A completely positive map 
between C*-algebras
is a linear transformation whose matrix
amplifications 
are positive for every integer 
. The amplification applies 
to every matrix entry (Anantharaman
and Popa, p. 313). The same definition applies between operator
systems, using the positive cones inherited at
every matrix size (Paulsen et al. 2011, preprint
p. 5). A star-homomorphism is completely
positive, while positivity at the scalar level alone does not suffice. For example,
matrix transposition on 
is positive but not completely
positive. Its amplification sends the positive
semidefinite matrix of matrix rank one associated
to the vector 
to a linear
operator with a negative eigenvalue.
Here, 
and 
are the standard basis vectors of 
. Completely positive contractive approximations through
finite-dimensional C*-algebras
characterize nuclearity. The definition
of nuclear dimension also imposes order-zero
conditions on the restrictions of the return maps to specified summands (Winter and
Zacharias 2010).
Completely Positive Map
See also
C*-Algebra, Nuclear C-Star-Algebra, Nuclear Dimension, Operator System, Order-Zero Map, Positive MapExplore with Wolfram|Alpha
References
Anantharaman, C. and Popa, S. An Introduction to
Factors. Undated draft, p. 313. https://www.math.ucla.edu/~popa/Books/IIun.pdf.Paulsen,
V. I.; Todorov, I. G.; and Tomforde, M. "Operator System Structures
on Ordered Spaces." Proc. London Math. Soc. 102, 25-49, 2011.
https://doi.org/10.1112/plms/pdq011.
Author's preprint, p. 5. https://arxiv.org/abs/0904.3783v2.Winter,
W. and Zacharias, J. "The Nuclear Dimension of 
-Algebras." Adv. Math. 224, 461-498, 2010.
https://doi.org/10.1016/j.aim.2009.12.005.
Cite this as:
Weisstein, Eric W. "Completely Positive Map." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CompletelyPositiveMap.html