Let $V$ be a closed subspace of $B(H,K)$ such that $xy^*z \in V$ for all $x,y,z \in V$. Let $I$ be an ideal $(IV^*V+VV^*I \subset I)$ of $V$. Let $C(V), D(V)$ denotes the $C^{\ast}$-algebra generated by $VV^{\ast}$ and $V^*V$ respectively. We define $A(V)$, the linking $C^*-$ algebra of $V$ as follows:
$$A(V) = \begin{bmatrix} C(V) & V\\ V^* & D(V) \end{bmatrix}$$
Is $A(I)$ ideal of $A(V)$?
Note that
$$A(I)A(V) = \begin{bmatrix} C(V)C(I)+VI^* & C(V)I+VD(I)\\ V^*C(I)+D(V)I^* & V^*I+D(V)D(I) \end{bmatrix}$$
The main issue is with entries which lies on main diagonal.Anti diagonal entries are controllable.