A decomposable linear map, in the operator-algebra sense, is a linear map between C*-algebras that is a linear combination of completely positive maps. Four such maps suffice: with all completely positive (Pisier 2020).

For , its decomposable norm is the infimum of over completely positive maps for which

is a completely positive map from to . Every decomposable linear map is a completely bounded map and satisfies (Pisier 2020, Okayasu 2026).

Equality for every linear map characterizes an injective von Neumann algebra (Okayasu 2026). This decomposition into completely positive maps is distinct from decomposition of a positive map as a sum of a completely positive map and a completely copositive map.

See also

Completely Bounded Map, Completely Positive Map, Injective von Neumann Algebra

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References

Okayasu, R. "A Solution to Haagerup's Problem on Decomposable Maps." 30 Oct 2026. https://arxiv.org/abs/2610.41746v1.Pisier, G. Tensor Products of C*-Algebras and Operator Spaces: The Connes-Kirchberg Problem. Cambridge, England: Cambridge University Press, 2020. https://doi.org/10.1017/9781108782081.

Cite this as:

Weisstein, Eric W. "Decomposable Linear Map." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DecomposableLinearMap.html

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