On banach spaces which containl 1(τ) and types of measures on compact spaces

Abstract

Two closely related results are presented, one of them concerned with the connection between topological and measure-theoretic properties of compact spaces, the other being a non-separable analogue of a result of Peŀczynski's about Banach spaces containingL 1. Let τ be a regular cardinal satisfying the hypothesis that κω<τ whenever κ<τ. The following are proved: 1) A compact spaceT carries a Radon measure which is homogeneous of type τ, if and only if there exists a continuous surjection ofT onto [0, 1]τ. 2) A Banach spaceX has a subspace isomorphic tol 1(τ) if and only ifX ∗ has a subspace isomorphic toL 1({0, 1}τ). An example is given to show that a more recent result of Rosenthal's about Banach spaces containingl 1 does not have an obvious transfinite analogue. A second example (answering a question of Rosenthal's) shows that there is a Banach spaceX which contains no copy ofl 1 (ω1), while the unit ball ofX ∗ is not weakly∗ sequentially compact.