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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, $$\Delta_ b(A)$$ 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 $$d_ 1d_ 2$$ of two derivations $$d_ 1$$, $$d_ 2$$, and obtained the following results:
1. Let A be an ultraprime normed algebra and let $$d_ 1,d_ 2\in D_ b(A)$$ then $$dist(d_ 1d_ 2,\Delta_ b(A))\geq (C^ 2/6)\| d_ 1\| \| d_ 2\|$$ if $$\| M_{a,b}\| \geq C\| a\| \| b\|$$ (a,b$$\in A)$$, where $$M_{a,b}(x)=axb$$, $$x\in A.$$
2. Let A be an ultrasemiprime normed algebra and $$d\in D_ b(A)$$, then $$dist(d^ 2,\Delta_ b(A))\geq (C/2)\| d\|^ 2$$ if $$C>0$$ satisfies $$\| M_{a,a}\| \geq C\| a\|^ 2$$ for all $$a\in A.$$
3. Let A be a von Neumann algebra. If $$d_ 1,d_ 2\in D(A)$$ then $$dist(d_ 1d_ 2,\Delta (A))\leq (1/2)\| d_ 1\| \| d_ 2\|$$. In particular, for any $$d\in \Delta (A)$$, $$dist(d^ 2,\Delta_ b(A))=(1/2)\| d\|^ 2.$$
As a consequence of these results, the author obtains a partial answer to Mathieu’s question; if $$d_ 1=d_ 2=d\in D_ b(A)$$ then $$(1/2)\| d\|^ 2\leq \| d^ 2\| \leq \| d\|^ 2$$.
Reviewer: J.C.Rho (Seoul)

##### MSC:
 47B47 Commutators, derivations, elementary operators, etc. 46L57 Derivations, dissipations and positive semigroups in $$C^*$$-algebras
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##### References:
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