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Completely Bounded Map

A completely bounded map T:A->B between C*-algebras is a linear map whose entrywise matrix amplifications T_n:M_n(A)->M_n(B) have uniformly bounded operator norm. Its completely bounded norm is

||T||_(cb)=sup_(n>=1)||T_n||.

For C*-algebras, the norm on M_n(A) is its C*-algebra norm (Pisier 2020, pp. 3-4).

A completely positive map is completely bounded, and its completely bounded norm equals the norm of its value at the identity when the domain is unital (Pisier 2020).

See also

Completely Positive Map, Decomposable Linear Map, Operator Algebra

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References

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. "Completely Bounded Map." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CompletelyBoundedMap.html

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