 In 1943, Gelfand and Neumark published an important paper on an important class of normed rings, which are now known as abstract \(C^*\)-algebras. Their paper was influenced by Murray and von Neumann’s work on rings of operators, which was discussed in the previous section. In their paper, Gelfand and Neumark focus attention on abstract normed \(*\)-rings. They show that any \(C^*\)-algebra can be given a concrete representation in a Hilbert space (which need not be separable). That is to say, there is an isomorphic mapping of the elements of a \(C^*\)-algebra into the set of bounded operators of the Hilbert space. Four years later, Segal (1947a) published a paper that served to complete the work of Gelfand and Neumark by specifying the definitive procedure for constructing concrete (Hilbert space) representations of an abstract \(C^*\)-algebra. It is called the GNS construction (after Gelfand, Neumark, and Segal). That same year, Segal (1947b) published an algebraic formulation of quantum mechanics, which was substantially influenced by (though deviating somewhat from) von Neumann’s (1963, Vol. III, No. 9) algebraic formulation of quantum mechanics, which is cited in the previous section. It is worth noting that although \(C^*\)-algebras satisfy Segal’s postulates, the algebra that is specified by his postulates is a more general structure known as a Segal algebra. Every \(C^*\)-algebra is a Segal algebra, but the converse is false since Segal’s postulates do not require an adjoint operation to be defined. If a Segal algebra is isomorphic to the set of all self-adjoint elements of a \(C^*\)-algebra, then it is a special or exceptional Segal algebra. Although the mathematical theory of Segal algebras has been fairly well developed, a \(C^*\)-algebra is the most important type of algebra that satisfies Segal’s postulates.