Chapter 6 Von Neumann algebras 6.1 The algebras l (X) In this chapter we discuss quantum measure theory. This is formally rather similar to quantum topology; in both cases we treat the given structure (topological or measure theoretic) in terms of the algebra of complex-valued functions which are compatible with that structure. The algebras which arise in the latter case are actually a special case of the former: every von Neumann algebra is a C*-algebra (although this is not always the best way to think about them).
Chapter 6 Von Neumann algebras 6.1 The algebras l (X) In this chapter we discuss quantum measure theory. This is formally rather similar to quantum topology; in both cases we treat the given structure (topological or measure theoretic) in terms of the algebra of complex-valued functions which are compatible with that structure. The algebras which arise in the latter case are actually a special case of the former: every von Neumann algebra is a C*-algebra (although this is not always the best way to think about them).
This paper explores the intricate connections between various algebraic structures—Banach algebras, C*-algebras, and von Neumann algebras—and their profound implications for the theory of quantum entanglement and quantum information. We establish a comprehensive framework that bridges the abstract concepts of functional analysis with the concrete demands of quantum mechanics. Beginning with the foundational properties of Banach algebras, we progressively introduce C*-algebras as a natural setting for quantum observables and states, followed by von Neumann algebras, which are essential for describing quantum measurement and infinite-dimensional quantum systems. A central focus is to delineate how quantum entanglement, a cornerstone resource in quantum information science, can be rigorously characterized and quantified within this hierarchical algebraic framework. We integrate recent game-theoretic approaches to quantum information, providing new perspectives on quantum relative entropy, mutual information, and channel capacity. By demonstrating how the rich structure of C*-algebras acts as a pivotal bridge, we illuminate the mathematical coherence underlying the transition from general algebraic concepts to the specialized domain of quantum information processing, offering a unified view of quantum phenomena. The work also delves into the operational justifications for these algebraic descriptions, particularly in the context of infinite quantum systems.
The axioms of measurements introduced by Ludwig are formulated and studied in the framework of operator algebras. It is shown that a concrete C*-algebra with identity satisfies the axiom of sensitivity increase of effects if and only if it is a von Neumann algebra; although a von Neumann algebra satisfies the axiom of decompossability of ensembles, however, the axiom of components of the mixtures of two ensembles is true only if a von Neumann algebra is a factor of type In (n < + ∞). It is also verified that the set of decision effects, which is proved to be a subset of projections of a von Neumann algebra, has similar lattice structure of quantum mechanics, and its connection with quantum logic in the sense of Varadarjan is also figured out.
In "Self-adjoint Operators as Functions I: Lattices, Galois Connections, and the Spectral Order" [arXiv:1208.4724], it was shown that self-adjoint operators affiliated with a von Neumann algebra N can equivalently be described as certain real-valued functions on the projection lattice P(N) of the algebra, which we call q-observable functions. Here, we show that q-observable functions can be interpreted as generalised quantile functions for quantum observables interpreted as random variables. More generally, when L is a complete meet-semilattice, we show that L-valued cumulative distribution functions and the corresponding L-quantile functions form a Galois connection. An ordinary CDF can be written as an L-CDF composed with a state. For classical probability, one picks L=B(Ω), the complete Boolean algebra of measurable subsets modulo null sets of a measurable space Ω. For quantum probability, one uses L=P(N), the projection lattice of a nonabelian von Neumann algebra N. Moreover, using some constructions from the topos approach to quantum theory, we show that there is a joint sample space for all quantum observables, despite no-go results such as the Kochen-Specker theorem. Specifically, the spectral presheaf Σ of a von Neumann algebra N, which is not a mere set, but a presheaf (i.e., a 'varying set'), plays the role of the sample space. The relevant meet-semilattice L in this case is the complete bi-Heyting algebra of clopen subobjects of Σ. We show that using the spectral presheaf Σ and associated structures, quantum probability can be formulated in a way that is structurally very similar to classical probability.
The mathematics of classical probability theory was subsumed into classical measure theory by Kolmogorov in 1933.Quantum theory as nonclassical probability theory was incorporated into the beginnings of noncommutative measure theory by von Neumann in the early thirties, as well.To precisely this end, von Neumann initiated the study of what are now called von Neumann algebras and, with Murray, made a first classification of such algebras into three types.The nonrelativistic quantum theory of systems with finitely many degrees of freedom deals exclusively with type I algebras.However, for the description of further quantum systems, the other types of von Neumann algebras are indispensable.The paper reviews quantum probability theory in terms of general von Neumann algebras, stressing the similarity of the conceptual structure of classical and noncommutative probability theories and emphasizing the correspondence between the classical and quantum concepts, though also indicating the nonclassical nature of quantum probabilistic predictions.In addition, differences between the probability theories in the type I, II and III settings are explained.A brief description is given of quantum systems for which probability theory based on type I algebras is known to be insufficient.These illustrate the physical significance of the previously mentioned differences.
This short chapter contains basics of the mathematical formalism for thequantum measurement theory. In this book we proceed mainly withthe von Neumann measurement theory in which observables are given byHermitian operators and the state update by projections. However, we alsomention the measurement formalism based on quantum instruments, sinceit gives the general framework for quantum measurements. This formalismis used only in Chapters 10 and 18. The latter chapter is devoted to quantum-likemodeling – the applications of the mathematical formalism and methodologyof quantum mechanics (QM) to cognition, psychology, and decision making.Surprisingly, in such applications even the simplest effects can’t be described bythe von Neumann theory. One should use quantum instruments (compare withquantum physics where the main body of theory can be presented solelywithin the von Neumann measurement theory).
Observables of a quantum system, described by self-adjoint operators in a von Neumann algebra or affiliated with it in the unbounded case, form a conditionally complete lattice when equipped with the spectral order. Using this order-theoretic structure, we develop a new perspective on quantum observables. In this first paper (of two), we show that self-adjoint operators affiliated with a von Neumann algebra can equivalently be described as certain real-valued functions on the projection lattice of the algebra, which we call q-observable functions. Bounded self-adjoint operators correspond to q-observable functions with compact image on non-zero projections. These functions, originally defined in a similar form by de Groote, are most naturally seen as adjoints (in the categorical sense) of spectral families. We show how they relate to the daseinisation mapping from the topos approach to quantum theory. Moreover, the q-observable functions form a conditionally complete lattice which is shown to be order-isomorphic to the lattice of self-adjoint operators with respect to the spectral order. In a subsequent paper, we will give an interpretation of q-observable functions in terms of quantum probability theory, and using results from the topos approach to quantum theory, we will provide a joint sample space for all quantum observables.