Isotone functions on partially ordered linear algebras with a multiplicative diagonal map

Abstract

Abstract. The diagonal of the product of two triangular matrices is the product of the diagonals of each matrix. This idea is used to characterize Dedekind o-complete lattice ordered linear algebras which admit isotone functions with familiar functional and order properties as possessed by the real-valued logarithm or root function.1. Motivated by the work of Kadison and Singer [5] on triangular operator algebras, the authors [3] characterized abstract partially ordered linear algebras (polas) which have the order properties similar to an algebra of real upper triangular matrices. The main idea of the characterization we used in [3] is that in an algebra of upper triangular matrices the diagonal of the product of two matrices is equal to the product of the diagonals. We use the same idea in this paper to characterize the polas which admit an isotone function with familiar functional and order properties as possessed by the real-valued logarithm function or root function on the real line.A dsc-pola, denoted by A, is a real linear associative algebra which satisifes the following two conditions: (1) It is partially ordered so that it is a directed partially ordered linear space and 0 < xv whenever 0 < x, y E A.