Dichotomy theorems for random matrices and closed ideals of operators on (\bigoplus_{n=1}^\infty\ell_1^n)_{c_0}.

Laustsen, Niels Jakob and Odell, Edward and Schlumprecht, Thomas and Zsak, Andras (2012) Dichotomy theorems for random matrices and closed ideals of operators on (\bigoplus_{n=1}^\infty\ell_1^n)_{c_0}. Journal of the London Mathematical Society, 86 (1). pp. 235-258. ISSN 1469-7750

Abstract

We prove two dichotomy theorems about sequences of operators into L_1 given by random matrices. In the second theorem we assume that the entries of each random matrix form a sequence of independent, symmetric random variables. Then the corresponding sequence of operators either uniformly factor the identity operators on l_1^k for k = 1,2, ..., or uniformly approximately factor through c_0. The first theorem has a slightly weaker conclusion still related to factorization properties but makes no assumption on the random matrices. Indeed, it applies to operators defined on an arbitrary sequence of Banach spaces. These results provide information on the closed ideal structure of the Banach algebra of all operators on the c_0-direct sum of the finite-dimensional l_1-spaces l_1^1, l_1^2,..., l_1^n,...