Banach lattice AM-algebras

TL;DR

This paper characterizes Banach lattice algebras that can be embedded into continuous function spaces and explores the uniqueness of product operations compatible with their order and algebraic structure.

Contribution

It provides an analogue of Kakutani's theorem for Banach lattice algebras and shows the existence of alternative compatible product operations.

Findings

Characterization of Banach lattice algebras embeddable in C(K)

Existence of different product operations on Banach lattice algebras with identity

Uniqueness of pointwise multiplication on C(K) spaces

Abstract

An analogue of Kakutani's representation theorem for Banach lattice algebras is provided. We characterize Banach lattice algebras that embed as a closed sublattice-algebra of $C(K)$ precisely as those with a positive approximate identity $(e_\gamma)$ such that $x^{*}(e_\gamma)\to \|x^{*}\|$ for every positive functional $x^{*}$. We also show that every Banach lattice algebra with identity other than $C(K)$ admits different product operations which are compatible with the order and the algebraic identity. This complements the classical result, due to Martignon, that on $C(K)$ spaces pointwise multiplication is the unique compatible product.