Joint similarity problems and the generation of operator algebras with bounded length (Q1265584)

The author obtains an example of a pair of commuting Hilbert space operators \((T_1,T_2)\) such that each of them is similar to a contraction but the pair \((T_1,T_2)\) is not similar to a pair of contractions. The same ideas allows to obtain a pair of commuting unitarizable representations \((\pi_1,\pi_2)\) on the free group with \(N\geq 2\) generators such that \((\pi_1,\pi_2)\) is not similar to a pair of unitary representations. In connection with these examples the author introduces a notion of a length for an operator algebra generated by two subalgebras and studies the length of the maximal tensor product of two \(C^\ast\)-algebras.