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Norm-linear and norm-additive operators between uniform algebras. (English) Zbl 1171.47032
Let $$A\subseteq C(X)$$ and $$B\subseteq C(Y)$$ be uniform algebras with Choquet boundaries $$\delta A$$ and $$\delta B$$, respectively, and let $${\mathbb R}_+$$ denote the set of nonnegative reals numbers. The authors call a map $$T:A\to B$$ norm-additive in modulus if
$\big\| |Tf|+|Tg|\big\|=\big\| |f|+|g|\big\|\qquad (f,g\in A).$
It is then shown that an $${\mathbb R}_+$$-homogeneous, norm-additive in modulus surjection $$T:A\to B$$ is induced on the Choquet boundary by a continuous map $$\tau:\delta A\to\delta B$$ in the sense that $$|(Tf)(\tau(x))|=|f(x)|$$ for every $$x\in \delta A$$ and every $$f\in A$$. Actually, $$\tau$$ is a homeomorphism whenever $$T$$ is bijective. The key idea is to show that, for each $$x\in\delta A$$, the set $E_x:=\bigcap_{f\in A\atop |f(x)|=\|f\|} |T(f)|^{-1}\{\|Tf\|\}$ is a singleton and belongs to $$\delta B$$.
Several applications follow where norm conditions on a surjection $$T:A\to B$$, which force it to be an algebra isomorphism, are studied. The list is too long to be given here, but as a sample example, we note that it is shown that if a surjection $$T$$ between uniform algebras satisfies (i) $$\|Tf+\alpha Tg\|=\|f+\alpha g\|$$, $$|\alpha|=1$$, and (ii) $$T$$ is unital and $$T(\sqrt{-1})=\sqrt{-1}$$, then $$T$$ is an isometric algebra isomorphism. The same conclusion holds if (ii) is replaced by (ii’) $$T$$ preserves the peripheral spectrum of $$\mathbb C$$-peaking functions from $$A$$.

##### MSC:
 47B49 Transformers, preservers (linear operators on spaces of linear operators) 46J10 Banach algebras of continuous functions, function algebras
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