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Contractive linear preservers of absolutely compatible pairs between \(\mathrm{C}^*\)-algebras. (English) Zbl 1434.46033
Summary: Let \(a\) and \(b\) be elements in the closed ball of a unital \(\mathrm{C}^*\)-algebra \(A\) (if \(A\) is not unital we consider its natural unitization). We shall say that \(a\) and \(b\) are domain (respectively, range) absolutely compatible (\(a\triangle _d b\), respectively, \(a\triangle _r b\), in short) if \(\big | |a| -|b| \big | + \big | 1-|a|-|b| \big | =1\) (respectively, \(\big | |a^*| -|b^*| \big | + \big | 1-|a^*|-|b^*| \big | =1\)), where \(|a|^2= a^* a\). We shall say that \(a\) and \(b\) are absolutely compatible (\(a\triangle b\) in short) if they are both range and domain absolutely compatible. In general, \(a\triangle _d b\) (respectively, \(a\triangle _r b\) and \(a\triangle b\)) is strictly weaker than \(ab^*=0 \) (respectively, \(a^* b =0\) and \(a\perp b\)). Let \(T: A\rightarrow B\) be a non-expansive bounded linear mapping between \(\mathrm{C}^*\)-algebras. We prove that, if \(T\) preserves domain absolutely compatible elements (i.e., \(a\triangle _d b\Rightarrow T(a)\triangle _d T(b)\)), then \(T\) is a triple homomorphism. A similar statement is proved when \(T\) preserves range absolutely compatible elements. It is finally shown that \(T\) is a triple homomorphism if, and only if, \(T\) preserves absolutely compatible elements.

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