Following an idea of Choi, we obtain a decomposition theorem for k-positive linear maps from M m (C) to M n (C), where 2 ≤ k < min{m, n}. As a consequence, we give an affirmative answer to Kye's conjecture (also solved independently by Choi) that every 2-positive linear map from M 3 (C) to M 3 (C) is decomposable.
We provide a variety of results for (quasi)convex, law-invariant functionals defined on a general Orlicz space, which extend well-known results in the setting of bounded random variables. First, we show that Delbaen's representation of convex functionals with the Fatou property, which fails in a general Orlicz space, can be always achieved under the assumption of lawinvariance. Second, we identify the range of Orlicz spaces where the characterization of the Fatou property in terms of norm lower semicontinuity by Jouini, Schachermayer and Touzi continues to hold. Third, we extend Kusuoka's representation to a general Orlicz space. Finally, we prove a version of the extension result by Filipović and Svindland by replacing norm lower semicontinuity with the (generally non-equivalent) Fatou property. Our results have natural applications to the theory of risk measures.
Suppose that (Fn) ∞ n=0 is a sequence of regular families of finite subsets of N such that F0 contains all singletons, and (θn) ∞ n=1 is a nonincreasing null sequence in (0, 1). The mixed Tsirelson space T (F0, (θn, Fn) ∞ n=1 ) is the completion of c00 with respect to the implicitly defined normwhere x F 0 = sup F ∈F F x ℓ 1 and the last supremum is taken over all sequences (Ei) k i=1 in [N] <∞ such that max Ei < min Ei+1 and {min Ei : 1 ≤ i ≤ k} ∈ Fn. In this paper, we compute the Bourgain ℓ 1 -index of the space T (F0, (θn, Fn) ∞ n=1 ). As a consequence, it is shown that if η is a countable ordinal not of the form ω ξ for some limit ordinal ξ, then there is a Banach space whose ℓ 1 -index is ω η .
Abstract. We study minimality properties of partly modified mixed Tsirelson spaces. A Banach space with a normalized basis (e k ) is said to be subsequentially minimal if for every normalized block basis (x k ) of (e k ), there is a further block basis (y k ) of (x k ) such that (y k ) is equivalent to a subsequence of (e k ). Sufficient conditions are given for a partly modified mixed Tsirelson space to be subsequentially minimal, and connections with Bourgain's 1 -index are established. It is also shown that a large class of mixed Tsirelson spaces fails to be subsequentially minimal in a strong sense.
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