In this letter we show that the field of Operator Space Theory provides a general and powerful mathematical framework for arbitrary Bell inequalities, in particular regarding the scaling of their violation within quantum mechanics. We illustrate the power of this connection by showing that bipartite quantum states with local Hilbert space dimension n can violate a Bell inequality by a factor of order $frac{sqrt{n}}{log^2n}$ when observables with n possible outcomes are used. Applications to resistance to noise, Hilbert space dimension estimates and communication complexity are given.
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