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Abstract

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 $x^{**}$ in $\mathcal {B}_1(S^1)$ we associate a positive Borel measure $\mu _{x^{**}}$ on the Cantor set. We use the measures $\{\mu _{x^{**}}: x^{**}\in \mathcal {B}_1(S^1) \}$ to characterize the operators $T:X\to S^1$, defined on a space $X$ with an unconditional basis, which preserve a copy of $S^1$. In particular, if $X=S^1$, we show that $T$ preserves a copy of $S^1$ if and only if $\{\mu _{T^{**}(x^{**})}:x^{**}\in \mathcal {B}_1(S^1)\}$ is non-separable as a subset of $\mathcal {M}(2^\mathbb {N})$.