Choda–Dykema conjecture on purely infinite reduced free products

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Let (A,φ)(A,\varphi) and (B,ψ)(B,\psi) be nontrivial C∗C^*-probability spaces inducing faithful GNS representations. Form their reduced free product (A,φ)∗(B,ψ)(A,\varphi)*(B,\psi). Choda–Dykema conjecture. If (A,φ)∗(B,ψ)(A,\varphi)*(B,\psi) is simple and at least one of φ\varphi or ψ\psi is nontracial, then (A,φ)∗(B,ψ)(A,\varphi)*(B,\psi) is purely infinite and simple. This conjecture predicts pure infiniteness for simple reduced free products whenever one of the input states is nontracial; the source presents it as a conjecture of Choda and Dykema, and no resolution is supplied here.

References

Primary source

Miles Gould, “The Selfless Dichotomy”, arXiv:2606.09654 (2026).

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