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^* \) -algebra. We study the structure of the ideal space of the operator space injective tensor product \ (V ^tmin B \) via two maps: \ Φ (I, J) = (qI ^tmin qJ) and Δ (I, J) = I ^tmin B + V ^tmin J. \ We show that \ (Φ\) 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 ^tmin B \). We prove that if \ (Φ= Δ\), then \ (Φ\) induces a homeomorphism between the space of minimal primal ideals of \ (V ^tmin B \) and the product of the spaces of minimal primal ideals of \ (V \) and \ (B \)