 The algebraic approach has proven most effective in quantum statistical mechanics. It is extremely useful for characterizing many important macroscopic quantum effects including crystallization, ferromagnetism, superfluidity, structural phase transition, Bose-Einstein condensation, and superconductivity. A good introductory presentation is Sewell (1986), and for a more advanced discussion see Bratteli and Robinson (1979, 1981). In algebraic quantum statistical mechanics, an infinite quantum system is defined by specifying an abstract algebra of observables. A particular state may then be used to specify a concrete representation of the algebra as a set of bounded operators in a Hilbert space. Among the most important types of states that are considered in algebraic statistical mechanics are the equilibrium states, which are often referred to as “KMS states” (since they were first introduced by the physicists Kubo, Martin, and Schwinger). There is a continuum of KMS states since there is at least one KMS state for each possible temperature value \(\tau\) of the system, for \(0\le \tau \le +\infty\). Given an automorphism group, each KMS state corresponds to a representation of the algebra of observables that defines the system, and each of these representations is unitarily inequivalent to any other. It turns out that each representation that corresponds to a KMS state is a factor: if \(\tau = 0\) then it is a type-I factor, if \(\tau = +\infty\) then it is a type-II factor, and if \(0\lt \tau \lt +\infty\) then it is a type-III factor. Thus, type-III factors play a predominant role in algebraic quantum statistical mechanics.