Completely positive contraction - Imported Knowlpedia Content

Definition

Let $A$ and $B$ be $C^*$ -algebras. A completely positive contraction, often abbreviated CPC map, is a completely positive map $\phi:A\to B$ satisfying

\|\phi\|\leq 1.

Thus every matrix amplification $\phi^{(n)}:M_n(A)\to M_n(B)$ is positive, while the contraction condition is imposed on the ordinary operator norm. Complete positivity then implies $\|\phi\|_{\mathrm{cb}}=\|\phi\|$ , in the sense of the completely bounded norm, so every CPC map is completely contractive. Neither multiplicativity nor unitality is required. In particular, a unital CPC map need not preserve products.

Norm tests

If $A$ is unital and $\phi$ is completely positive, then

\|\phi\|=\|\phi\|_{\mathrm{cb}}=\|\phi(1_A)\|.

Consequently such a map is CPC exactly when $\phi(1_A)\leq 1_B$ , provided $B$ is unital. A unital completely positive map is automatically contractive. For a nonunital domain, the definition still uses $\|\phi\|\leq1$ ; one may test it after passing to an appropriate unitization Brown–Ozawa, §1.5.

Examples and boundary cases

Every contractive $*$ -homomorphism is CPC. If $V:K\to H$ is a contraction, the compression $\phi(T)=V^*TV$ from $B(H)$ to $B(K)$ is CPC; it is unital exactly when $V$ is an isometry. Multiplying the identity map by a scalar $\lambda>1$ preserves complete positivity but fails the contraction condition, so it is not CPC.

Role in approximation

Finite-dimensional approximation properties are commonly formulated using CPC maps because positivity controls order at every matrix level while contractivity prevents norms from growing during composition. Nuclearity, for example, can be expressed through point-norm approximations of the identity by CPC maps factoring through matrix algebras Brown–Ozawa, §2.3.

References

Nathanial P. Brown and Narutaka Ozawa, C-Algebras and Finite-Dimensional Approximations*, Graduate Studies in Mathematics 88, American Mathematical Society, 2008. AMS DOI record. Relevant: §1.5 on completely positive maps and §2.3 on nuclear approximation.