On Banach lattice algebras of type 1 (Q1426890)

It is known that a finite-dimensional real Banach algebra lattice with unit, such that every positive element is regular, is norm and lattice isomorphic to \(\mathbb R\), and it is unknown whether the result holds in the infinite-dimensional case. The aim of this paper is to prove its validity for Banach algebra lattices of type 1. A Banach algebra lattice \(A\) with unit \(e\) is called of type 1 if \(a(e+a)^{-1} \geq 0\) for every \(a\geq 0\).