### Lifting Properties for Some Quotients of L1-Spaces and Other Spaces L-Summand in Their Bidual.

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We study the Complex Unconditional Metric Approximation Property for translation invariant spaces ${C}_{\Lambda}\left(\right)$ of continuous functions on the circle group. We show that although some “tiny” (Sidon) sets do not have this property, there are “big” sets Λ for which ${C}_{\Lambda}\left(\right)$ has (ℂ-UMAP); though these sets are such that ${L}_{\Lambda}^{\infty}\left(\right)$ contains functions which are not continuous, we show that there is a linear invariant lifting from these ${L}_{\Lambda}^{\infty}\left(\right)$ spaces into the Baire class 1 functions.

We are interested in Banach space geometry characterizations of quasi-Cohen sets. For example, it turns out that they are exactly the subsets E of the dual of an abelian compact group G such that the canonical injection $C\left(G\right)/{C}_{{E}^{c}}\left(G\right)\hookrightarrow L{\xb2}_{E}\left(G\right)$ is a 2-summing operator. This easily yields an extension of a result due to S. Kwapień and A. Pełczyński. We also investigate some properties of translation invariant quotients of L¹ which are isomorphic to subspaces of L¹.

give estimates for the approximation numbers of composition operators on the Hp spaces, 1 ≤ p < ∞

We give new proofs that some Banach spaces have Pełczyński's property (V).

We study the canonical injection from the Hardy-Orlicz space ${H}^{\Psi}$ into the Bergman-Orlicz space ${}^{\Psi}$.

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