Some thoughts on Andoʹs theorem and Parrottʹs example Original Research Article

In this short note, we present an elementary proof of Andoʹs theorem within a restricted class image of homomorphisms modeled after Parrottʹs example. We also show by explicit estimation that the cb-norm of the contractive homomorphism ρ of the tri-disc algebra, induced by the commuting triple of Parrott, exceeds 1. Indeed, we construct a polynomial P with matrix coefficients with the property short parallelρ(P)short parallel>short parallelPshort parallel∞. In particular, we show that there are contractive homomorphisms of the tri-disc algebra which are not even 2-contractive.