A remark on the minimal dilation of the semigroup generated by a normal UCP-map

TL;DR

This paper compares different methods for constructing the minimal dilation of semigroups generated by normal UCP-maps on von Neumann algebras, clarifying the relations among these approaches.

Contribution

It clarifies the relationship between the Bhat-Skeide and Muhly-Solel constructions for minimal dilations of such semigroups.

Findings

Established the relation between Bhat-Skeide and Muhly-Solel constructions.

Compared three known methods for minimal dilation construction.

Provided insights into the structure of dilations for normal UCP-maps.

Abstract

There are known three ways to construct the minimal dilation of the discrete semigroup generated by a normal unital completely positive map on a von Neumann algebra, which are given by Arveson, Bhat-Skeide and Muhly-Solel. In this paper, we clarify the relation of the constructions by Bhat-Skeide and Muhly-Solel.