Operators on the Stopping Time Space

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Mar 26, 2015 - FA] 26 Mar 2015. OPERATORS ON THE STOPPING TIME SPACE. DIMITRIS APATSIDIS. Abstract. Let S1 be the stopping time space and ...
arXiv:1503.07756v1 [math.FA] 26 Mar 2015

OPERATORS ON THE STOPPING TIME SPACE DIMITRIS APATSIDIS Abstract. Let S 1 be the stopping time space and B1 (S 1 ) be the Baire-1 elements of the second dual of S 1 . To each element x∗∗ in the space B1 (S 1 ) we associate a positive Borel measure µx∗∗ on the Cantor set. We use the measures {µx∗∗ : x∗∗ ∈ B1 (S 1 )} to characterize the operators T : X → 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 {µx∗∗ : x∗∗ ∈ B1 (S 1 )} is non separable as a subset of M(2N ).

1. Introduction. The Stopping Time space S 1 , was introduced by H. P. Rosenthal as the unconditional analogue of L1 (2N ), where by 2N we denote the Cantor set and L1 (2N ) is the Banach space of equivalence classes of measurable functions on 2N which are absolutely integrable on 2N , with respect to the Haar measure. The space S 1 belongs to the wider class of the spaces S p , 1 ≤ p < ∞ which we are about to define. We denote by 2