Joint similarity problems and the generation of operator algebras with bounded length
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There is a pair of commuting operators (T1, T2) on Hilbert space such that each T1 and T2 is similar to a contraction but the pair (T1, T2) is not similar to a pair of contractions. There is a pair of commuting unitarizable representations (1, 2) on the free group with N 2 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 a C *-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.