Atomic decompositions, two stars theorems, and distances for the Bourgain-Brezis-Mironescu space and other big spaces

Given a Banach space $E$ with a supremum-type norm induced by a collection of operators, we prove that $E$ is a dual space and provide an atomic decomposition of its predual. We apply this result, and some results obtained previously by one of the authors, to the function space $mathcal{B}$ introduced recently by Bourgain, Brezis, and Mironescu. This yields an atomic decomposition of the predual $mathcal{B}_ast$, the biduality result that $mathcal{B}_0^ast = mathcal{B}_ast$ and $mathcal{B}_ast^ast = mathcal{B}$, and a formula for the distance from an element $f in mathcal{B}$ to $mathcal{B}_0$.