Free complex Banach lattices

TL;DR

This paper extends the concept of free Banach lattices to the complex setting, establishing their universal properties and exploring their spectral theory, thus broadening the theoretical framework of Banach lattice constructions.

Contribution

It introduces the construction of free complex Banach lattices and demonstrates their universal property, paralleling the real case, with additional insights into their spectral theory.

Findings

Existence of free complex Banach lattices with universal extension property

Examples of non-isomorphic spaces with lattice isometric free lattices

Analysis of spectral properties of lattice homomorphisms

Abstract

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 $FBL_{\mathbb C}[E]$ containing a linear isometric copy of $E$ and satisfying the following universal property: for every complex Banach lattice $X_{\mathbb C}$, every operator $T:E\rightarrow X_{\mathbb C}$ admits a unique lattice homomorphic extension $\hat{T}:FBL_{\mathbb C}[E]\rightarrow X_{\mathbb C}$ with $\|\hat{T}\|=\|T\|$. The free complex Banach lattice $FBL_{\mathbb C}[E]$ 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 $FBL_{\mathbb C}[E]$ and $FBL_{\mathbb C}[F]$ are lattice isometric. The spectral theory of induced lattice homomorphisms on $FBL_{\mathbb C}[E]$ is also explored.