Let $H$ and $K$ be Hilbert spaces. Recall that a closed subspace $X\subset B(H,K)$ is called a ternary ring of operators(TRO) provided $xy^*z \in X$ for all $x,y,z \in X$
It seems TROs does not have its own norm, it inherits the norm from $B(H,K)$. Is this understanding correct?
Suppose $X$ is a $*-$Banach space and we want to prove that it is a TRO so we need to find an isometry $i: X \to B(H,K)$ for some $H$ and $K$. Also we need to show that $xy^*z \in X$ for all $x,y,z \in X$. Is this correct?