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Composition operators between algebras of differentiable functions

Authors: Joaquín M. Gutiérrez and José G. Llavona
Journal: Trans. Amer. Math. Soc. 338 (1993), 769-782
MSC: Primary 46G20; Secondary 26E15, 46E25, 47B38
MathSciNet review: 1116313
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Abstract: Let $ E$, $ F$ be real Banach spaces, $ U \subseteq E$ and $ V \subseteq F$ nonvoid open subsets and $ {C^k}(U)$ the algebra of real-valued $ k$-times continuously Fréchet differentiable functions on $ U$, endowed with the compact open topology of order $ k$. It is proved that, for $ m \geq p$, the nonzero continuous algebra homomorphisms $ A:{C^m}(U) \to {C^p}(V)$ are exactly those induced by the mappings $ g:V \to U$ satisfying $ \phi \circ g \in {C^p}(V)$ for each $ \phi \in {E^\ast}$, in the sense that $ A(f) = f \circ g$ for every $ f \in {C^m}(U)$. Other homomorphisms are described too. It is proved that a mapping $ g:V \to {E^{\ast \ast}}$ belongs to $ {C^k}(V,({E^{\ast \ast}},{w^\ast}))$ if and only if $ \phi \circ g \in {C^k}(V)$ for each $ \phi \in {E^\ast}$. It is also shown that if a mapping $ g:V \to E$ verifies $ \phi \circ g \in {C^k}(V)$ for each $ \phi \in {E^\ast}$, then $ g \in {C^{k - 1}}(V,E)$.

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Keywords: Differentiable mappings between Banach spaces, algebras of differentiable functions, homomorphisms, composition operators
Article copyright: © Copyright 1993 American Mathematical Society