Let $H$ and $K$ be Hilbert spaces and $B(H,K)$ denotes the space of bounded operators from $H$ to $K$. Recall that a ternary ring of operators (TRO) $V$ is a closed subspace of $B(H,K)$ which is closed under the operation $(x,y,z) \to xy^*z$. Moreover, $V$ is called commutative if $ab^*c=cb^*a$ for all $a,b,c \in V$.
Is there any definition of center of TRO In literature?
What about defining $$C = \{ v\in V : av^*c = cv^*a \ (a,c\in V) \}. $$ This is evidently a closed linear subspace of $V$. Then, given $d,e,f\in C$ and $a,c\in V$, $$a(de^*f)^*c = (af^*e)d^*c = cd^*(af^*e) = cd^*(ef^*a) = c(d^*ef^*)a = c(f^*ed^*)a = c(de^*f)^* a$$ using that $d\in C$, then $f\in C$, then $e\in C$. Thus $de^*f\in C$, so $C$ is a sub-TRO of $V$, and clearly $C$ is commutative. If $V$ were itself commutative, then $C=V$. This seems like a reasonable definition of "center" to me.
I don't know of a reference...
