Isomorphisms from Extremely Regular Subspaces of C0K into C0S,X Spaces

Abstract EN

For a locally compact Hausdorff space K and a Banach space X, let C0K,X 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 ℝ, we denote C0K,X as C0K.

If AK be an extremely regular subspace of C0K and T:AK⟶C0S,X 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 c0, then the cardinality of K is less than that of S.

Moreover, if TT−1<3 and AK is also a subalgebra of C0K, the cardinality of the αth derivative of K is less than that of the αth derivative of S, for each ordinal α.

Finally, if λX>1 and TT−1<λX, then K is a continuous image of a subspace of S.

Here, λX is the geometrical parameter introduced by Jarosz in 1989: λX=infmaxx+λy:λ=1:x=y=1.

As a consequence, we improve classical results about into isomorphisms from extremely regular subspaces already obtained by Cengiz.