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I guess you don't need approximate identity assumption to prove it (in any case what does approximate identity mean for a TRO?). I am not sure which paper of Kirchberg you are referring to, but indeed Kirchberg has proved that the sum of a closed left ideal and a closed right ideal is closed (for C-algebras) and this fact generalizes to TROs (C-spaces in Kirchberg's terminology) via the linking algebra construction. For the proof, see Section 4 of Kirchberg's paper "On restricted perturbations..." JFA 1995 (https://mathscinet.ams.org/mathscinet-getitem?mr=1322640).