Meaning
A selected correspondence is spatially effective when a concrete corner operator realizes it as the graph of an onto algebra isomorphism.
For the selected A–B correspondence, one needs w in its linking algebra with w∗w=e_A and ww∗=e_B, such that a↦waw∗ maps A onto B. The support equations and onto condition turn the abstract correspondence into a spatial graph realization on the selected support.
Example (illustrative)
For A=B=M₂(C), a unitary w implements a↦waw∗, an onto automorphism. This is an elementary graph correspondence. A proper isometric embedding into a larger destination need not have the required onto property.
Scope and limits
Abstract correspondence data alone do not supply w or prove spatial effectivity. The assertion is relative to the selected corners.
Source
GAT I — Grassmannian Gluing, Spectral Trichotomy, and Conditional Sectorial Type II. GAT I, Definition 5.6.