Similarity Problems and Length

This is a survey of the author's recent results on the Kadison and Halmos similarity problems and the closely connected notion of " length " of an operator algebra. We start by recalling a well known conjecture formulated by Kadison [Ka] in 1955. Kadison's similarity problem. Let A be a unital C *-algebra and let u: A → B(H) (H Hilbert) be a unital homomorphism (i.e. we have u(1) = 1 and u(ab) = u(a)u(b) ∀ a, b ∈ A). Show that if u is bounded, then u is similar to a *-homomorphism, i.e. ∃ ξ: H → H invertible such that u ξ : a → ξ −1 u(a)ξ is a *-homomorphism (= C *-representation). Explicitly, the conclusion means that ∀ a ∈ A ξ −1 u(a *)ξ = (ξ −1 u(a)ξ) * , when this holds, Kadison calls u " orthogonalizable ". Many partial results are known, mainly due to Erik Christensen ([C1–C4]) and Uffe Haagerup ([H1]). In particular, they established (see [C3] and [H1]) this conjecture for cyclic homomorphisms, i.e. when u admits a cyclic vector h in H (= a vector h such that u(A)h = H) or more generally when u admits a finite cyclic set h 1 ,. .. , h n (so that we have u(A)h 1 + · · · + u(A)h n = H). In addition, the Kadison conjecture is known in the following cases: (i) A is commutative.