Let $V$ be a Ternary rings of operators(TRO) i.e. closed subspace of $B(H,K)$ such that $xy^\*z \in V$ for all $x,y,z \in V$. A subspace $I$ of $V$ is called a left (right)TRO ideal provided $VV^\*I \subset I$$(IV^\*V \subset V)$.



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> Sum of closed left and right ideals is closed provided one of the ideal has bounded approximate identity.
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I was informed that results on sum of closed TRO ideals is known from some old paper(~1974) of Kirchberg but unfortunately I could not find it. Can someone please help me to trace that paper? Thank you so much.



Looking for an old paper of Kirchberg
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>).


Added: Associated with a closed left TRO ideal $I$ is a closed left ideal
$$L:=\left[\begin{matrix} [VI^\*] & I \\ I^\* & [V^\*I]\end{matrix}\right]$$
of the linking $C^\*$-algebra
$$A:=\left[\begin{matrix} [VV^\*] & V \\ V^\* & [V^\*V]\end{matrix}\right].$$
Likewise for a right TRO ideal.


