A similarity degree characterization of nuclear \(C^*\)-algebras (Q863787)

Summary: We show that a \(C^*\)-algebra \(A\) is nuclear iff there is a number \(\alpha<3\) and a constant \(K\) such that, for any bounded homomorphism \(u:A\to B(H)\), there is an isomorphism \(\xi:H\to H\) satisfying \(\|\xi^{-1}\|\,\|\xi\|\leq K\|u\|^\alpha\) and such that \(\xi^{-1}u(\cdot)\xi\) is a *-homomorphism. In other words, an infinite dimensional \(A\) is nuclear iff its length (in the sense of our previous work on the Kadison similarity problem) is equal to 2.