Coordinate systems and distributional embeddings in Bourgain-Rosenthal-Schechtman spaces: a framework for operator reduction

For every $1\le \alpha <{\omega }_1$, we construct an explicit unconditional finite-dimensional decomposition (FDD) $(X_{\lambda }{)}_{\lambda \in {\mathcal{T}}_{\alpha }}$ of the Bourgain-Rosenthal-Schechtman space $R_{\alpha }^{p,0}$ by blocking its standard martingale difference sequence (MDS) basis. This FDD has strong reproducing properties and supports a theory of distributional representations between the spaces $R_{\alpha }^{p,0}$, $1\le \alpha <{\omega }_1$. We use this framework to prove an approximate orthogonal reduction: every bounded linear operator on a limit space $R_{\alpha }^{p,0}$ is, via a distributional embedding and up to arbitrary precision, reduced to a scalar FDD-diagonal operator. As a consequence, the standard MDS bases of the limit spaces $R_{\alpha }^{p,0}$ satisfy the factorization property.