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Injective von Neumann Algebra

An injective von Neumann algebra is a von Neumann algebra N subset= B(H) that is the range of a norm-one bounded projection from the algebra B(H) of bounded linear operators on a Hilbert space H. This property is independent of the faithful representation (Pisier 2020, Okayasu 2026).

Okayasu (2026) characterizes injectivity using decomposable linear maps from l_3^infty=C^3 with coordinatewise operations and the maximum norm. The von Neumann algebra N is injective iff every such linear map T:l_3^infty->N satisfies

||T||_(dec)=||T||_(cb).

The norm on the right is the completely bounded map norm. Okayasu (2026) developed the argument with Claude assistance and reports checking all proofs.

See also

Completely Bounded Map, Decomposable Linear Map, 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. "Injective von Neumann Algebra." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/InjectivevonNeumannAlgebra.html

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