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Stability of linear mappings in quasi-Banach modules. (English) Zbl 1154.39028

Let 𝒜 be a unital C * -algebra and 𝒜 1 + be the positive part of the unit ball 𝒜 1 of 𝒜. Let 𝒳 be a quasi-normed 𝒜-module with quasi-norm · 𝒳 and 𝒴 be a p-Banach 𝒜-module with p-norm · 𝒴 (i.e. 0<p1 and x+y 𝒴 p x 𝒴 p +y 𝒴 p for x,y𝒴). The authors prove the Hyers-Ulam-Rassias stability of linear mappings from 𝒳 to 𝒴, associated to the Cauchy functional equation and a generalized version of the Jensen equation. In the first case they assume that a function f:𝒳𝒴 fulfils

f(ux+y)-uf(x)-f(y) 𝒴 ε(x 𝒳 r +y 𝒳 r )

for x,y𝒳 and for all u from the unitary group U(𝒜) if r>1, and for all u𝒜 1 + {i} if r<1. In the second case they assume that f(0)=0 and for some integer N>1 we have

Nfax+y N-af(x)-f(y) 𝒴 ε(x 𝒳 r +y 𝒳 r )

for x,y𝒳 and for all a from the unit sphere of 𝒜. In both cases the authors show that there is a unique 𝒜-linear mapping L:𝒳𝒴 satisfying

f(x)-L(x) 𝒴 K(ε,p,r,N)x 𝒳 r (x𝒳)

with certain stability constants K(ε,p,r,N) which always tend to zero as ε0.

MSC:
39B82Stability, separation, extension, and related topics
39B52Functional equations for functions with more general domains and/or ranges
46L05General theory of C * -algebras