Positive linear map - Imported Knowlpedia Content

Definition

Let $A$ and $B$ be $C^*$ -algebras. A complex-linear map $\Phi:A\to B$ is positive if

\Phi(A_+)\subseteq B_+,

where $A_+$ and $B_+$ are their positive cones. Equivalently, $a\geq0$ implies $\Phi(a)\geq0$ . Positivity forces $\Phi(a^*)=\Phi(a)^*$ . When $A$ is unital, every positive map is bounded and

\lVert\Phi\rVert=\lVert\Phi(1_A)\rVert.

If also $\Phi(1_A)=1_B$ , the map is unital positive. Positivity does not require multiplicativity. The definition compares the canonical orders on the self-adjoint parts of the two algebras and does not impose any condition at matrix levels larger than one.

Matrix levels

For each $n$ , entrywise application gives $\Phi^{(n)}:M_n(A)\to M_n(B)$ . The map is $n$ -positive when $\Phi^{(n)}$ is positive, and completely positive when this holds for every $n$ . Thus ordinary positivity is only the first matrix level. Every $*$ -homomorphism is completely positive, whereas a merely positive map need not be $2$ -positive.

Standard example and counterexample

The transpose map $x\mapsto x^{\mathsf T}$ on $M_n(\mathbb C)$ is unital and positive: it preserves the eigenvalues of positive matrices. For $n\geq2$ , however, it is not $2$ -positive and hence is not completely positive. In contrast, maps of the form

\Phi(a)=V^*\pi(a)V

for a $*$ -representation $\pi$ are completely positive. This distinction is essential in operator-algebraic quantum theory.

Order and norm behavior

A positive map is order preserving on self-adjoint elements: $a\leq b$ implies $\Phi(a)\leq\Phi(b)$ . A unital positive map sends every self-adjoint contraction to a self-adjoint contraction. Stronger Schwarz inequalities require stronger hypotheses: the Kadison inequality for a unital positive map applies to self-adjoint elements, while the full inequality $\Phi(a)^*\Phi(a)\leq\Phi(a^*a)$ follows from unital $2$ -positivity.

References

Vern Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge University Press, 2002. Cambridge DOI record. Relevant: Chapters 2 and 3 on positive, $n$ -positive, and completely positive maps.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter IV on positive maps and operator inequalities.