The three types of normal sequential effect algebras

Abstract

A sequential effect algebra (SEA) is an effect algebra equipped with a sequential product operation modeled after the Lüders product $(a,b)\mapsto \sqrt{a}b\sqrt{a}$ on C*-algebras. A SEA is called normal when it has all suprema of directed sets, and the sequential product interacts suitably with these suprema. The effects on a Hilbert space and the unit interval of a von Neumann or JBW algebra are examples of normal SEAs that are in addition convex, i.e. possess a suitable action of the real unit interval on the algebra. Complete Boolean algebras form normal SEAs too, which are convex only when $0=1$. We show that any normal SEA $E$ splits as a direct sum $E\equiv E_b\oplus E_c\oplus E_{ac}$ of a complete Boolean algebra $E_b$, a convex normal SEA $E_c$, and a newly identified type of normal SEA $E_{ac}$ we dub purely almost-convex. Along the way we show, among other things, that a SEA which contains only idempotents must be a Boolean algebra; and we establish a spectral theorem using which we settle for the class of normal SEAs a problem of Gudder regarding the uniqueness of square roots. After establishing our main result, we propose a simple extra axiom for normal SEAs that excludes the seemingly pathological a-convex SEAs. We conclude the paper by a study of SEAs with an associative sequential product. We find that associativity forces normal SEAs satisfying our new axiom to be commutative, shedding light on the question of why the sequential product in quantum theory should be non-associative.

arXiv comment20 pages. V2: Journal version with several textual improvements

References in corpus (20)

• Higher-order interference and single-system postulates characterizing quantum theory
• Dynamical phase transitions, temporal orthogonality and the dynamics of observables in one dimensional ultra-cold quantum gases: from the continuum to the lattice
• Characterization of the Sequential Product on Quantum Effects
• Sequential product on standard effect algebra $\mathcal{E}(H)$
• Blocks of homogeneous effect algebras
• Uniqueness and order in sequential effect algebras
• No-free-information principle in general probabilistic theories
• An effect-theoretic reconstruction of quantum theory
• Sequential Product Spaces are Jordan Algebras
• Space in Monoidal Categories
• The Uniqueness Problem of Sequence Product on Operator Effect Algebra $\epsilon (H)$
• Type-Decomposition of a Pseudo-Effect Algebra
• Every State on Interval Effect Algebra is Integral
• Strongly symmetric spectral convex bodies are Jordan algebra state spaces
• Commutativity in Jordan Operator Algebras
• Cohomology of Effect Algebras
• Remarks on the sequential effect algebras
• On the properties of spectral effect algebras
• A characterisation of ordered abstract probabilities
• Three characterisations of the sequential product

Cited by in corpus (3)

• A computer scientist's reconstruction of quantum theory
• Spectral resolutions in effect algebras
• A Categorical Construction of the Real Unit Interval