On contractively complemented subspaces of separableL ₁-preduals

Abstract

It is shown that for anL ₁-predual spaceX and a countable linearly independent subset of ext(B _X*) whose norm-closed linear spanY inX* isω*-closed, there exists aω*-continuous contractive projection fromX* ontoY. This result combined with those of Pelczynski and Bourgain yields a simple proof of the Lazar-Lindenstrauss theorem that every separableL ₁-predual with non-separable dual contains a contractively complemented subspace isometric toC(Δ), the Banach space of functions continuous on the Cantor discontinuum Δ.

It is further shown that ifX* is isometric tol ₁ and (e* _n ) is a basis forX* isometrically equivalent to the usuall ₁-basis, then there exists aω*-convergent subsequence (e ^* _{m n} ) of (e* _n ) such that the closed linear subspace ofX* generated by the sequence ($\left( {e_{m_{2n} }^ * - e_{m_{2n - 1} }^ * } \right)$) is the range of aω*-continuous contractive projection inX*. This yields a new proof of Zippin’s result thatc ₀ is isometric to a contractively complemented subspace ofX.