Let $H$ and $K$ be Hilbert spaces. Recall that a closed subspace $V\subset B(H,K)$ is called a ternary ring of operators(TRO) if $xy^*z \in V$ for all $x,y,z \in V$.
Let $H$ and $K$ be Hilbert spaces and let $\mathscr{I}_1(H,K)$ denotes the space of trace class operators. Is $\mathscr{I}_1(H,K)$ a TRO?
The trace class operators $H \to K$ are not closed in $B(H,K)$ (with respect to the norm topology). Indeed, every finite-dimensional operator is trace class. Then choose a compact operator that is not trace class. Since every compact operator (between Hilbert spaces!) is the norm-limit of finite rank operators, we can conclude the first claim.
On the other hand, the identity $xy^*z \in \mathscr{T}(H,K)$ does hold for all $x,y,z \in \mathscr{T}(H,K)$ because the trace class operators form an ideal.