Abstract
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 (π1,π2) on the free group withN≥2 generators such that (π1,π2) 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.
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Partially supported by the N.S.F.