Classifications of Baire-$1$ functions and $c_ 0$-spreading models

Abstract:

We prove that if for a bounded function f defined on a compact space K there exists a bounded sequence $({f_n})$ of continuous functions, with spreading model of order $\xi$, $1 \leq \xi < {\omega _1}$, equivalent to the summing basis of ${c_0}$, converging pointwise to f, then ${r_{{\text {ND}}}}(f) > {\omega ^\xi }$ (the index ${r_{{\text {ND}}}}$ as defined by A. Kechris and A. Louveau). As a corollary of this result we have that the Banach spaces ${V_\xi }(K)$, $1 \leq \xi < {\omega _1}$, which previously defined by the author, consist of functions with rank greater than ${\omega ^\xi }$. For the case $\xi = 1$ we have the equality of these classes. For every countable ordinal number $\xi$ we construct a function S with ${r_{{\text {ND}}}}(S) > {\omega ^\xi }$. Defining the notion of null-coefficient sequences of order $\xi$, $1 \leq \xi < {\omega _1}$, we prove that every bounded sequence $({f_n})$ of continuous functions converging pointwise to a function f with ${r_{{\text {ND}}}}(f) \leq {\omega ^\xi }$ is a null-coefficient sequence of order $\xi$. As a corollary to this we have the following ${c_0}$-spreading model theorem: Every nontrivial, weak-Cauchy sequence in a Banach space either has a convex block subsequence generating a spreading model equivalent to the summing basis of ${c_0}$ or is a null-coefficient sequence of order 1.