# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Mathieu, Rings of quotients of ultraprime Banach algebras. With applications to elementary operators, to appear in the Proceedings of the Conference on Banach algebras and Automatic Continuity (1989) · Zbl 0701.46027  Sakai, C (1971)  DOI: 10.2307/2032686 · Zbl 0082.03003 · doi:10.2307/2032686  DOI: 10.1007/BF01442873 · Zbl 0648.46052 · doi:10.1007/BF01442873  Williams, Pacific J. Math 38 pp 273– (1971) · Zbl 0205.42102 · doi:10.2140/pjm.1971.38.273  DOI: 10.2307/1996301 · Zbl 0252.46085 · doi:10.2307/1996301  Fong, Canad. J. Math 31 pp 845– (1979) · doi:10.4153/CJM-1979-080-x  DOI: 10.1016/0021-8693(81)90120-4 · Zbl 0463.16023 · doi:10.1016/0021-8693(81)90120-4  DOI: 10.2307/2038786 · Zbl 0257.46103 · doi:10.2307/2038786  Lanski, Pacific J. Math 134 pp 275– (1988) · Zbl 0614.16028 · doi:10.2140/pjm.1988.134.275
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.