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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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