[Paper Review] Completely positive maps of order zero

Wilhelm Winter, Joachim Zacharias|ArXiv.org|Mar 19, 2009

Advanced Operator Algebra Research16 references131 citations

TL;DR

This paper introduces and characterizes completely positive maps of order zero between C*-algebras—maps that preserve orthogonality of positive elements. It establishes a structure theorem showing such maps are in one-to-one correspondence with *-homomorphisms from the cone over the domain algebra to the target, enabling a functional calculus and proving that tensor products and compositions with tracial functionals preserve order zero. The key contribution is that order zero maps induce ordered semigroup morphisms between Cuntz semigroups, providing a bridge for lifting semigroup-level maps to C*-algebra maps.

ABSTRACT

We say a completely positive contractive map between two C*-algebras has order zero, if it sends orthogonal elements to orthogonal elements. We prove a structure theorem for such maps. As a consequence, order zero maps are in one-to-one correspondence with *-homomorphisms from the cone over the domain into the target algebra. Moreover, we conclude that tensor products of order zero maps are again order zero, that the composition of an order zero map with a tracial functional is again a tracial functional, and that order zero maps respect the Cuntz relation, hence induce ordered semigroup morphisms between Cuntz semigroups.

Motivation & Objective

• To develop a general structure theory for completely positive contractive (c.p.c.) maps of order zero between C*-algebras, extending prior results limited to finite-dimensional domains.
• To establish a one-to-one correspondence between c.p.c. order zero maps and *-homomorphisms from the cone over the domain algebra to the target algebra.
• To show that order zero maps preserve key structures such as tensor products, tracial functionals, and the Cuntz subequivalence relation.
• To demonstrate that order zero maps naturally induce morphisms between Cuntz semigroups, enabling a new framework for lifting semigroup-level maps to C*-algebra maps.

Proposed method

• Define order zero maps as c.p.c. maps that preserve orthogonality of positive elements: if a ⊥ b in A+, then φ(a) ⊥ φ(b) in B+.
• Prove a structure theorem showing every c.p.c. order zero map φ: A → B admits a factorization φ = h^{1/2} π_φ h^{1/2}, where h ∈ B+ commutes with the image of φ and π_φ is a *-homomorphism from A to the multiplier algebra of the C*-algebra generated by φ(A).
• Establish the correspondence between c.p.c. order zero maps and *-homomorphisms from the cone algebra C₀(A) to B, using the universal property of the cone.
• Use the factorization to show that tensor products of order zero maps are again order zero, and that compositions with tracial functionals remain tracial.
• Prove that order zero maps preserve the Cuntz subequivalence relation, hence induce well-defined ordered semigroup morphisms between Cuntz semigroups W(A) and W(B).
• Extend results to the biduals, showing that the bitransposed map φ** is also order zero, and that the structure theorem holds in the von Neumann algebra setting with normality preserved.

Experimental results

Research questions

• RQ1 How can one characterize completely positive contractive maps of order zero between C*-algebras in a general, non-finite-dimensional setting?
• RQ2 What is the precise structural form of a c.p.c. order zero map, and how does it relate to *-homomorphisms and the cone construction?
• RQ3 Do order zero maps preserve tensor products and tracial functionals under composition?
• RQ4 Can order zero maps induce well-defined morphisms between Cuntz semigroups, and if so, how does this relate to lifting maps from the semigroup level to the C*-algebra level?
• RQ5 Is there a functional calculus for order zero maps analogous to the continuous functional calculus for normal elements?

Key findings

• Every c.p.c. order zero map φ: A → B is uniquely factorizable as φ = h^{1/2} π_φ h^{1/2}, where h ∈ B+ commutes with the image of φ and π_φ is a *-homomorphism from A to M(C), with C = C*(φ(A)).
• There is a canonical one-to-one correspondence between c.p.c. order zero maps from A to B and *-homomorphisms from the cone algebra C₀(A) to B.
• Tensor products of c.p.c. order zero maps are again order zero, including amplifications to matrix algebras.
• Compositions of c.p.c. order zero maps with positive tracial functionals are again tracial functionals, and the same holds for 2-quasitraces.
• Order zero maps induce well-defined, order-preserving, semigroup morphisms W(φ): W(A) → W(B) between Cuntz semigroups.
• The bitransposed map φ** of any c.p.c. order zero map φ is also order zero, and the structure theorem extends to the biduals.