REPRESENTATIONS OF FUNCTION ALGEBRAS, ABSTRACT OPERATOR-SPACES, AND BANACH-SPACE GEOMETRY

Let G be the closed unit ball of some norm on Cn, 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. © 1992.