Baire classes of $L_{1}$-preduals and $C^{*}$-algebras

Abstract

Let $X$ be a separable real or complex $L_{1}$-predual such that its dual unit ball $B_{X^{*}}$ has the set $\operatorname{ext}B_{X^{*}}$ of its extreme points of type $F_{\sigma}$. We identify intrinsic Baire classes of $X$ with the spaces of odd or homogeneous Baire functions on $\operatorname{ext}B_{X^{*}}$. Further, we answer a question of S. A. Argyros, G. Godefroy and H. P. Rosenthal by showing that there exists a separable $C^{*}$-algebra $X$ (the so-called CAR-algebra) for which the second intrinsic Baire class of $X^{**}$ does not coincide with the second Baire class of $X^{**}$.