Abstract: In this work, we made progress on the problem that l_r⊗l_p⊗l_q is a Banach algebra under schur product. Our results extend Tonge's results. We also obtained estimates for the norm of the random quadralinear form A:l_r^M×l_p^N×l_q^K×l_s^H→C, defined by: A(e_i, e_j, e_k, e_s)=a_ijks, where the (a_ijks)'s are uniformly bounded, independent, mean zero random variables. We proved that under some conditions l_r⊗l_p⊗l_q⊗l_s is not a Banach algebra under schur product.

Key words: Random tensors, Schur product, Banach algebra

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