By Gilles Pisier
Those notes revolve round 3 similarity difficulties, showing in 3 assorted contexts, yet all facing the distance B(H) of all bounded operators on a fancy Hilbert area H. the 1st one bargains with workforce representations, the second with C* -algebras and the 3rd one with the disc algebra. We describe them intimately within the advent which follows. This quantity is dedicated to the heritage essential to comprehend those 3 difficulties, to the ideas which are recognized in a few targeted instances and to various comparable suggestions, effects, counterexamples or extensions which their research has generated. whereas the 3 difficulties appear various, it truly is attainable to put them in a standard framework utilizing the foremost suggestion of "complete boundedness", which we found in element. utilizing this suggestion, the 3 difficulties can all be formulated as asking no matter if "boundedness" implies "complete boundedness" for linear maps pleasant definite extra algebraic identities.
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