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We generalize the results of [11] and [12] for the unit ball \( \mathbb{B}_d \) of ℂd. In particular, we show that under the weight condition (B) the weighted H∞-space on \( \mathbb{B}_d \) is isomorphic to l∞ and thus complemented in the corresponding weighted L∞-space. We construct concrete, generalized Bergman projections accordingly. We also consider the case where the domain is the entire space ℂd. In addition, we show that for the polydisc \( \mathbb{D}^d \)d, the weighted H∞-space is never isomorphic to l∞.

LetX be a Banach space with a sequence of linear, bounded finite rank operatorsR n:X→X such thatR nRm=Rmin(n,m) ifn≠m and lim n→∞ R n x=x for allx∈X. We prove that, ifR n−Rn −1 factors uniformly through somel p and satisfies a certain additional symmetry condition, thenX has an unconditional basis. As an application, we study conditions on Λ ⊂ ℤ such thatL Λ=closed span $$\left\{ {z^k :k \in \Lambda } \right\} \subset L_1 \left( \mathbb{T} \right)$$ , where $$\mathbb{T} = \left\{ {z \in \mathbb{C}:\left| z \right| = 1} \right\}$$ , has an unconditional basis. Examples include the Hardy space $$H_1 = L_{\mathbb{Z}_ + } $$ .

We discuss and prove three space properties and basis extension theorems of the following kind: LetY be a separableL 1-space andX⊄Y a non-reflexive subspace such thatY/X has a basis. ThenX has a basis. IfY is a separableC(K)-space andX⊄Y is such thatY/X is nonreflexive, then every basis ofX can be extended to a basis ofY.

We show that the Poulsen simplex space is primary by investigating special monotone bases inL 1-predual spaces.

We show that everyL 1-predual space is complemented in a simplex space. This answers a question raised by Lazar and Lindenstrauss.