Weak and pointwise compactness in the space of bounded continuous functions

Abstract

Let T T be a completely regular Hausdorff space, C b ( T ) {C_b}(T) the space of bounded continuous real-valued functions on T T , M ( T ) M(T) the Banach space dual of C b ( T ) {C_b}(T) . Let H \mathcal {H} denote the family of subsets of C b ( T ) {C_b}(T) which are uniformly bounded and relatively compact for the topology J p {\mathfrak {J}_p} of pointwise convergence. The basic question considered here is: what is the largest subspace Z Z of M ( T ) M(T) such that every member of H \mathcal {H} is relatively σ ( C b , Z ) \sigma ({C_b},Z) -compact? Classical results of Grothendieck and Ptak show that Z = M ( T ) Z = M(T) if T T is pseudocompact. In general, M t ⊂ Z ⊂ M s ; {M_t} \subset Z \subset {M_s}; assuming Martin’s Axiom, a deep result of Talagrand improves the lower bound to M τ {M_\tau } . It is frequently, but not always, true that Z = M s ; Z = {M_s}; counterexamples are given which use Banach spaces in their weak topologies to construct the underlying T T ’s.