Vector lattice generated by finite rank operators, with applications to tensor products of Banach lattices

Let $X$ and $Y$ be two Banach lattices. While the spaces of all regular operators from $X$ to $Y$ and of all bounded finite-rank operators are not lattices, it is known that lattice operations of finite-rank operators do exist. We investigate the vector lattice generated by finite rank operators among all operators. The adjoint map that sends $T$ into $T^*$ is a lattice isometry on this space (with respect to the regular norm). Also, the map that sends $T$ into $jT$, where $j$ is the canonical embedding of $Y$ into $Y^{**}$, is a lattice isometry on this space. Our approach provides an alternative construction of the lattice injective tensor product of $X$ and $Y$.