E-dilation of strongly commuting CP-semigroups (the nonunital case)

TL;DR

This paper extends the dilation theory of strongly commuting CP-semigroups on B(H) by removing the unitality assumption, using a novel construction inspired by Ptak's work from the 1980s.

Contribution

It proves that every pair of strongly commuting CP-semigroups on B(H) admits an E-dilation without requiring unitality, broadening previous results.

Findings

Every strongly commuting CP-semigroup pair on B(H) has an E-dilation.

The proof employs a construction originally designed for unitary dilation of two-parameter contraction semigroups.

The approach differs significantly from the unital case, leveraging Ptak's 1980s construction.

Abstract

In a previous paper, we showed that every strongly commuting pair of CP_0-semigroups on a von Neumann algebra (acting on a separable Hilbert space) has an E_0-dilation. In this paper we show that if one restricts attention to the von Neumann algebra B(H) then the unitality assumption can be dropped, that is, we prove that every pair of strongly commuting CP-semigroups on B(H) has an E-dilation. The proof is significantly different from the proof for the unital case, and is based on a construction of Ptak from the 1980's designed originally for constructing a unitary dilation to a two-parameter contraction semigroup.