 Initially, it appeared that Segal’s assessment of the relative merits of von Neumann algebras and \(C^*\)-algebras with respect to physics was substantiated by a seminal paper, (Haag and Kastler 1964). Among other things, Haag and Kastler introduced the key axioms of the algebraic approach to QFT. They also argued that unitarily inequivalent representations are “physically equivalent” to each other. However, the use of physical equivalence to show that unitarily inequivalent representations are not physically significant has been challenged; see Kronz and Lupher (2005), Lupher (2018), and Ruetsche (2011). The prominent role of type-III factor von Neumann algebras within the algebraic approach to quantum statistical mechanics and QFT raises further doubts about Segal’s assessment.