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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)$.

Sum of closed left and right ideals is closed provided one of the ideal has bounded approximate identity.

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.

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  • $\begingroup$ googling for Kirchberg ternary ideal I came across somthing by David P. Blecher where Krichberg is mentioned with "...and, we believe Kirchberg, (although we are not sure if this work is in print)... $\endgroup$ Aug 5 '21 at 19:19
  • $\begingroup$ Further links to Kirchberg can be found on page 44 here where also a "forthcoming book of Kirchberg" is mentioned $\endgroup$ Aug 5 '21 at 19:26
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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).

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.

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  • $\begingroup$ It seems we need to use $ V^*VI^* \subset I^*$ for a left TRO ideal $I$. Is it obvious? $\endgroup$
    – Math Lover
    Aug 24 '21 at 15:57
  • $\begingroup$ Thank you. I miscalculated. In the easy case where $I=Vp$ for some projection $p\in [V^*V]$, the left ideal $L$ should be $A(1\oplus p)$ i.e., $L:=\left[\begin{matrix} [VV^*] & I \\ V^* & [V^*I]\end{matrix}\right]$. $\endgroup$ Aug 25 '21 at 0:21
  • $\begingroup$ So we don't get the bijection between arbitrary left(right) ideals? $\endgroup$
    – Math Lover
    Aug 25 '21 at 0:35
  • $\begingroup$ The latter formula holds true for an arbitrary left TRO ideal. I just wanted to show how to find the right formula. $\endgroup$ Aug 25 '21 at 1:09

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