Joint similarity problems and the generation of operator algebras with bounded length

There is a pair of commuting operators (T ₁,T ₂) on Hilbert space such that eachT ₁ andT ₂ is similar to a contraction but the pair (T ₁,T ₂) is not similar to a pair of contractions. There is a pair of commuting unitarizable representations (₁,₂) on the free group withN2 generators such that (₁,₂) is not similar to a pair of unitary representations. In connection with these examples, we introduce and study a notion of length for aC ^*-algebra (or an operator algebra) generated by two subalgebras, which is analogous to the minimum length of a word in the generators of a group.Partially supported by the N.S.F.