Pre-Markov Operators

TL;DR

This paper introduces the concept of Pre-Markov operators between unital $f$-algebras, characterizes when they are algebra homomorphisms, and explores their extreme points and conditions for lattice homomorphisms.

Contribution

It defines Pre-Markov operators, characterizes algebra homomorphisms among them, and analyzes their extreme points and lattice homomorphism conditions.

Findings

Pre-Markov operator is an algebra homomorphism iff its second adjoint is an extreme point.

Characterization of extreme points of contractive mappings.

Conditions under which order bounded algebra homomorphisms are lattice homomorphisms.

Abstract

A positive linear operator $T$ between two unital $f$-algebras, with point separating order duals, $A$ and $B$ is called a Markov operator for which $% T\left( e_{1}\right) =e_{2}$ where $e_{1},e_{2}$ are the identities of $A$ and $B$ respectively. Let $A$ and $B$ be semiprime $f$-algebras with point separating order duals such that their second order duals $A^{\sim \sim }$ and $B^{\sim \sim }$ are unital $f$-algebras. In this case, we will call a positive linear operator $T:A\rightarrow B$ \ to be a Pre-Markov operator, if the second adjoint operator of $T$ is a Markov operator. A positive linear operator $T$ between two semiprime $f$-algebras, with point separating order duals, $A$ and $B$ is said to be contractive if $Ta\in B\cap \left[ 0,I_{B}\right] $ whenever $a\in A\cap \left[ 0,I_{A}\right] $, where $I_{A}$ and $I_{B}$ are the identity operators on $A$ and $B$ respectively. In this paper we characterize pre-Markov algebra homomorphisms. In this regard, we show that a pre-Markov operator is an algebra homomorphism if and only if its second adjoint operator is an extreme point in the collection of all Markov operators from $A^{\sim \sim }$ to $B^{\sim \sim }$. Moreover we characterize extreme points of contractive mappings from $A$ to $B$. In addition, we give a condition that makes an order bounded algebra homomorphism is a lattice homomorphism.