# 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)... Commented Aug 5, 2021 at 19:19
• Further links to Kirchberg can be found on page 44 here where also a "forthcoming book of Kirchberg" is mentioned Commented Aug 5, 2021 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? Commented Aug 24, 2021 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., $L:=\left[\begin{matrix} [VV^*] & I \\ V^* & [V^*I]\end{matrix}\right]$. Commented Aug 25, 2021 at 0:21