Realization of conditionally monotone independence and monotone products of completely positive maps

TL;DR

This paper develops an operator algebra framework for conditional monotone independence, demonstrating how to construct monotone products of completely positive maps that preserve positivity, and proving an embedding result for monotone products of C*-algebras.

Contribution

It introduces an operator algebra model for conditional monotone independence and defines monotone products of maps that maintain complete positivity.

Findings

Established an operator algebra model for conditional monotone independence.

Proved an embedding theorem for monotone products of C*-algebras.

Defined monotone products of maps that preserve complete positivity.

Abstract

The paper gives an operator algebras model for the conditional monotone independence, introduced by T. Hasebe. The construction is used to prove an embedding result for the N. Muraki's monotone product of C*-algebras. Also, the formulas from the definition of conditional monotone independence are used to define the monotone product of maps which is shown to preserve complete positivity, a similar to the results from the case of free products.