An operator system is a complex linear subspace 
of 
, the bounded operators
on a complex Hilbert
space 
, that contains the identity
operator 
and is closed under adjoints (Paulsen et al.
2011, preprint p. 5). The vector space 
of 
matrices with entries in
inherits its positive cone
from 
, where 
is a positive integer.
Closure under multiplication is not required.
For example, 
is an operator system for any square complex matrix 
(Mohari 2023). A completely
positive map preserves these positive cones at
every matrix size.
Operator System
See also
C*-Algebra, Completely Positive Map, Operator Algebra, Self-Adjoint ElementExplore with Wolfram|Alpha
References
Mohari, A. "Unital Completely Positive Maps and Their Operator Systems." 2 Aug 2023. https://arxiv.org/abs/1006.5198v8.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. Authors' preprint, p. 5. https://arxiv.org/abs/0904.3783v2.Cite this as:
Weisstein, Eric W. "Operator System." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/OperatorSystem.html