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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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