Closure of tensor products in the σ-C*-algebra category

Determine whether either the minimal or maximal C*-algebraic tensor product of two σ-C*-algebras yields a σ-C*-algebra, thereby ensuring functoriality of these tensor products in the σ-C*-algebra setting.

Background

Defining a symmetric monoidal structure on categories of σ-C*-algebras requires that tensor products of σ-C*-algebras remain inside the category. This closure is unclear for both minimal and maximal tensor products and impacts functoriality and monoidal structures in the noncommutative measurable setting.