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

Okayasu (2026) characterizes injectivity using decomposable linear maps from with coordinatewise operations and the maximum norm. The von Neumann algebra is injective iff every such linear map satisfies

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

Explore with Wolfram|Alpha

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

Subject classifications