Jan Hamhalter

Quantum measure theory

Preface. 1: Introduction. 2: Operator Algebras. 2.1. C*-Algebras. 2.2. Von Neumann Algebras. 2.3. Jordan Algebras And Ordered Structures. 3: Gleason Theorem. 3.1. Reduction To Three-Dimensional Space. 3.2. Regularity Of Frame Functions On R3. 3.3. Boundedness Of Frame Functions. 3.4. Historical Remarks And Comments. 4: Completeness Criteria. 4.1. Functiona1 Completeness Criteria. 4.2. Algebraic Completeness Criteria. 4.3. Measure Theoretic Completeness Criteria. 4.4. Historical Remarks And Comments. 5: Generalized Gleason Theorem. 5.1. The Mackey-Gleason Problem. 5.2. Reduction To Scalar Quasi-Functionals. 5.3. Linear Extensions Of Measures On Type In Algebras. 5.4. Linear Extensions Of Measures On Infinite Algebras. 5.5. Linear Extensions Of Measures On Finite Algebras. 5.6. Historical Remarks And Comments. 6: Basic Principles Of Quantum Measure Theory. 6.1. Boundedness Of Completely Additive Measures. 6.2. Yosida-Hewitt Decompositions Of Quantum Measures. 6.3. Convergence Theorems. 6.4. Historical Remarks And Comments. 7: Applications Of Gleason Theorem. 7.1. Multiform Gleason Theorem And Decoherence. 7.2. Velocity Maps And Derivations. 7.3. Approximate Hidden Variables. 7.4. Historical Remarks And Comments. 8: Orthomorphisms Of Projections. 8.1. Orthomorphisms Of Projection Lattices. 8.2. Countable Additivity Of *-Homomorphisms. 8.3. Historical Remarks And Comments. 9: Restrictions And Extensions Of States. 9.1. Restriction Properties Of Pure States. 9.2. Gleason Type Theorems For Quantum Logics. 9.3. Historical Remarks And Comments. 10: Jauch-Piron States. 10.1. Basic Properties Of Jauch States. 10.2. Nonsingularity Of Jauch-Piron States. 10.3. Countable Additivity Of States. 10.4. Historical Remarks And Comments. 11: Independence Of Quantum Systems. 11.1. Independence In Classical And Quantum Theory. 11.2. Independence Of C*-Algebras. 11.3. Independence Of Von Neumann Algebras. 11.4. Historical Remarks And Comments. Bibliography. Index.

States on orthoalgebras

The orthoalgebras, introduced by Foulis and Randall and studied by various authors, have recently become a significant mathematical structure of the logicoalgebraic foundation of quantum theories. In this paper we give a coherent account of states (=finitely additive measures) on orthoalgebras. In the first section we review basic properties of (and constructions with) orthoalgebras and develop a useful “pasting technique” (Theorem 1.12 and Proposition 1.16) applied later in this paper (and possibly elsewhere, too). We also exhibit orthoalgebras with rather interesting and “exotic” state spaces (Example 1.20 and Proposition 1.21). In the second section we construct orthoalgebras with preassigned state space properties. We prove a state representation theorem (Theorem 2.1) and obtain an orthoalgebraic version of Shultzs theorem (Theorem 2.7). In the third section we make a thorough analysis of the extension problem for states on orthoalgebras. We first study the orthoalgebras whose state spaces are finite dimensional. For these orthoalgebras we find a necessary and sufficient condition to allow extensions of states over larger orthoalgebras (Theorem 3.4). Then we prove that all Hilbertian orthoalgebras as well as all Boolean orthoalgebras allow extensions of states over larger orthoalgebras (Theorems 3.10 and 3.12).

Nonlinear maps on von Neumann algebras preserving the star order

Star order is defined on a C*-algebra in the following way: a ⪯ b if a*a = a*b and aa* = ba*. Let 𝒜 be a von Neumann algebra without Type I2 direct summand. Let 𝒜 n be the set of all normal elements of 𝒜. Suppose that ϕ: 𝒜 n  → 𝒜 n is a continuous bijection that preserves the star order on 𝒜 n in both directions. Further, let there is a function f : ℂ → ℂ and an invertible central element c in 𝒜 such that ϕ(λ1) = f(λ)c for all λ ∈ ℂ. We show that there is a unique Jordan *-isomorphism ψ: 𝒜 → 𝒜 such that Ramifications of this result as well as optimality of the assumptions are discussed.

Quantum structures and operator algebras

Publisher Summary This chapter summarizes both classical and recent results concerning the extension of fundamental principles of measure theory to the projection logics of operator algebras. It is noted that in standard measure theory, the basic concept is that of a measure on a σ-field, A, of subsets of a set Ω. A is a Boolean algebra with respect to the set theoretic operations and the corresponding set of A-measurable functions on Ω constitutes a commutative algebra with respect to the arithmetic operations. This concept, the core of Kolmogorovian probability theory, has proved to be extremely useful, and it plays an important role both in theoretical and in the light of quantum theory. Classical measure theory however needed modification and extension so as to be made suitable as a framework for quantum probability. As a consequence, noncommutative measure theory also called quantum measure theory evolved, which is essentially based on operators rather than scalar measurable functions.

De Morgan Property for Effect Algebras of von Neumann Algebras

It is shown that the unit interval of a von Neumann algebra is a Sum Brouwer–Zadeh algebra when equipped with another unary operation sending each element to the complement of its range projection. The main result of this Letter says that a von Neumann algebra is finite if and only if the corresponding Brouwer–Zadeh structure is de Morgan or, equivalently, if the range projection map preserves infima in the unit interval. This provides a new characterization of finiteness in the Murray–von Neumann structure theory of von Neumann algebras in terms of Brouwer–Zadeh structures.

JAUCH–PIRON STATES AND σ-ADDITIVITY

We study σ-additivity of the physically plausible Jauch–Piron states on a von Neumann algebra M. Amongst other consequences we extend our earlier results [5] by showing that geometric conditions much weaker than pureness imply σ-additivity for a Jauch–Piron state and, further, that if M is properly infinite and the continuum hypothesis is assumed to be true then all Jauch–Piron factor states are σ-additive.

Extension of Jauch-Piron states on Jordan algebras

A state ρ on a JW -algebra or von Neumann algebra M is said to be a Jauch–Piron state if whenever e and f are projections in M with ρ( e ) = ρ( f ) = 0 then ρ( e ∨ f ) = 0.

States on W*−algebras and orthogonal vector measures

We show that every state on a If*-algebra sf without type I2 direct summand is induced by an orthogonal vector measure on sf . This result may find an application in quantum stochastics (1, 7). Particularly, it allows us to find a simple formula for the transition probability between two states on sf (3, 8), 1. Preliminaries

Linear maps preserving maximal deviation and the Jordan structure of quantum systems

In the algebraic approach to quantum theory, a quantum observable is given by an element of a Jordan algebra and a state of the system is modelled by a normalized positive functional on the underlying algebra. Maximal deviation of a quantum observable is the largest statistical deviation one can obtain in a particular state of the system. The main result of the paper shows that each linear bijective transformation between JBW algebras preserving maximal deviations is formed by a Jordan isomorphism or a minus Jordan isomorphism perturbed by a linear functional multiple of an identity. It shows that only one numerical statistical characteristic has the power to determine the Jordan algebraic structure completely. As a consequence, we obtain that only very special maps can preserve the diameter of the spectra of elements. Nonlinear maps preserving the pseudometric given by maximal deviation are also described. The results generalize hitherto known theorems on preservers of maximal deviation in the case of self-adjoint parts of von Neumann algebras proved by Molnar.

Determinacy of States and Independence of Operator Algebras

The aim of this paper is to summarize, deepen,and comment upon recent results concerning states onoperator algebras and their extensions. The first partis focused on the relationship between pure states and singly generated subalgebras. Among otherswe show that every pure state ρ on a separablealgebra A is uniquely determined by some element of Awhich exposes ρ. The main part of this paper is the second section, dealing with characterizationof various types of independence conditions arising inthe axiomatics of quantum field theory. These twotopics, seemingly different, are connected by a common extension technique based on determinacy ofpure states.