Operators on the Stopping Time Space

Let $S^1$ be the stopping time space and ${\mathcal{B}}_1(S^1)$ be the Baire-1 elements of the second dual of $S^1$. To each element $x^{\ast \ast }$ in the space ${\mathcal{B}}_1(S^1)$ we associate a positive Borel measure ${\mu }_{x^{\ast \ast }}$ on the Cantor set. We use the measures ${\mu_{x^{}}: x^{}\in$\mathcal{B}$_1(S^1) }$tocharacterizetheoperators$T:X\to S^1$, defined on a space $X$ with an unconditional basis, which preserve a copy of $S^1$. In particular, we show that $T$ preserves a copy of $S^1$ if and only if the set $\{{\mu }_{x^{\ast \ast }}:x^{\ast \ast }\in {\mathcal{B}}_1(S^1)\}$ is non separable as a subset of $\mathcal{M}(2^{\mathbb{N}})$.