In mathematics a positive map is a map between C*-algebras that sends positive elements to positive elements. A completely positive map is one which satisfies a stronger, more robust condition.
Let
A
and
B
be C*-algebras. A linear map
ϕ
:
A
→
B
is called positive map if
ϕ
maps positive elements to positive elements:
a
≥
0
⟹
ϕ
(
a
)
≥
0
.
Any linear map
ϕ
:
A
→
B
induces another map
id
⊗
ϕ
:
C
k
×
k
⊗
A
→
C
k
×
k
⊗
B
in a natural way. If
C
k
×
k
⊗
A
is identified with the C*-algebra
A
k
×
k
of
k
×
k
-matrices with entries in
A
, then
id
⊗
ϕ
acts as
(
a
11
⋯
a
1
k
⋮
⋱
⋮
a
k
1
⋯
a
k
k
)
↦
(
ϕ
(
a
11
)
⋯
ϕ
(
a
1
k
)
⋮
⋱
⋮
ϕ
(
a
k
1
)
⋯
ϕ
(
a
k
k
)
)
.
We say that
ϕ
is k-positive if
id
C
k
×
k
⊗
ϕ
is a positive map, and
ϕ
is called completely positive if
ϕ
is k-positive for all k.
Positive maps are monotone, i.e.
a
1
≤
a
2
⟹
ϕ
(
a
1
)
≤
ϕ
(
a
2
)
for all self-adjoint elements
a
1
,
a
2
∈
A
s
a
.
Since
−
∥
a
∥
A
1
A
≤
a
≤
∥
a
∥
A
1
A
every positive map is automatically continuous w.r.t. the C*-norms and its operator norm equals
∥
ϕ
(
1
A
)
∥
B
. A similary statement with approximate units holds for non-unital algebras.
The set of positive functionals
→
C
is the dual cone of the cone of positive elements of
A
.
Every *-homomorphism is completely positive.
For every operator
V
:
H
1
→
H
2
between Hilbert spaces, the map
L
(
H
1
)
→
L
(
H
2
)
,
A
↦
V
A
V
∗
is completely positive. Stinespring's theorem says that all completely positive maps are compositions of *-homomorphisms and these special maps.
Every positive functional
ϕ
:
A
→
C
(in particular every state) is automatically completely positive.
Every positive map
C
(
X
)
→
C
(
Y
)
is completely positive.
The transposition of matrices is a standard example of a positive map that fails to be 2-positive. Let T denote this map on
C
n
×
n
. The following is a positive matrix in
C
2
×
2
⊗
C
2
×
2
:
The image of this matrix under
I
2
⊗
T
is
[
(
1
0
0
0
)
T
(
0
1
0
0
)
T
(
0
0
1
0
)
T
(
0
0
0
1
)
T
]
=
[
1
0
0
0
0
0
1
0
0
1
0
0
0
0
0
1
]
,
which is clearly not positive, having determinant -1. Moreover, the eigenvalues of this matrix are 1,1,1 and -1.
Incidentally, a map Φ is said to be
co-positive if the composition Φ
∘
T is positive. The transposition map itself is a co-positive map.
