A similarity degree characterization of nuclear C*-algebras

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We show that a C*-algebra A is nuclear iff there is a constant K and α<3 such that, for any bounded homomorphism u A → B(H), there is an isomorphism ξ H→ H satisfying ξ⁻¹ξ ≤ Kuα and such that ξ⁻¹ u(.) ξ is a *-homomorphism. In other words, an infinite dimensional A is nuclear iff its length (in ths sense of our previous work on the Kadison similarity problem) is equal to 2.