On a Question of N. Th. Varopoulos and the constant C 2 ( n )

Rajeev Gupta, S. Ray · Annales de l’institut Fourier · 2018

Let ℂ k [ Z 1 , ... , Z n ] denote the set of all polynomials of degree at most k in n complex variables and 𝒞 n denote the set of all n -tuple T = ( T 1 , ... , T n ) of commuting contractions on some Hilbert space ℍ . The interesting inequality K G ℂ ≤ lim n → ∞ C 2 ( n ) ≤ 2 K G ℂ , where C k ( n ) = sup ∥ p ( T ) ∥ : ∥ p ∥ 𝔻 n , ∞ ≤ 1 , p ∈ ℂ k [ Z 1 , ... , Z n ] , T ∈ 𝒞 n and K G ℂ is the complex Grothendieck constant, is due to Varopoulos. We answer a long–standing question by showing that the limit lim n → ∞ C 2 ( n ) K G ℂ </

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