2023/05/22 by Rajeev Gupta, Gupta, Rajeev, Gadadhar Misra +3
Mathematics · #Advanced Banach Space Theory #FOS: Mathematics #Functional Analysis (math.FA)
paper · pdf · doi:10.48550/arxiv.2305.13270
openalex publication_date 2023/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate a Grothendieck-type inequality for pairs of Banach spaces E,F assuming E is finite-dimensional and study the associated Grothendieck-type constant. We prove that if there is a C >0 such that ‖A⊗ idF‖_Em\check⊗F→ En^*⊗F\leqslant C ‖A‖Em→ En^* for all m,n∈ℕ, where dim En=n, then both F and F^* must have finite cotype. Moreover, assuming that F has the bounded approximation property and that the conjecture in \citePisierDuality has an affirmative answer, we show that (En^*)n\geqslant 1 satisfies G.T. uniformly. We show that the Grothendieck-type constant defined for a pair of Banach spaces (E,F) is closely related to another interesting quantity introduced recently in \citeXOR games and GT comparing the projective and injective norms on the tensor product of two finite-dimensional Banach spaces E and F. We also study analogously the constants appearing in these extremal problems by restricting only to non-negative tensors. For contractive little Parrott homomorphisms \varrhoV : H^∞(Ω) → Mn, where Ω is the dual unit ball of a finite dimensional Banach space (E,‖⋅‖), we prove the sharp estimate ‖\varrhoV‖cb≤√(γ(E)), γ(E) being the positive Grothendieck constant associated with the pair (E, ℓn2). %\stwith extremal cases achieving equality. This yields a new proof of \cite[Theorem 2.1]Davidchoi using the lower bound KG+(ℓ_∞4,ℓ22) ≥ 1.1658 obtained in this paper.