Completely positive maps of order zero

2009/03/19 by Wilhelm Winter, Winter, Wilhelm, Joachim Zacharias +1 · 6 citations
Mathematics · Physics and Astronomy · #46L05 #46L85 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Noncommutative and Quantum Gravity Theories #Operator Algebras (math.OA) #math.FA #math.OA #msc:46L05 #msc:46L85

paper · pdf · doi:10.48550/arxiv.0903.3290

13 pages

arxiv created 2009/03/19 · openalex publication_date 2009/03/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Localizing the Elliott Conjecture at Strongly Self-absorbing C*-algebras --An Appendix
• The nuclear dimension of C*-algebras

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