[Paper Review] Normal CP-Maps Admit Weak Tensor Dilations

Rolf Gohm, Michael Skeide|arXiv (Cornell University)|Nov 7, 2003

Algebraic structures and combinatorial models9 references4 citations

TL;DR

This paper proves that every normal completely positive (CP) map between C*-algebras admits a weak tensor dilation, meaning it can be represented as a composition of a *-homomorphism and a conditional expectation on a larger C*-algebra. The key contribution is establishing the existence of such dilations under normality conditions, extending dilation theory to a broader class of CP-maps via tensor product constructions.

ABSTRACT

1 Introduction and statement of the main result A dilation of a CP-map (a completely positive mapping) S between C ∗ – algebras A and B is a homomorphism j from A to another C ∗ –algebra D containing B and a conditional expectation P: D → B such that S is recovered as S = P ◦ j.

Motivation & Objective

• To extend dilation theory to normal CP-maps between C*-algebras.
• To address the lack of general dilation results for normal CP-maps in non-unital or infinite-dimensional settings.
• To establish the existence of weak tensor dilations as a structural characterization of normal CP-maps.
• To provide a framework for representing normal CP-maps through homomorphisms and conditional expectations in larger C*-algebras.

Proposed method

• Construct a *-homomorphism j from the domain C*-algebra A into a larger C*-algebra D containing the codomain B.
• Define a conditional expectation P: D → B such that the composition P ◦ j recovers the original CP-map S.
• Utilize the normality of S to ensure compatibility with inductive limits and weak operator topologies.
• Employ tensor product structures to build the ambient C*-algebra D as a weak tensor dilation space.
• Ensure the dilation respects the order and positivity structure inherent in CP-maps.
• Verify that the resulting construction satisfies the axioms of a weak tensor dilation in the C*-algebraic setting.

Experimental results

Research questions

• RQ1 Can every normal CP-map between C*-algebras be dilated via a *-homomorphism and a conditional expectation in a larger C*-algebra?
• RQ2 What structural conditions on CP-maps allow for weak tensor dilations in non-unital or infinite-dimensional C*-algebras?
• RQ3 How does normality of the CP-map influence the existence and construction of such dilations?
• RQ4 Is there a canonical or universal way to construct weak tensor dilations for normal CP-maps?

Key findings

• Every normal CP-map between C*-algebras admits a weak tensor dilation, generalizing classical dilation theorems.
• The dilation is constructed via a *-homomorphism into a larger C*-algebra D and a conditional expectation onto the original codomain B.
• The construction relies on the normality of the CP-map to ensure compatibility with the weak topology and inductive limits.
• The resulting dilation preserves the complete positivity and linearity of the original map.
• The ambient C*-algebra D is built using tensor product techniques to support the dilation structure.
• The result establishes a structural characterization of normal CP-maps through their dilations in the C*-algebra framework.