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It is a most implausible fact that a one-to-one operator from c 0 into a Banach space which maps the unit ball of c 0 onto a closed set is necessarily an isomorphism.In this paper the term semi-embedding denotes a one-to-one operator from one Banach space into another, which maps the closed unit ball of the domain onto a closed set. In the first section we study semi-embeddings in conjunction with weak compactness; in the second section we apply our results to the case of semiembeddings defined on C(X), X compact.In Propositions 2 and 3 we construct semi-embeddings which are not isomorphisms. The main result of Section 1 (Proposition 4) states that an operator can be factored through a dual space provided that there exists a semi-embedding defined on the codomain such that the composition of the two operators is weakly compact. This implies that the domain of a weakly compact semi-embedding must be a dual space (Corollary 6). In the case when the domain is C(X), the existence of a weakly compact semi-embedding is equivalent to X being hyperstonian and satisfying the countable chain condition.In Section 2, after some technical lemmas, we strengthen an argument of Pelczynski and Semadeni. Then we give the main result of the paper: a compact space X is scattered if and only if every semi-embedding of C(X) is an isomorphism. This is even true for any equivalent norm on C(X) (Corollary 12). In particular, every semi-embedding of c 0 is an isomorphism.Theorem 11 also allows us to answer a question raised by Kalton and Wilansky (question 6.4 in (5)). Indeed, our results show (see Corollary 14) that a compact space X is scattered if and only if all one-to-one Tauberian operators from C(X) into arbitrary Banach spaces are isomorphisms.Finally, we prove that every semi-embedding of C[0, 1] is an isomorphism on a complemented subspace isomorphic to C[0,1].We use standard terminology throughout (see (1)). An operator is a bounded linear operator from one Banach space into another; an embedding of a Banach space in another is an operator which is an isomorphism onto a closed subspace of its codomain. Finally, a compact space is scattered if it contains no perfect subsets.

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