Completely positive maps of order zero

By W. Winter, J. Zacharias. 2009

187 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/74fecf2038fd6f118f6f1aa35a5973e0e7ad968f

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Citations in the corpus

Tracially Z-absorbing C*-algebras (2012, background): “2 of [27] along with some additional notation:”
Nuclear dimension of simple stably projectionless C∗-algebras (2019, methodology): “These results were inspired by the structure theory for order zero maps developed in [68] and the existence of supporting order zero maps proved in [8, Lemma 1.”
Nuclear dimension and $$\mathcal Z$$Z-stability (2014, unknown): “The structure theorem from [46] for order zero maps ψ : A → B provides a supporting -homomorphism π from A into the multiplier algebra M(C(ψ(A))) ⊆ B and a positive element h ∈ M(C(ψ(A))) ∩ ψ(A) such that ψ(a) = π(a)h = hπ(a) for all a ∈ A.”
Perturbations of nuclear C*-algebras (2009, methodology): “In particular, we will use the following two facts from [53] and [54] (see also [55], where these assertions are established when F is not finite dimensional).”
Nuclear dimension and -stability of pure C ∗ -algebras (2010, methodology): “We collect below some facts about order zero maps from [42] (see [36, Proposition 3.”
Rokhlin dimension for compact group actions (2014, background): “See [30] for more details and further results.”
Covering Dimension of C*-Algebras and 2-Coloured Classification (2015, background): “Building on the work of Wolff ([101]), the structure theory for these maps was developed in [99].”