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On the distance of the composition of two derivations to the generalized derivations. (English) Zbl 0731.47037
Let A be a ring, an additive map $\delta$ : $A\to A$ is said to be a generalized derivation if there exists a derivation h of A such that $\delta$ satisfies $\delta (xy)=\delta (x)y+xh(y)$ (x,y$\in A)$. Let $\Delta$ (A) denotes the set of all generalized derivations; if A is a normed algebra, $\Delta\sb b(A)$ denotes the setof all $\delta$ in $\Delta$ (A) which are also bounded linear operators on A. In this note, the author estimates the distance of the composition $d\sb 1d\sb 2$ of two derivations $d\sb 1$, $d\sb 2$, and obtained the following results: 1. Let A be an ultraprime normed algebra and let $d\sb 1,d\sb 2\in D\sb b(A)$ then $dist(d\sb 1d\sb 2,\Delta\sb b(A))\ge (C\sp 2/6)\Vert d\sb 1\Vert \Vert d\sb 2\Vert$ if $\Vert M\sb{a,b}\Vert \ge C\Vert a\Vert \Vert b\Vert$ (a,b$\in A)$, where $M\sb{a,b}(x)=axb$, $x\in A.$ 2. Let A be an ultrasemiprime normed algebra and $d\in D\sb b(A)$, then $dist(d\sp 2,\Delta\sb b(A))\ge (C/2)\Vert d\Vert\sp 2$ if $C>0$ satisfies $\Vert M\sb{a,a}\Vert \ge C\Vert a\Vert\sp 2$ for all $a\in A.$ 3. Let A be a von Neumann algebra. If $d\sb 1,d\sb 2\in D(A)$ then $dist(d\sb 1d\sb 2,\Delta (A))\le (1/2)\Vert d\sb 1\Vert \Vert d\sb 2\Vert$. In particular, for any $d\in \Delta (A)$, $dist(d\sp 2,\Delta\sb b(A))=(1/2)\Vert d\Vert\sp 2.$ As a consequence of these results, the author obtains a partial answer to Mathieu’s question; if $d\sb 1=d\sb 2=d\in D\sb b(A)$ then $(1/2)\Vert d\Vert\sp 2\le \Vert d\sp 2\Vert \le \Vert d\Vert\sp 2$.
Reviewer: J.C.Rho (Seoul)

47B47Commutators, derivations, elementary operators, etc.
46L57Derivations, dissipations and positive semigroups in $C^*$-algebras
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