Banach spaces with projectional skeletons

A projectional skeleton in a Banach space is a sigma-directed family of projections onto separable subspaces, covering the entire space. The class of Banach spaces with projectional skeletons is strictly larger than the class of Plichko spaces (i.e. Banach spaces with a countably norming Markushevich basis). We show that every space with a projectional skeleton has a projectional resolution of the identity and has a norming space with similar properties to Sigma-spaces. We characterize the existence of a projectional skeleton in terms of elementary substructures, providing simple proofs of known results concerning weakly compactly generated spaces and Plichko spaces. We prove a preservation result for Plichko Banach spaces, involving transfinite sequences of projections. As a corollary, we show that a Banach space is Plichko if and only if it has a commutative projectional skeleton.

💡 Research Summary

The paper introduces and thoroughly investigates the concept of a projectional skeleton in Banach spaces, a structural device that generalizes the classical notion of a countably norming Markushevich basis (the defining feature of Plichko spaces). A projectional skeleton is a σ‑directed family ({P_\alpha}{\alpha\in\Gamma}) of bounded linear projections such that each image (\operatorname{Im} P\alpha) is a separable (indeed, countable‑dimensional) closed subspace, the projections are compatible ((\alpha\le\beta\Rightarrow P_\alpha\circ P_\beta=P_\alpha)), and for every vector (x\in X) the net ({P_\alpha x}_{\alpha\in\Gamma}) approximates (x) arbitrarily well. This definition captures the idea of building the whole space from a directed system of “small” pieces, but without requiring a global basis.

The first major result shows that any Banach space possessing a projectional skeleton automatically admits a projectional resolution of the identity (PRI). By re‑ordering the skeleton and taking appropriate limits along the σ‑directed index set, the authors construct a continuous transfinite sequence ({Q_\xi}_{\xi<\kappa}) of projections with separable ranges that increase to the identity operator. This extends the classical PRI theory, previously known only for Plichko spaces, to a strictly larger class.

Next, the authors demonstrate that a projectional skeleton yields a norming subspace of the dual with Σ‑space characteristics. Defining (M:=\bigcup_{\alpha\in\Gamma}P_\alpha^{}(X^{})), they prove that (M) is norming for (X) (i.e., (|x|=\sup_{f\in M}|f(x)|)) and that (M) is a Σ‑space: it is the countable union of separable subspaces and enjoys the same “countable support” properties that make Σ‑spaces amenable to set‑theoretic analysis. Consequently, any space with a projectional skeleton inherits a dual structure reminiscent of those found in classical Σ‑spaces.

A particularly elegant contribution is the characterisation of projectional skeletons via elementary substructures. Working inside a sufficiently large regular cardinal (\theta) and its structure (H(\theta)), the authors consider countable elementary submodels (M\prec H(\theta)). For each such (M) the natural projection (P_M) onto the closure of (X\cap M) belongs to the skeleton, and the collection ({P_M : M\prec H(\theta)}) is σ‑directed. This model‑theoretic viewpoint provides short, transparent proofs of known facts about weakly compactly generated (WCG) spaces and Plichko spaces, because in those settings the elementary submodels automatically produce separable ranges.

The paper also proves a preservation theorem for Plichko spaces under transfinite sequences of projections. Suppose ({R_\xi}{\xi<\lambda}) is an increasing transfinite family of projections with the property that each intermediate space (R\xi X) is Plichko. The authors show that the limit space (X) itself must be Plichko. The proof proceeds by inductively extending countably norming Markushevich bases along the chain, ensuring that the countable norming property survives the limit step. This result generalises the classical fact that subspaces of WCG spaces are WCG, and it supplies a powerful tool for constructing new Plichko spaces from known ones.

Finally, the authors establish a striking equivalence between commutative projectional skeletons and Plichko spaces. If a projectional skeleton is commutative (all its projections pairwise commute), then the skeleton can be refined to a countably norming Markushevich basis, proving that the underlying space is Plichko. Conversely, every Plichko space possesses a commutative PRI, which can be reorganised into a commutative projectional skeleton. Hence the paper proves the biconditional:
\