The ideal structure of the minimal Tensor Product of Ternary Rings of Operators

Let $V$ be a ternary ring of operator and $B$ a $C^{\ast }$-algebra. We study the structure of the ideal space of the operator space injective tensor product $V{\otimes }^{\mathrm{t}\mathrm{m}\mathrm{i}\mathrm{n}}B$ via two maps: $$\mathrm{\Phi }(I,J)=\ker (q_I{\otimes }^{\mathrm{t}\mathrm{m}\mathrm{i}\mathrm{n}}{q}_J)\text{and}\mathrm{\Delta }(I,J)=I{\otimes }^{\mathrm{t}\mathrm{m}\mathrm{i}\mathrm{n}}B+V{\otimes }^{\mathrm{t}\mathrm{m}\mathrm{i}\mathrm{n}}J\mathrm{.}$$ We show that $\mathrm{\Phi }$ is continuous with respect to the hull-kernel topology, and that its restriction to primitive and prime ideals defines a homeomorphism onto dense subsets of the respective ideal spaces of $V{\otimes }^{\mathrm{t}\mathrm{m}\mathrm{i}\mathrm{n}}B$. We prove that if $\mathrm{\Phi }=\mathrm{\Delta }$, then $\mathrm{\Phi }$ induces a homeomorphism between the space of minimal primal ideals of $V{\otimes }^{\mathrm{t}\mathrm{m}\mathrm{i}\mathrm{n}}B$ and the product of the spaces of minimal primal ideals of $V$ and $B$