Kadison similarity problem

Kadison's similarity problem, posed in 1955, asks whether every bounded unital algebra homomorphism $\pi$ from a unital $C^*$-algebra $A$ into the bounded operators $B(H)$ on a complex Hilbert space is similar to a $*$-homomorphism: is there an invertible $S\in B(H)$ such that $a\mapsto S\pi(a)S^{-1}$ is a $*$-homomorphism? The question originates in Kadison's paper on the orthogonalization of operator representations [Kadison1955OperatorAlgebras], and can be viewed as an operator-algebraic analogue of unitarizability questions for bounded group representations.

Many important cases are settled affirmatively. Haagerup solved the problem for cyclic representations [Haagerup1983SimilarityCyclic], and the answer is positive for nuclear $C^*$-algebras and for the many classes of algebras known to have finite similarity degree, a framework treated systematically in [Pisier2001SimilarityProblems]; a bounded homomorphism is similar to a $*$-homomorphism precisely when it is completely bounded.

For an arbitrary $C^*$-algebra the problem remains open; a positive solution amounts to showing that every bounded unital homomorphism of a $C^*$-algebra is automatically completely bounded.