Pre-Markov Operators

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 $$ 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.