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

TL;DR

This paper investigates the ideal structure of the minimal tensor product of ternary rings of operators and C*-algebras, establishing homeomorphisms between ideal spaces under certain conditions.

Contribution

It characterizes the ideal space of the tensor product via specific maps and shows when these maps induce homeomorphisms, advancing understanding of tensor product ideal structures.

Findings

 is continuous with respect to the hull-kernel topology.

 restricts to a homeomorphism on primitive and prime ideals.

When  = , it induces a homeomorphism between minimal primal ideals.

Abstract

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 \( V \otimes^{\mathrm{tmin}} B \) via two maps: \[ \Phi(I, J) = \ker(q_I \otimes^{\mathrm{tmin}} q_J) \quad \text{and} \quad \Delta(I, J) = I \otimes^{\mathrm{tmin}} B + V \otimes^{\mathrm{tmin}} J. \] We show 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 \( B \)