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Let Y be a Banach space, (Ω, Σ; μ) a probability space and φ a finite Young function. It is shown that the Y-valued Orlicz heart H φ(μ, Y) is isometrically isomorphic to the l-completed tensor product $$H_{\varphi }\left(\mu \right){\tilde{\otimes }}_{l}Y$$ of the scalar-valued Orlicz heart Hφ(μ) and Y, in the sense of Chaney and Schaefer. As an application, a characterization is given of the equality of $$\left(H_{\varphi }\left(\mu \right){\tilde{\otimes }}_{l}Y\right)*$$ and $$H_{\varphi }\left(\mu \right)*{\tilde{\otimes }}_{l}Y*$$ in terms of the Radon-Nikodým property on Y. Convergence of norm-bounded martingales in H φ(μ, Y) is characterized in terms of the Radon-Nikodým...

Spaces of cone absolutely summing maps are generalizations of Bochner spaces L p(μ, Y), where (Ω, Σ, μ) is some measure space, 1 ≤ p ≤ ∞ and Y is a Banach space. The Hiai-Umegaki space $${ℒ}^1\left[\sum ,cbf\left(X\right)\right]$$ of integrably bounded functions F: Ω → cbf(X), where the latter denotes the set of all convex bounded closed subsets of a separable Banach space X, is a set-valued analogue of L 1(μ, X). The aim of this work is to introduce set-valued cone absolutely summing maps as a generalization of $${ℒ}^1\left[\sum ,cbf\left(X\right)\right]$$ , and to derive necessary...