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Llavona, José G. and Gutiérrez, Joaquín M.
(1993)
*Composition operators between algebras of differentiable functions.*
Transactions of the American Mathematical Society, 338
(2).
pp. 769-782.
ISSN 0002-9947

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Official URL: http://www.ams.org/journals/tran/1993-338-02/S0002-9947-1993-1116313-5/S0002-9947-1993-1116313-5.pdf

## Abstract

Let E, F be real Banach spaces, U subset-or-equal-to E and V subset-equal-to F non-void open subsets and C(k)(U) the algebra of real-valued k-times continuously Frechet differentiable functions on U, endowed with the compact open topology of order k. It is proved that, for m greater-than-or-equal-to p, the nonzero continuous algebra homomorphisms A: C(m)(U) --> C(p)(V) are exactly those induced by the mappings g: V --> U satisfying phi . g is-an-element-of C(p)(V) for each phi is-an-element-of E*, in the sense that A(f) = fog for every f is-an-element-of C(m)(U). Other homomorphisms are described too. It is proved that a mapping g: V --> E** belongs to C(k)(V, (E**, w*)) if and only if phi . g is-an-element-of C(k)(V) for each phi is-an-element-of E*. It is also shown that if a mapping g: V --> E verifies phi . g is-an-element-of C(k)(V) for each phi is-an-element-of E*, then g is-an-element-of C(k-1)(V, E).

Item Type: | Article |
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Uncontrolled Keywords: | Differentiable mappings between banach spaces; Algebras of differentiable functions; Homomorphisms; Composition operators |

Subjects: | Sciences > Mathematics > Functional analysis and Operator theory |

ID Code: | 16383 |

Deposited On: | 17 Sep 2012 09:12 |

Last Modified: | 27 Jul 2018 10:16 |

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