Operators on the stopping time space

Dimitris Apatsidis¹

¹ Faculty of Applied Sciences National Technical University of Athens Department of Mathematics, Zografou Campus 157 80, Athens, Greece

Studia Mathematica, Tome 228 (2015) no. 3, pp. 235-258

Résumé

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 $\mathcal {M}(2^\mathbb {N})$.

Détail

DOI : 10.4064/sm228-3-3

Keywords: stopping time space mathcal baire elements second dual each element ** mathcal associate positive borel measure ** cantor set measures ** ** mathcal characterize operators defined space unconditional basis which preserve copy particular preserves copy only ** ** ** mathcal non separable subset mathcal mathbb