Partially Ordered Linear Algebras With 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 partially ordered linear algebras which have order properties similar to an algebra of real triangular matrices.

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

“…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.…”

“…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.…”

“…Since A is linear, it follows trivially that A is isotone, i.e., A(_y) > A(z) whenever y > z, y, z E A. This map was studied in [3]. Example 2.4 in [3] shows that in some dsc-pola there may not exist a DP map; however, if it exists, it is unique [3, , rather than A(-), was used.)…”

“…This map was studied in [3]. Example 2.4 in [3] shows that in some dsc-pola there may not exist a DP map; however, if it exists, it is unique [3, , rather than A(-), was used.) Also, the reader should note that the concept of a DP map is related to the so-called "diagonal process" defined by Kadison and Singer; see [4, p. 387].…”

“…In [3], various algebraic (order) conditions on A were given which imply that a diagonal projection map is multiplicative. In this paper, we first prove the following theorem.…”