Completely positive maps of order zero

Winter, W. and Zacharias, J. (2009) Completely positive maps of order zero. Munster Journal of Mathematics, 2, pp. 311-324.



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We say a completely positive contractive map between two C<sup>*</sup>-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.

Item Type:Articles
Glasgow Author(s) Enlighten ID:Zacharias, Professor Joachim
Authors: Winter, W., and Zacharias, J.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Munster Journal of Mathematics
ISSN (Online):1867-5786
Copyright Holders:Copyright © 2009 Munster J. of Math.
First Published:First published in Munster Journal of Mathematics 2:311-324
Publisher Policy:Reproduced in accordance with the copyright policy of the publisher

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