Unital completely positive map - Imported Knowlpedia Content

Definition

Let $A$ and $B$ be unital $C^*$ -algebras. A unital completely positive map, or UCP map, is a completely positive linear map $\Phi:A\to B$ satisfying

\Phi(1_A)=1_B.

Thus every matrix amplification $\Phi^{(n)}:M_n(A)\to M_n(B)$ is positive, and $\Phi$ preserves the distinguished unit. Multiplicativity, injectivity, surjectivity, and normality are not part of the definition. UCP maps are the morphisms commonly used for operator systems and for Heisenberg-picture quantum operations.

Contractivity and the Schwarz inequality

Every UCP map has norm one and is contractive. It also satisfies the Kadison–Schwarz inequality

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

Complete positivity is stronger than needed for this inequality—unital $2$ -positivity suffices—but it supplies stable matrix-level control. Compositions and convex combinations of UCP maps are again UCP Paulsen, Chapters 2–3.

The elements $a$ for which equality holds both for $a^*a$ and $aa^*$ form the multiplicative domain of $\Phi$ . On that $C^*$ -subalgebra, $\Phi$ behaves multiplicatively on both sides.

Stinespring form

When the codomain is $\mathcal B(H)$ , Stinespring's theorem writes a UCP map as

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

where $\pi:A\to\mathcal B(K)$ is a unital representation and $V:H\to K$ is an isometry. Conversely, every such compression is UCP. Unitality is exactly what changes the general Stinespring operator $V$ into an isometry, since $\Phi(1_A)=V^*V$ Paulsen, Chapter 4.

Examples and distinctions

Every unital $*$ -homomorphism is UCP. A state on a unital $C^*$ -algebra is precisely a UCP map $A\to\mathbb C$ . More generally, a conditional expectation between unital $C^*$ -algebras with a common unit is UCP.

The normalized trace map

M_n(\mathbb C)\longrightarrow\mathbb C,\qquad a\longmapsto \frac{1}{n}\operatorname{Tr}(a),

is UCP but is not multiplicative for $n>1$ . This shows why UCP maps should not be confused with $*$ -homomorphisms.

Conventions and scope

Unitality presupposes specified units in both domain and codomain. A completely positive map between nonunital $C^*$ -algebras may instead be contractive, nondegenerate, or extend unitally to suitable unitizations, but it is not literally UCP under the core definition. For von Neumann algebras, the additional adjective normal requires ultraweak continuity and gives the separate notion of a normal UCP map.

References

Vern Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge University Press, 2002. DOI record. Relevant: Chapters 2–4 on UCP maps, matrix positivity, the Schwarz inequality, and Stinespring dilation.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter IV on completely positive maps and operator-algebraic expectations.