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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 δ : AA is said to be a generalized derivation if there exists a derivation h of A such that δ satisfies δ(xy)=δ(x)y+xh(y) (x,yA). Let Δ (A) denotes the set of all generalized derivations; if A is a normed algebra, Δ b (A) denotes the setof all δ in Δ (A) which are also bounded linear operators on A.

In this note, the author estimates the distance of the composition d 1 d 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 D b (A) then dist(d 1 d 2 ,Δ b (A))(C 2 /6)d 1 d 2 if M a,b Cab (a,bA), where M a,b (x)=axb, xA·

2. Let A be an ultrasemiprime normed algebra and dD b (A), then dist(d 2 ,Δ b (A))(C/2)d 2 if C>0 satisfies M a,a Ca 2 for all aA·

3. Let A be a von Neumann algebra. If d 1 ,d 2 D(A) then dist(d 1 d 2 ,Δ(A))(1/2)d 1 d 2 . In particular, for any dΔ(A), dist(d 2 ,Δ 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 =dD b (A) then (1/2)d 2 d 2 d 2 .

Reviewer: J.C.Rho (Seoul)

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