On the index and dilations of completely positive semigroups

TL;DR

This paper calculates the index of unital completely positive semigroups using generator structures and shows their minimal dilations are conjugate to CAR/CCR flows, advancing understanding of their structure and classification.

Contribution

It provides explicit index calculations for unital CP semigroups and links their minimal dilations to well-known flow models, extending previous theoretical results.

Findings

Index of unital CP semigroups expressed via generator structures

Minimal dilations are cocycle conjugate to CAR/CCR flows

Applicable to all unital CP semigroups on matrix algebras

Abstract

It is known that every semigroup of normal completely positive maps $P = {P_t: t\geq 0}$ of $B(H)$, satisfying $P_t(1) = 1$ for every $t\geq 0$, has a minimal dilation to an E_0-semigroup acting on $B(K)$ for some Hilbert space K containing H. The minimal dilation of P is unique up to conjugacy. In a previous paper a numerical index was introduced for semigroups of completely positive maps and it was shown that the index of P agrees with the index of its minimal dilation to an E_0-semigroup. However, no examples were discussed, and no computations were made. In this paper we calculate the index of a unital completely positive semigroup whose generator is a bounded operator $ L: B(H)\to B(H) $ in terms of natrual structures associated with the generator. This includes all unital CP semigroups acting on matrix algebras. We also show that the minimal dilation of the semigroup $P={\exp{tL}: t\geq 0}$ to an \esg\ is is cocycle conjugate to a CAR/CCR flow.