Abstract
Let G be the closed unit ball of some norm on C n , and let A ( G ) be the closure of the polynomials in the sup norm. We prove that if n ⩾ 5 then there is a contractive representation of A ( G ) as operators on a Hilbert space which is not completely contractive. Our technique involves introducing a numerical invariant α ( X ) for a normed space X which measures the difference between the minimal operator space structure which can be assigned to X , MIN( X ), and the maximal structure, MAX( X ). We estimate α ( X ) using Banach space techniques. We also prove that if X is any infinite dimensional subspace of the space of continuous functions on a compact Hausdorff space, then there exists a bounded linear map on X which is not completely bounded.
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