Illinois Journal of Mathematics

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

Pavel Ludvík and Jiří Spurný

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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 $X^{**}$.

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Illinois J. Math., Volume 58, Number 1 (2014), 97-112.

First available in Project Euclid: 1 April 2015

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Zentralblatt MATH identifier

Primary: 46B25: Classical Banach spaces in the general theory 26A21: Classification of real functions; Baire classification of sets and functions [See also 03E15, 28A05, 54C50, 54H05]
Secondary: 46B04: Isometric theory of Banach spaces 46A55: Convex sets in topological linear spaces; Choquet theory [See also 52A07] 46L05: General theory of $C^*$-algebras


Ludvík, Pavel; Spurný, Jiří. Baire classes of $L_{1}$-preduals and $C^{*}$-algebras. Illinois J. Math. 58 (2014), no. 1, 97--112. doi:10.1215/ijm/1427897169.

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