The Ideal Structure of the Minimal Tensor Product of Ternary Rings of Operators

Let \(V\) be a ternary ring of operator, and let \(B\) be a \(C^*\)-algebra. We study the structure of the ideal space of the operator space injective tensor product \(V \otimes^{\mathrm{tmin}} B\) via two maps:\begin{align*}\Phi(I, J) = \ker(q_I \otimes^{\mathrm{tmin}} q_J) \quad \text{and} \quad \Delta(I, J) = I \otimes^{\mathrm{tmin}} B + V \otimes^{\mathrm{tmin}} J.\end{align*}We prove that \(\Phi\) 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 \(V \otimes^{\mathrm{tmin}} B\). We prove that if \(\Phi = \Delta\), then \(\Phi\) induces a homeomorphism between the space of minimal primal ideals of \(V \otimes^{\mathrm{tmin}} B\) and the product of the spaces of minimal primal ideals of \(V \) and \(B\).