# Looking for an old paper of Kirchberg

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.

• 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)... Aug 5 '21 at 19:19
• Further links to Kirchberg can be found on page 44 here where also a "forthcoming book of Kirchberg" is mentioned Aug 5 '21 at 19:26

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.
• It seems we need to use $V^*VI^* \subset I^*$ for a left TRO ideal $I$. Is it obvious? Aug 24 '21 at 15:57
• 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., $L:=\left[\begin{matrix} [VV^*] & I \\ V^* & [V^*I]\end{matrix}\right]$. Aug 25 '21 at 0:21