A decomposable linear map, in the operator-algebra sense, is a linear map between C*-algebras that is a linear
combination of completely positive maps.
Four such maps suffice: 
with all 
completely positive
(Pisier 2020).
For 
,
its decomposable norm is the infimum of 
over completely
positive maps 
for which
 |
is a completely positive map from 
to 
. Every decomposable linear map is a completely
bounded map and satisfies 
(Pisier 2020, Okayasu 2026).
Equality for every linear map 
characterizes an injective
von Neumann algebra 
(Okayasu 2026). This decomposition into completely
positive maps is distinct from decomposition of a positive
map as a sum of a completely positive map
and a completely copositive map.
