Completely bounded map

Let $A$ and $B$ be $C^*$ -algebras and let $\phi:A\to B$ be linear. For each $n\geq1$ , its matrix amplification is the entrywise map

\phi^{(n)}:M_n(A)\longrightarrow M_n(B),\qquad [a_{ij}]\longmapsto[\phi(a_{ij})],

where the domain and codomain carry their canonical matrix $C^*$ -algebra norms. The map $\phi$ is completely bounded if

\lVert\phi\rVert_{\mathrm{cb}} =\sup_{n\geq1}\lVert\phi^{(n)}\rVert<\infty.

The number $\lVert\phi\rVert_{\mathrm{cb}}$ is its completely bounded norm. It measures $\phi$ simultaneously at every matrix level, rather than only on $A$ . No positivity, multiplicativity, or unitality is part of the definition.

Fundamental examples

Every $*$ -homomorphism is completely contractive. Every bounded scalar-valued functional is completely bounded with the same norm. A completely positive map is completely bounded; when its domain is unital, $\lVert\phi\rVert_{\mathrm{cb}}=\lVert\phi(1)\rVert$ . These examples explain why the matrix norm, rather than only the ordinary operator norm, is natural for maps between operator algebras.

Structure and consequences

Compositions satisfy

\lVert\psi\mathbin{\circ}\phi\rVert_{\mathrm{cb}} \leq\lVert\psi\rVert_{\mathrm{cb}}\lVert\phi\rVert_{\mathrm{cb}},

so completely bounded maps form the morphisms of the operator-space category. For maps into $B(H)$ , Wittstock's decomposition theorem expresses every completely bounded map as a linear combination of completely positive maps; equivalently, it admits a Stinespring-type factorization with bounded coefficients.

Bounded versus completely bounded

Ordinary boundedness controls only the level $n=1$ . The transpose map on $M_k(\mathbb C)$ has operator norm $1$ but completely bounded norm $k$ , so amplification can reveal behavior invisible at the first level. On a fixed finite-dimensional operator space every bounded map is completely bounded, but there is no dimension-free comparison between the bounded and completely bounded norms.

References

Vern Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge University Press, 2002. DOI record. Relevant: Chapters 3 and 8 on completely positive maps, completely bounded maps, amplifications, and the cb norm.