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 FBL C [ E ] containing a linear isometric copy of E and satisfying the following universal property: for every complex Banach lattice X C , every operator T : E → X C admits a unique lattice homomorphic extension T ˆ : FBL C [ E ] → X C with ‖ T ˆ ‖ = ‖ T ‖. The free complex Banach lattice FBL 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 FBL C [ E ] and FBL C [ F ] are lattice isometric. The spectral theory of induced lattice homomorphisms on FBL C [ E ] is also explored. [ABSTRACT FROM AUTHOR]