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 FBLC[E] containing a linear isometric copy of E and satisfying the following universal property: for every complex Banach lattice XC, every operator T : E -XC admits a unique lattice homomorphic extension T circumflex accent : T circumflex accent II = IIT II. The free complex Banach FBLC[E] -XC with II lattice FBLC[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 FBLC[E] and FBLC[F] are lattice isometric. The spectral theory of induced lattice homomorphisms on FBLC[E] is also explored.(c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons .org /licenses /by -nc -nd /4 .0/).
展开▼
