Isomorphisms from Extremely Regular Subspaces of into Spaces

For a locally compact Hausdorff space K and a Banach space X , let be the Banach space of all X -valued continuous functions defined on K , which vanish at infinite provided with the sup norm. If X is , we denote as . If be an extremely regular subspace of and is an into isomorphism, what can be said about the set-theoretical or topological properties of K and S ? Answering the question, we will prove that if X contains no copy of , then the cardinality of K is less than that of S . Moreover, if and is also a subalgebra of , the cardinality of the α th derivative of K is less than that of the α th derivative of S , for each ordinal Finally, if and , then K is a continuous image of a subspace of S . Here, is the geometrical parameter introduced by Jarosz in 1989: As a consequence, we improve classical results about into isomorphisms from extremely regular subspaces already obtained by Cengiz.