Lattice structures of ordered Banach algebras (Q1203484)

The central purpose of the present paper is to investigate conditions which assure that a Banach algebra \(A\) possessing a bounded approximate identity and ordered by a closed multiplicative cone \(A_+\) is isomorphic to a sublattice and subalgebra of the set of all continuous real-valued functions on a compact Hausdorff space. For example, for the existence of such a representation it suffices that \(A\) be an almost \(f\)-algebra, or that every element can be written as a difference of two positive elements with product zero. We emphasize that for the second criterion no lattice property has to be assumed. In this context it seems to be interesting to consider the following analogous property: \vskip3mm (DDP) \quad For all \(z\in A\) there exists \(x,y\in A_+\) with \(z= x-y\) and \([0,x ]\cap [0,y ]= \{0\}\). \vskip3mm Positive elements \(x\), \(y\) with the property \([0,x ]\cap [0,y ]= \{0\}\) are called disjoint and a partially ordered vector space possesses the disjoint decomposition property iff condition DDP is fulfilled. We show in the first section that a partially ordered vector space is a vector lattice if and only if it possesses the DDP and the well known Riesz decomposition property, briefly RDP, which is defined by the validity of the equation \([0,x ]+ [0,y ]= [0,x+ y]\) for all positive \(x,y\in A\). The main results about the lattice properties of ordered Banach algebras are presented in Section 2. Our main tool is a representation theorem for Banach algebras possessing a bounded approximate identity which are ordered by a multiplicative cone containing all squares. The proof of the representation theorem is given in Section 3. As an interesting consequence we obtain that an almost \(f\)-Banach lattice algebra \(A\) is isometrically isomorphic to the set \(C_0 (X, \mathbb{R})\) of all continuous real-valued functions vanishing at infinity on a locally compact space if and only if \(A\) possesses a bounded approximate identity with norm bound 1. In the fourth section we show that the positive cones of certain almost \(f\)-Banach lattice algebras are uniquely determined by their Banach algebra multiplications.