 Hilbert space conservatism dismisses the availability of a plethora of UIRs as a mathematical artifact with no physical relevance.  In contrast, algebraic imperialism argues that instead of choosing a particular Hilbert space representation, one should stay on the abstract algebraic level. Haag &amp; Kastler (1964) try to give an operationalist justification for their claim that no diverging physical content is encoded by the different unitarily inequivalent representations. The core point of the argument is that the topology used to distinguish these representations as different, namely the uniform operator topology or “norm topology”, is inappropriately fine-grained (where a topology defines what is meant by the neighborhood of an element). Therefore, it classifies representations as inequivalent, which are in fact empirically and therefore “physically equivalent.” The whole argument depends decisively on a theorem by Fell (1960), according to which a finite number of measurements, performed with some inaccuracy, can never distinguish unitarily inequivalent representations. They are “weakly equivalent,” a notion introduced by Fell (1960), using a weak operator topology. Invoking Fell’s theorem and equating weak equivalence and physical equivalence, Haag &amp; Kastler (1964) reason that the “relevant object is the abstract algebra and not the representation. The selection of a particular (faithful) representation is a matter of convenience without physical implications. It may provide a more or less handy analytical apparatus.”