Free complex Banach lattices

The construction of the free Banach lattice generated by a real Banach space is extended to the complex setting. It is shown that for every complex Banach space $E$ there is a complex Banach lattice $FB{L}_{\mathbb{C}}[E]$ containing a linear isometric copy of $E$ and satisfying the following universal property: for every complex Banach lattice $X_{\mathbb{C}}$, every operator $T:E\to X_{\mathbb{C}}$ admits a unique lattice homomorphic extension $\widehat{T}:FB{L}_{\mathbb{C}}[E]\to X_{\mathbb{C}}$ with $\mathrm{\parallel }\widehat{T}\mathrm{\parallel }=\mathrm{\parallel }T\mathrm{\parallel }$. The free complex Banach lattice $FB{L}_{\mathbb{C}}[E]$ is shown to have analogous properties to those of its real counterpart. However, examples of non-isomorphic complex Banach spaces $E$ and $F$ can be given so that $FB{L}_{\mathbb{C}}[E]$ and $FB{L}_{\mathbb{C}}[F]$ are lattice isometric. The spectral theory of induced lattice homomorphisms on $FB{L}_{\mathbb{C}}[E]$ is also explored.