Some special compacta K and related spaces C(K) of continuous functions
by
Witold Marciszewski
Vrije Universiteit Amsterdam and University of Warsaw

Two special classes of compact spaces K and Banach spaces C(K) of real-valued continuous functions on K (with the sup norm) will be discussed.

For a scattered Eberlein compact space (i.e., a weakly compact subset of a Banach space) K, we prove that if K is of scattered height at most \omega+1, then K is a Uniform Eberlein compact space (i.e., a weakly compact subset of a Hilbert space). This is a joint result with Murray Bell.

We characterize compacta K for which the Banach space C(K) is isomorphic to the classical Banach space c₀(\Gamma) for some set \Gamma. Such compacta K form a proper subclass of the class of Eberlein compact spaces of finite scattered height. In particular, we exhibit an Eberlein compact space L of weight \omega_\omega and with the empty third derived set L⁽³⁾ and such that the space C(L) is not isomorphic to any c₀(\Gamma). A consequence: the weakly compactly generated (WCG) Banach spaces C(L) and c₀(\omega_\omega) are bilipschitz isomorphic and nonisomorphic.

For a subset A of [0, 1], let S_A be the quotient space of the ``double arrow space'' S (that is, of [0, 1]×{0, 1} with the order topology given by the lexicographical order) obtained by identifying points (t, 0) and (t, 1), t in A. Since the spaces C(S_A) provide interesting examples in Banach space theory, recently, Godefroy has posed the problems of topological classification of spaces S_A and of the isomorphical classification of Banach spaces C(S_A). Some partial results in this direction will be presented.

Date received: February 28, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cajz-98.