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Cocycle Conjugacy

Cocycle conjugacy of actions alpha:G->Aut(A) and beta:G->Aut(B) on unital C*-algebras means that there are a unital star-isomorphism Phi:A->B and unitaries u_g in B such that u_e=1, u_(gh)=u_gbeta_g(u_h), and

Phi(alpha_g(a))=u_gbeta_g(Phi(a))u_g^*.

The identity for u_(gh) is the cocycle condition relative to beta. If every u_g=1, the actions are conjugate without a cocycle perturbation. Equivariant Jiang-Su stability uses cocycle conjugacy with the action that is trivial on the new Jiang-Su tensor factor.

See also

C*-Algebra, Jiang-Su Stability, Unitary Element

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References

Blackadar, B. Operator Algebras: Theory of C*-Algebras and von Neumann Algebras. Berlin, Germany: Springer, 2006. https://doi.org/10.1007/3-540-28517-2.Schafhauser, C.; Tikuisis, A.; and White, S. "Nuclear C^*-Algebras: 99 Problems." 12 Jun 2025. https://arxiv.org/abs/2506.10902.

Cite this as:

Weisstein, Eric W. "Cocycle Conjugacy." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CocycleConjugacy.html

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