mathbb{K}_{\upvarphi}$ for $\mathbb{K}_{\upvarphi,\upvarphi}$. Each one-dimentional $\mathscr{A}$-bimodule has the form $\mathbb{K}_{\upvarphi,\uppsi}$ for some $\upvarphi,\uppsi\in\Phi_{\mathscr{A}}\cup\{0\}$. Let $\mathscr{A}$ be an algebra, and let $E$ he an $\mathscr{A}$-bimodule. Then $E$ is symmetric (or commutative) if

[TABLE]

For example, the $\mathscr{A}$-himodule $\mathbb{K}_{\upvarphi,\uppsi}$ is symmetric if and only if $\upvarphi=\uppsi$.

A linear functional $d$ on $\mathscr{A}$ is a point derivation at $\upvarphi$ if

[TABLE]

The definition of a $(\upvarphi,\uppsi)$-derivation is derived from the definition of a point derivation (see [2], Proposition 1.8.10).

It is known that each derivation on a commutative and semi-simple Banach algebras (commutative $C^{*}$-algebras) is zero ([5], Corollary 2.2.3 and Corollary 2.2.8), but we can study $(\upvarphi,\uppsi)$-derivations on such algebras.

In Section 2, Theorem 2.2, is an extention of ([2], Theorem 2.8.63). Indeed, we show that if $\mathscr{A}$ is a weakly amenable commutative Banach algebras, $\upvarphi\in Hom(\mathscr{A})$ is onto and $X$ is a Banach $\mathscr{A}$-bimodule with the property that ”$a.x\neq 0\Leftrightarrow x.a\neq 0$” $(a\in\mathscr{A},x\in X)$, then each $\upvarphi$-derivation from $\mathscr{A}$ to $X$ is zero. In Theorem 2.5, we prove that each $C^{*}$-algebra which has a separating family of normal tracial states is commutative.

In Section 3, we survey about Johnson’s theorem for Jordan $(\upvarphi,\uppsi)$-derivations on abelian $C^{*}$-algebras. Indeed, we show that every bounded Jordan $(\upvarphi,\uppsi)$-derivation from a commutative $C^{*}$-algebra $\mathscr{A}$ into a Banach $\mathscr{B}-\mathscr{C}$-bimodule $X$ is a $(\upvarphi,\uppsi)$-derivation.

In Section 4, Theorem 4.4, we prove that each $(\upvarphi,\uppsi)$-derivation on finite von Neumann algebra $\mathscr{M}$, is $(\upvarphi,\uppsi)$-inner, this is an extention of ([4], Theorem 4.1.6). We introduce the concept of $(\upvarphi,\uppsi)$-finite von Neumann algebras and we use Theorem LABEL:3, to prove that every (approximately) 2-local $(\upvarphi,\uppsi)$-derivation on finite and $(\upvarphi,\uppsi)$-finite von Neumann algebras with condition \{\big{(}\Delta(u+v)-\Delta(u)-\Delta(v)\big{)}^{*}\}\subseteq\uppsi(\mathscr{M}), is a $(\upvarphi,\uppsi)$-derivation.

2. $(\upvarphi,\uppsi)$-derivations

Suppose that $\mathscr{A},\mathscr{B},\mathscr{C}$ are Banach algebras and let $\upvarphi\in Hom(\mathscr{A},\mathscr{B})$, $\uppsi\in Hom(\mathscr{A},\mathscr{C})$ 111 If $\mathscr{A},\mathscr{B}$ are Banach algebras and $\upvarphi\in Hom(\mathscr{A},\mathscr{B})$, then $\mathscr{B}$ is an $\mathscr{A}$-bimodule with actions

$a~{}\smallblacktriangleright~{}b:=\upvarphi(a)b,\hskip 7.11317ptb~{}\smallblacktriangleleft~{}a:=b\upvarphi(a),\hskip 14.22636pt(a\in\mathscr{A},b\in\mathscr{B})$

and let $E$ be a Banach $\mathscr{B}-\mathscr{C}$-module 222If $\mathscr{B}$, $\mathscr{C}$ are Banach algebras, then $E$ is a Banach $\mathscr{B}-\mathscr{C}$-module, if it is a left Banach $\mathscr{B}$-module and right Banach $\mathscr{C}$-module. Clearly it satisfy the following condition

$\|b~{}\cdot~{}x~{}\cdot~{}c\|_{E}\leqslant\|b\|_{\mathscr{B}}\|x\|_{E}\|c\|_{\mathscr{C}}$

..

Definition 2.1**.**

A linear operator $\delta:\mathscr{A}\longrightarrow E$ is a $(\upvarphi,\uppsi)$-derivation if it satisfies $\delta(ab)=\delta(a)\cdot\uppsi(b)+\upvarphi(a)\cdot\delta(b)$, ($a,b\in\mathscr{A}$). A $(\upvarphi,\uppsi)$-derivation $\delta$ is called $(\upvarphi,\uppsi)$-inner derivation if there exists $x\in E$ such that $\delta(a)=x\cdot\uppsi(a)-\upvarphi(a)\cdot x$ ($a\in\mathscr{A}$).

Let $\mathcal{Z}_{\upvarphi,\uppsi}^{1}(\mathscr{A},E)$, denote the set of all continuous $(\upvarphi,\uppsi)$-derivations from $\mathscr{A}$ to $E$ and let $\mathcal{N}_{\upvarphi,\uppsi}(\mathscr{A},E)$, denote the set of all inner $(\upvarphi,\uppsi)$-derivations from $\mathscr{A}$ to $E$. The first cohomology group $H_{\upvarphi,\uppsi}^{1}(\mathscr{A},E)$ is defined to be the quotient space $\mathcal{Z}_{\upvarphi,\uppsi}^{1}(\mathscr{A},E)/\mathcal{N}_{\upvarphi,\uppsi}^{1}(\mathscr{A},E)$. In the case that $\upvarphi=\uppsi$, we use the notations $\mathcal{Z}_{\upvarphi}^{1}(\mathscr{A},E)$, $\mathcal{N}_{\upvarphi}(\mathscr{A},E)$ and $H_{\upvarphi}^{1}(\mathscr{A},E)$.

Theorem 2.2**.**

Let $\mathscr{A},\mathscr{B}$ be Banach algebras, $\upvarphi\in\mathrm{Hom}(\mathscr{B},\mathscr{A})$, $\mathscr{C}=\varphi(\mathscr{B})$ is closed and $\mathscr{H}^{1}_{\upvarphi}(\mathscr{B},\mathscr{C}^{*})=\{0\}$. Then

• (i)

$\varphi(\mathscr{B})=\overline{\varphi(\mathscr{B}^{2})}$;

• (ii)

If $\mathscr{A}$ is commutative and $\upvarphi(\mathscr{B})=\mathscr{A}$, then $\mathcal{Z}^{1}(\mathscr{A},E)=\{0\}$ for each Banach $\mathscr{A}$-bimodule $E$ with following condition

[TABLE]

Proof.

(i) Asuume that $\upvarphi(\mathscr{B})\smallsetminus\overline{\upvarphi(\mathscr{B}^{2})}\neq\emptyset$. Take $\upvarphi(b_{0})\in\upvarphi(\mathscr{B})\smallsetminus\overline{\upvarphi(\mathscr{B}^{2})}$, choose $\lambda_{0}\in\mathscr{C}^{*}$ with $\lambda_{0}|_{\overline{\varphi(\mathscr{B}^{2})}}\equiv 0$ and $\langle\upvarphi(b_{0}),\lambda_{0}\rangle=1$. Define

[TABLE]

Certainly $D$ is a continuous linear map and since $\lambda_{0}|_{\overline{\varphi(\mathscr{B}^{2})}}\equiv 0$, we have $D(b_{1}b_{2})=\langle\upvarphi(b_{1}b_{2}),\lambda_{0}\rangle\lambda_{0}=0$ ( $b_{1},b_{2}\in\mathscr{B}$). So

[TABLE]

Therefore $D(b_{1}b_{2})=\upvarphi(b_{1})D(b_{2})+D(b_{1})\upvarphi(b_{2})$ and so $D\in\mathcal{Z}^{1}_{\upvarphi}(\mathscr{B},\mathscr{C}^{*})$. Now we have

[TABLE]

But for $\delta_{\lambda}(b)=[\lambda,\upvarphi(b_{0})]$, ($\lambda\in\mathscr{C}^{*}$) we have

[TABLE]

Therefore $D$ is not $\upvarphi$-inner derivation, but $\mathscr{H}^{1}_{\upvarphi}(\mathscr{B},\mathscr{C}^{*})=\{0\}$.

(ii) Assume that $D\in\mathcal{Z}^{1}(\mathscr{A},E)$ with $D\neq 0$, the partition (i) and equality $\upvarphi(\mathscr{B})=\mathscr{A}$, implies that $\overline{\upvarphi(\mathscr{B}^{2})}=\upvarphi(\mathscr{B})$ and $D^{\prime}=D\circ\upvarphi\in\mathcal{Z}_{\upvarphi}^{1}(\mathscr{B},E)$. So there is a $b_{0}\in\mathscr{B}$ such that

[TABLE]

Hence $\upvarphi(b_{0}).D^{\prime}(b_{0})+D^{\prime}(b_{0}).\upvarphi(b_{0})\neq 0$ It follows by condition (2.1) that $\upvarphi(b_{0}).D^{\prime}(b_{0})\neq 0$. Put $N=\overline{L.S\{a.x-x.a|a\in\mathscr{A},x\in E\}^{\|\cdot\|}}$. Since $N$ is a closed linear subspace of $E$ and $\upvarphi(b_{0}).D^{\prime}(b_{0})\notin N$ there is a $\lambda\in E^{*}$ such that $\lambda|_{N}\equiv 0$ and $\lambda(\upvarphi(b_{0}).D^{\prime}(b_{0}))\neq 0$. It follows by ([1], Proposition 2.6.6) that there is a $R_{\lambda}\in B_{\mathscr{A}}(E,\mathscr{A}^{*})$ such that \big{\langle}a,R_{\lambda}(x)\big{\rangle}=\big{\langle}a.x,\lambda\big{\rangle}. Clearly we have $R_{\lambda}(x).c=R_{\lambda}(x.c)=R_{\lambda}(c.x)=c.R_{\lambda}(x)$, ($c\in\mathscr{A},x\in E$). Indeed for each $a,c\in\mathscr{A}$ and $x\in E$ we have

[TABLE]

On the other hands

[TABLE]

And

[TABLE]

Therefore $R_{\lambda}\circ D^{\prime}\in\mathcal{Z}_{\upvarphi}^{1}(\mathscr{B},\mathscr{A}^{*})$. It follows by assumption $\mathscr{H}^{1}_{\upvarphi}(\mathscr{B},\mathscr{A}^{*})=\{0\}$ that there is a $\mu\in\mathscr{A}^{*}$ such that $R_{\lambda}\circ D^{\prime}(b)=[\mu,\upvarphi(b)]$ (i.e. $R_{\lambda}\circ D^{\prime}$ is a $\upvarphi$-inner derivation) and we have

[TABLE]

Which is impossible. ∎

Remark 2.3*.*

Let in condition $(ii)$ of Theorem 2.5, $\mathscr{A}$ be a subalgebra of $\mathscr{B}$ and we have $\varphi^{2}=\varphi$. then by changing condition $\mathscr{H}^{1}_{\varphi}(\mathscr{B},\mathscr{A}^{*})=\{0\}$ with $\mathscr{H}^{1}_{\varphi}(\mathscr{A},\mathscr{A}^{*})=\{0\}$, the proof is still valid.

Remark 2.4*.*

Let $\mathscr{B}$ be a unital $C^{*}$-algebra, $a\in\mathscr{B}_{sa}$ and let $\mathscr{A}=\mathfrak{A}^{*}(a,1_{\mathscr{B}})$ be the $C^{*}$-algebra generated by $\{a,1_{\mathscr{B}}\}$ and let $E$ be a Banach $\mathscr{B}$-bimodule. If $\mathcal{R}:\mathscr{B}\rightarrow\mathscr{A}$ be the restriction mapp 333the Restriction mapp from $\mathscr{B}$ onto $\mathscr{A}$. i.e. $\mathcal{R}^{2}=\mathcal{R},~{}~{}\mathcal{R}(a)=a,~{}(a\in\mathcal{A})$.

from $\mathscr{B}$ to $\mathscr{A}$, then $E$ is a Banach $\mathscr{A}$-bimodule with the following actions

[TABLE]

amd for each $\delta\in\mathcal{Z}^{1}(\mathscr{B},E)$, we have

[TABLE]

Now since every $C^{*}$-algebra is weakly amenable 444Each $C^{*}$-algebra $\mathscr{A}$ is weakly amenable. i.e. $\mathscr{H}^{1}(\mathscr{A},\mathscr{A}^{*})=\{0\}$.

we have

[TABLE]

On the other hand for any $b\in\mathscr{B}$,

[TABLE]

so $(ab\neq 0\Leftrightarrow ba\neq 0)$. If $\delta:\mathscr{B}\rightarrow\mathscr{B}$ be a derivation, then it follows by 2.2 that $\delta\circ\mathcal{R}=0$, therefore $\delta\equiv 0$. So there is no every where defined nonzero derivation on $C^{*}$-algebras.

Theorem 2.5**.**

Let $\mathscr{M}$ be a Banach algebra which has a separating family of normal tracial states and let $\mathscr{A}$ be a weakly amenable Banach subalgebra of $\mathscr{M}$ and let $\mathscr{M}^{2}=\mathscr{M}$. Then each derivation $\delta$ from $\mathscr{A}$ to $\mathscr{M}$ is zero.

Proof.

Let $\delta$ be a non-zero derivation from $\mathscr{A}$ to $\mathscr{M}$. Since $\mathscr{M}^{2}=\mathscr{M}$ ([2], 2.8.63), there is an element $x\in\mathscr{M}$ such that $\delta(x^{2})\neq 0$. It follows that $x.\delta(x)+\delta(x).x\neq 0$. Since there is a separating family of normal tracial states on $\mathscr{M}$, so for some normal tracial state $\tau$ on $\mathscr{M}$ we have

[TABLE]

Therefore $\tau(x.\delta(x))\neq 0$. Define $R_{\tau}:\mathscr{M}\rightarrow\mathscr{A}^{*}$ as follows

[TABLE]

Clearly $R_{\tau}$ is $\mathscr{A}$-bimodule map (i.e. $R_{\tau}(a.x)=a.R_{\tau}(x)$ and $R_{\tau}(x.a)=R_{\tau}(x).a$ holds for each $a\in\mathscr{A},x\in\mathscr{M}$). If $x_{\alpha}\longrightarrow x$, then

[TABLE]

It follows by closed graph theorem that $R_{\tau}$ is bounded. And so $D=R_{\tau}\circ\delta\in\mathcal{Z}^{1}(\mathscr{A},\mathscr{A}^{*})$. Indeed

[TABLE]

Now since $\mathscr{H}^{1}(\mathscr{A},\mathscr{A}^{*})=\{0\}$ (each $C^{*}$-algebra is weakly amenable), there is a $\lambda\in\mathscr{A}^{*}$ such that $R_{\tau}\circ\delta(x)=[\lambda,x]$ and we have

[TABLE]

This is a contradiction. So $\delta\equiv 0$. ∎

3. Jordan $(\upvarphi,\uppsi)$-Derivations on $C^{*}$-algebras

Let $\upvarphi\in\mathscr{H}^{1}(\mathscr{A},\mathscr{B})$ and $\uppsi\in\mathscr{H}^{1}(\mathscr{A},\mathscr{C})$. A linear map $\delta$ from a Banach algebra $\mathscr{A}$ to a Banach $\mathscr{B}-\mathscr{C}$-bimodule as called a Jordan $(\upvarphi,\uppsi)$-derivation proved that $\delta(a^{2})=\upvarphi(a)\delta(a)+\delta(a)\uppsi(a)$ for each $a\in\mathscr{A}$. Clearly $(\upvarphi,\uppsi)$-derivations are Jordan $(\upvarphi,\uppsi)$-derivations. Using the fact that $ab+ba=(a+b)^{2}-a^{2}-b^{2}$. It is easy to proved that the Joedan Jordan $(\upvarphi,\uppsi)$-derivation condition is equivalent to

[TABLE]

Proposition 3.1**.**

Every bounded Jordan $(\upvarphi,\uppsi)$-derivation $\delta$ from a von Neumann algebra $\mathscr{M}$ to a unital Banach $\mathscr{B}-\mathscr{C}$-bimodule $X$ is a $(\upvarphi,\uppsi)$-derivation.

Proof.

Clearly $X$ is a Banach $\mathscr{M}$-bimodule with the following actions

[TABLE]

Therefore $\delta$ is a bounded Jordan derivation from $\mathscr{M}$ to a unital Banach $\mathscr{M}$-bimodule $X$. It follows from ([3], Lemma 2.1) that $\delta$ is a derivation from $\mathscr{M}$ to $X$. Equivalently $\delta$ is a bounded linear mapping which satisfy the following condition

[TABLE]

So $\delta$ is a bounded linear mapping which satisfies the following condition

[TABLE]

Hence $\delta$ as a bounded $(\upvarphi,\uppsi)$-derivation ∎

Lemma 3.2**.**

(Main One) For unital Banach algebra $\mathscr{A}$, the following asserations are equivalent :

• (i)

Every bounded Jordan $(\upvarphi,\uppsi)$-derivation from $\mathscr{A}$ to any unital Banach $\mathscr{B}-\mathscr{C}$-bimodule is a $(\upvarphi,\uppsi)$-derivation.

• (ii)

Every bounded trilinear form $V:\mathscr{A}\times\mathscr{C}\times\mathscr{B}\rightarrow\mathbb{C}$ which satisfies

[TABLE]

Will also satisfy

[TABLE]

Proof.

$(i)\Rightarrow(ii)$

The projective tensor product $\mathscr{C}\widehat{\otimes}\mathscr{B}$ is a unital $\mathscr{C}-\mathscr{B}$-bimodule with the following actions

[TABLE]

And so $(\mathscr{C}\widehat{\otimes}\mathscr{B})^{*}$ is a unital $\mathscr{B}-\mathscr{C}$-bimodule with the following actions

[TABLE]

To each bounded trilinear map $V:\mathscr{A}\times\mathscr{C}\times\mathscr{B}\rightarrow\mathbb{C}$ which satisfies the relation (3.1), we associate a bounded linear map $\delta:\mathscr{A}\rightarrow(\mathscr{C}\widehat{\otimes}\mathscr{B})^{*}$ by the following definition

[TABLE]

Now we show that $\delta$ is a bounded Jordan $(\upvarphi,\uppsi)$-derivation.

[TABLE]

Therefore $\delta(a^{2})=\delta(a).\uppsi(a)+\upvarphi(a).\delta(a)$, ($a\in\mathscr{A}$). So $\delta$ is a bounded Jordan $(\upvarphi,\uppsi)$-derivation and by hypothesis, $\delta$ is a $(\upvarphi,\uppsi)$-derivation and we have

[TABLE]

$(ii)\Rightarrow(i)$ Let $E$ be a unital Banach $\mathscr{B}-\mathscr{C}$-bimodule and $\delta:\mathscr{A}\rightarrow E$ be a bounded Jordan $(\upvarphi,\uppsi)$-derivation.

We associate to each $\sigma\in E^{*}$, the bounded trilinear form $V_{\sigma}:\mathscr{A}\times\mathscr{C}\times\mathscr{B}\rightarrow\mathbb{C}$ given by V_{\sigma}(a,c,b):=\big{\langle}b.\delta(a).c,\sigma\big{\rangle}, $(a,d\in\mathscr{A},b\in\mathscr{B},c\in\mathscr{C})$ and we have

[TABLE]

Therefore $V_{\sigma}$ satisfy the condition (3.2), and so it satisfy the condition (3.3) and we have

[TABLE]

Therefore $V_{\sigma}$ ($\sigma\in E^{*}$) satisfy the condition (3.2), and so it satisfy the condition (3.3) and we have

[TABLE]

Hence $\delta$ is a $(\upvarphi,\uppsi)$-derivation. ∎

Theorem 3.3**.**

Let $\mathscr{A}$ be a $C^{*}$-algebra. Every bounded Jordan $(\upvarphi,\uppsi)$-derivation $\delta$ from $\mathscr{A}$ to a Banach $\mathscr{B}-\mathscr{C}$-bimodule $X$ is a $(\upvarphi,\uppsi)$-derivation.

Proof.

It suffix to change (Proposition 3.1 and Lemma 3.2) resoectively with (Proposition 2.2 and Lemma 2.3) at the proof of Theorem 2.4 in [3]. ∎

4. (Approximately) 2-Local $(\upvarphi,\uppsi)$-Derivations

Preliminaries

Definition 4.1**.**

A mapping $\Delta$ from a Banach algebra $\mathscr{A}$ into a Banach $\mathscr{B}-\mathscr{C}$-bimodule $E$ is bounded 2-local (respectively, approximately 2-local) $(\upvarphi,\uppsi)$-derivation, if for each $a,b\in\mathscr{A}$, there is a bounded $(\upvarphi,\uppsi)$-derivation $D_{a,b}$ (respectively, a sequence of bounded $(\upvarphi,\uppsi)$-derivations $\{D^{n}_{a,b}\}$) from $\mathscr{A}$ into $E$ such that $D(a)=D_{a,b}(a)$ and $D(b)=D_{a,b}(b)$ (respectively, $D(a)=\lim_{n\rightarrow\infty}D^{n}_{a,b}(a)$ and $D(b)=\lim_{n\rightarrow\infty}D^{n}_{a,b}(b)$).

Lemma 4.2**.**

Let $\Delta$ be a 2-local (or an approximately 2-local) $(\upvarphi,\uppsi)$-derivation of a Banach algebra $\mathscr{A}$ into Banach $\mathscr{B}-\mathscr{C}$-bimodule $E$. Then

• (i)

$\Delta(\lambda x)=\lambda\Delta(x)$* for any $\lambda\in\mathbb{C}$ and $x\in\mathscr{A}$;*

• (ii)

$\Delta(x^{2})=\Delta(x).\uppsi(x)+\upvarphi(x).\Delta(x)$* for any $x\in\mathscr{A}$.*

Proof.

We prove ths lemma only for approxmately 2-local $(\upvarphi,\uppsi)$-derivations o Baach algebras.

• (i)

For each $x\in\mathscr{A}$ ad $\lambda\i\mathbb{C}$, there iexists a sequece of $(\upvarphi,\uppsi)$-derivations $\{D^{n}_{x,\lambda x}\}$ such that

[TABLE]

Hece $\Delta$ is homogeneous.

• (ii)

For each $x\in\mathscr{A}$, there exists $(\upvarphi,\uppsi)$-derivations $\{D^{n}_{x,x^{2}}\}$ such that

[TABLE]

∎

Lemma 4.3**.**

Any additive 2-local $(\upvarphi,\uppsi)$-derivation $\Delta$ from a $C^{*}$-algebra $\mathscr{A}$ to a Banach $\mathscr{B}-\mathscr{C}$-bimodule $E$ is a $(\upvarphi,\uppsi)$-derivation.

Proof.

We conclude from Lemma 3.6 that each additive 2-local $(\upvarphi,\uppsi)$-derivation $\Delta$ from a $C^{*}$-algebra $\mathscr{A}$ to a Banach $\mathscr{B}-\mathscr{C}$-bimodule $E$ is linear and satisfies $\Delta(x^{2})=\Delta(x).\uppsi(x)+\upvarphi(x)\Delta(x)$ for any $x\in\mathscr{A}$, so $\Delta$ is a $(\upvarphi,\uppsi)$-derivation. ∎

Main Theorems

Theorem 4.4**.**

Let $\mathscr{M}$ be a finite von Neumann algebra and let $\upvarphi,\uppsi$ be continuous homomorphisms on $\mathscr{M}$. Then each $(\upvarphi,\uppsi)$-derivation on $\mathscr{M}$ is a $(\upvarphi,\uppsi)$-inner derivation. i.e. there is an element $a_{0}$ in $\mathscr{M}$ such that $\delta(a)=a_{0}\cdot\uppsi(a)-\upvarphi(a)\cdot a_{0}$ and $\|a_{0}\|\leqslant\|\delta\|$.

Proof.

Let $\mathscr{M}^{u}$ be the group of all unitary elements in $\mathscr{M}$. For $u\in\mathscr{M}$, put

[TABLE]

If $u,v\in\mathscr{M}^{u}$, then

[TABLE]

Hence $T_{u}T_{v}=T_{uv}$, ($u,v\in\mathscr{M}^{u}$). Let $\Delta$ be the set of all non-empty $\sigma(\mathscr{M},\mathscr{M}_{*})$-closed convex sets $\mathscr{K}$ in $\mathscr{M}$ satisfying the following conditions

[TABLE]

Since

[TABLE]

Therefore $\Delta$ is on-empty. Define an order in $\Delta$ by the set inclusion. Let \big{(}\mathscr{K}_{\alpha}\big{)}_{\alpha\in I} be linearly ordered decreasing subsets in $\Delta$. Then $\bigcap_{\alpha\in I}\mathscr{K}_{\alpha}\in\Delta$, because $\mathscr{K}_{\alpha}(\alpha\in I)$ is compact (Arzela Ascoli). Hence there is a minimal element $\mathscr{K}_{0}$ in $\Delta$ by Zorns lemma.

If $a,b\in\mathscr{K}_{0}$, then $a-b\in\mathscr{K}_{0}\smallsetminus\mathscr{K}_{0}$ and for $u\in\mathscr{M}^{u}$, we have

[TABLE]

Hence $\mathscr{K}_{0}\smallsetminus\mathscr{K}_{0}$ is invariant under the mapping

[TABLE]

i.e. \Phi^{u}\big{(}\mathscr{K}_{0}\smallsetminus\mathscr{K}_{0}\big{)}\subseteq\mathscr{K}_{0}\smallsetminus\mathscr{K}_{0}. Since $\mathscr{M}$ is a finite von Neumann algebra, there is a faithful family of normal tracial states $\uptau$ on $\mathscr{M}$ ([1], Theorem 6.3.10). For each $\tau\in\uptau$, define seminorm $P_{\tau}$ as follows

[TABLE]

Let $\lambda=\sup_{x\in\mathscr{K}_{0}}P_{\tau}(x)$ and let $a,b\in\mathscr{K}_{0}$, then for an arbitrary positive number $\varepsilon>0$, there is an element $u\in\mathscr{M}^{u}$ with $P_{\tau}(\frac{a+b}{2})>\lambda-\varepsilon$. Indeed, if there is $\varepsilon_{0}>0$ such that $\forall u\in\mathscr{M}^{u}$, $P_{\tau}(T_{u}(\frac{a+b}{2}))\leqslant\lambda-\varepsilon_{0}$ then $P_{\tau}(\frac{a+b}{2})=P_{\tau}(T_{id}(\frac{a+b}{2}))\leqslant\lambda-\varepsilon_{0}$, which for $a=b$ implies that $P_{\tau}(\frac{a+a}{2})\leqslant\lambda-\varepsilon_{0}$, and so $\lambda=\sup_{a\in\mathscr{K}_{0}}P_{\tau}(a)\leqslant\lambda-\varepsilon_{0}$ which is impossible. Since $P_{\tau}(T_{u}(a))\leqslant\lambda,P_{\tau}(T_{u}(b))\leqslant\lambda$ we have

[TABLE]

It follows from the last two equalities that

[TABLE]

So

[TABLE]

Hence $\mathscr{K}_{0}$ consists of only one elemente $a_{0}$. Since each element of $\mathscr{M}$ is a finite linear combination of unitary elements in $\mathscr{M}$, 555 If $\mathscr{A}$ is a $C^{*}$-algebra and $a\in\mathscr{A}$ be such that $\|a\|\leqslant 1$, Then $a=b+ic$ where $b,c\in\mathscr{A}_{sa}$ are self-adjoint elements and given by

$b=\frac{1}{2}(a+a^{*}),\hskip 14.22636ptc=\frac{1}{2i}(a-ia^{*}).$

We can decompose b and c as

$b=\frac{1}{2}(U_{b}+V_{b}),\hskip 14.22636ptc=\frac{1}{2}(U_{c}+V_{c}).$

where $U_{b},V_{b},U_{c},V_{c}$ are unitary and given by

$\displaystyle U_{b}=b+i\sqrt{1-b^{2}},$ $\displaystyle\hskip 14.22636ptV_{b}=b-i\sqrt{1-b^{2}}$

$\displaystyle U_{c}=c+i\sqrt{1-c^{2}},$ $\displaystyle\hskip 14.22636ptV_{c}=c-i\sqrt{1-c^{2}}$

we have

[TABLE]

Therefore $\delta(x)=a_{0}\cdot\uppsi(x)-\upvarphi(x)\cdot a_{0}$, ($x\in\mathscr{M}$). Clearly $a_{0}\in\mathscr{K}_{0}$ implies that $\|a_{0}\|=\sup_{x\in\mathscr{K}_{0}}\|x\|\leqslant\|\delta\|$. ∎

Definition 4.5**.**

A linear functional $\tau:\mathscr{M}\rightarrow\mathbb{C}$ is called $(\upvarphi,\uppsi)$-tracial, if

[TABLE]

A von Neumann algebra $\mathscr{M}$ is called $(\upvarphi,\uppsi)$-finite, if there exists a faithful family of normal $(\upvarphi,\uppsi)$-tracial states $\mathfrak{T}$ on $\mathscr{M}$.

Theorem 4.6**.**

Let $\mathscr{M}$ be a finite and $(\upvarphi,\uppsi)$-finite von Neumann algebra. Then each 2-local $(\upvarphi,\uppsi)$-derivation $\Delta$ on $\mathscr{M}$ with condition \{\big{(}\Delta(u+v)-\Delta(u)-\Delta(v)\big{)}^{*}\}\subseteq\uppsi(\mathscr{M}), is a $(\upvarphi,\uppsi)$-derivation.

Proof.

Let $\Delta$ be a 2-local $(\upvarphi,\uppsi)$-derivation and let $\mathfrak{T}$ be a faithful family of normal $(\upvarphi,\uppsi)$-tracial states on $\mathscr{M}$ and $\tau\in\mathfrak{T}$. For each $x,y\in\mathscr{M}$ there exists a $(\upvarphi,\uppsi)$-derivation $D_{x,y}$ on $\mathscr{M}$ such that $\Delta(x)=D_{x,y}(x)$ and $\Delta(y)=D_{x,y}(y)$. It follows from theorem 4.4 that $D_{x,y}$ is $(\upvarphi,\uppsi)$-inner, so there is an element $m\in\mathscr{M}$ such that

[TABLE]

Therefore

[TABLE]

So

[TABLE]

Based on the above analysis, the following equality can be obtained

[TABLE]

For arbitrary $u,v,w\in\mathscr{M}$, set $x=u+v,y=w$. So we conclude that

[TABLE]

Hence

[TABLE]

It folows from assumption that for each $u,v\in\mathscr{M}$, there is a $w\in\mathscr{M}$ such that \uppsi(w)=\big{(}\Delta(u+v)-\Delta(u)-\Delta(v)\big{)}^{*}. So

[TABLE]

Now since the family $\mathfrak{T}$ is faithful, we have

[TABLE]

So

[TABLE]

It follows that $\Delta$ is an additive 2-local $(\upvarphi,\uppsi)$-derivation, and Lemma 3.7 implies that $\Delta$ is a bounded Jordan $(\upvarphi,\uppsi)$-derivation. ∎

Remark 4.7*.*

The last theorem also hold, if we replace the condition \{\big{(}\Delta(u+v)-\Delta(u)-\Delta(v)\big{)}^{*}\}\subseteq\uppsi(\mathscr{M}), with \{\big{(}\Delta(u+v)-\Delta(u)-\Delta(v)\big{)}^{*}\}\subseteq\upvarphi(\mathscr{M}) one.

Remark 4.8*.*

The last theorem hold also for approxmately 2-local $(\upvarphi,\uppsi)$-derivations, if in addition, $(\upvarphi,\uppsi)$-tracial map $\tau$ is normal.