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Characterizing Jordan maps on \(C^*\)-algebras through zero products. (English) Zbl 1216.47063

The authors consider continuous bilinear maps preserving zero products. The main result is as follows.
Theorem. Let \({\mathcal A}\) be a \(C^*\)-algebra and let \(X\) be a Banach space. Assume that \(\varphi:{\mathcal A}\times{\mathcal A}\to X\) is a continuous bilinear map satisfying \(\varphi(a, b) = 0\) whenever \(ab = ba = 0\) for \(a, b\in{\mathcal A}\). Then
\[ \varphi(ab,cd)- \varphi(a,bcd)+ \varphi(da,bc)- \varphi(dab,c)=0,\qquad a,b,c,d\in{\mathcal A}, \]
and there exist continuous linear maps \(\Phi,\Psi:{\mathcal A}\to X\) such that
\[ \varphi(ab,c)- \varphi(b,ca)+ \varphi(bc,a)= \Phi(abc),\qquad a,b,c\in{\mathcal A}, \]
and
\[ \varphi(a,b)+ \varphi(b,a)= \Psi(a\circ b),\qquad a,b\in{\mathcal A}, \]
where \(a\circ b=\frac12(ab+ba)\) for \(a,b\in{\mathcal A}\).
As applications, the authors characterize, respectively, Jordan homomorphisms and derivations through zero products.

MSC:

47B48 Linear operators on Banach algebras
46L05 General theory of \(C^*\)-algebras
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