n-positive map - Imported Knowlpedia Content

Definition

Let $A$ and $B$ be $C^*$ -algebras and let $n\geq1$ be an integer. A bounded linear map $\Phi:A\to B$ is $n$ -positive if its amplification

\Phi^{(n)}=\operatorname{id}_{M_n}\otimes\Phi: M_n(A)\longrightarrow M_n(B),\qquad [a_{ij}]\longmapsto[\Phi(a_{ij})],

is a positive linear map between the corresponding matrix $C^*$ -algebras. No unitality is assumed. The case $n=1$ is ordinary positivity; $2$ -positivity is the first condition that tests interactions between two matrix entries. Complete positivity means $n$ -positivity for every positive integer $n$ , not merely for one fixed matrix size.

The positivity hierarchy

If $\Phi$ is $n$ -positive, then it is $m$ -positive for every $1\leq m\leq n$ , by embedding $M_m(A)$ as a corner of $M_n(A)$ . The converses fail in general. For maps out of $M_k(\mathbb C)$ , $k$ -positivity already implies complete positivity, but this finite-dimensional cutoff depends on the size of the domain Paulsen, Chapter 2.

Schwarz inequality

If $\Phi$ is unital and $2$ -positive, positivity of a suitable $2\times2$ operator matrix gives the Kadison–Schwarz inequality

\Phi(a)^*\Phi(a)\leq\Phi(a^*a)\qquad(a\in A).

Unitality is essential to this normalization; for nonunital or merely contractive maps the statement requires a modified hypothesis. Ordinary positivity alone yields Kadison's inequality for self-adjoint inputs under unitality, but not the full displayed inequality for arbitrary $a$ .

Examples and non-examples

Every $*$ -homomorphism is completely positive because each amplification is again a $*$ -homomorphism. Positive linear functionals are completely positive when viewed as maps into $\mathbb C$ . Matrix transposition on $M_k(\mathbb C)$ is positive but, for $k\geq2$ , is not $2$ -positive; its failure on an entangled rank-one projection shows why entrywise positivity is stronger than positivity of the original map.

References

Vern Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge University Press, 2002. DOI record. Relevant: Chapter 2 on $n$ -positive maps, complete positivity, and matrix-order tests.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter IV on positive and completely positive maps.