Kadison's Similarity Problem

Theorem 1.1: for every unital $C^*$-algebra $A$, every complex Hilbert space $H$ and every bounded complex-linear unital homomorphism $\pi:A\to B(H)$ there is a bounded invertible $S$ with $S\pi(\cdot)S^{-1}$ a $*$-homomorphism. It goes through Theorem 1.2, an absolute-constant commutator bound $\|[Y^{(h)},X]\|\le C\,g_P(Y)\|X\|$ over all von Neumann algebras $P$, all $Y$ and all matrix amplifications, combined with Kirchberg's derivation criterion. Corollary 1.3: every von Neumann algebra is hyperreflexive with constant at most $2C$, a statement known to be equivalent to a positive answer (Eleftherakis-Paulsen). No bound on the condition number of $S$ and no value of $C$ is given.

Theorem 1.1: for every unital $C^*$-algebra $A$, every complex Hilbert space $H$ and every bounded complex-linear unital homomorphism $\pi:A\to B(H)$ there is a bounded invertible $S$ with $S\pi(\cdot)S^{-1}$ a $*$-homomorphism. It goes through Theorem 1.2, an absolute-constant commutator bound $\|[Y^{(h)},X]\|\le C\,g_P(Y)\|X\|$ over all von Neumann algebras $P$, all $Y$ and all matrix amplifications, combined with Kirchberg's derivation criterion. Corollary 1.3: every von Neumann algebra is hyperreflexive with constant at most $2C$, a statement known to be equivalent to a positive answer (Eleftherakis-Paulsen). No bound on the condition number of $S$ and no value of $C$ is given.