Domains of Commutative C*-Subalgebras

Heunen, Chris; Lindenhovius, Bert

doi:10.1109/lics.2015.49

Cited by 4 publications

(5 citation statements)

References 43 publications

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“…This article extends the earlier conference proceedings version (Heunen and Lindenhovius 2015). Chris Heunen was supported by the Engineering and Physical Sciences Research Council Fellowship EP/L002388/1.…”

Section: Acknowledgementssupporting

confidence: 66%

Domains of commutative C*-subalgebras

Heunen

Lindenhovius

2019

Math. Struct. Comp. Sci.

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A C*-algebra is determined to a great extent by the partial order of its commutative C*-subalgebras. We study order-theoretic properties of this directed-complete partially ordered (dcpo). Many properties coincide: the dcpo is, equivalently, algebraic, continuous, meet-continuous, atomistic, quasi-algebraic or quasi-continuous, if and only if the C*-algebra is scattered. For C*-algebras with enough projections, these properties are equivalent to finite-dimensionality. Approximately finite-dimensional elements of the dcpo correspond to Boolean subalgebras of the projections of the C*-algebra. Scattered C*-algebras are finitedimensional if and only if their dcpo is Lawson-scattered. General C*-algebras are finite-dimensional if and only if their dcpo is order-scattered.

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Section: Acknowledgementssupporting

confidence: 66%

Domains of commutative C*-subalgebras

Heunen

Lindenhovius

2019

Math. Struct. Comp. Sci.

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“…This information is already enough to determine the piecewise structure of A, but as a Jordan algebra. (In fact, considering C(A) as a mere partially ordered set gives precisely the same information as considering it as a diagram [76]. This justifies Definition 2.1.)…”

Section: Definition 22 a Piecewise C*-algebra Consists Of A Set A Withmentioning

confidence: 58%

“…The assignment A → C(C(A)) is not functorial, does not leave commutative C*-algebras invariant, and of course only works for scattered C*-algebras A in the first place [76]. Hence there is no contradiction with Theorem 2.5.…”

Section: Domainsmentioning

confidence: 95%

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The Many Classical Faces of Quantum Structures

Heunen

2017

Entropy

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“…The intuition behind this criterion is that C ∈ C AF (A) if and only if it is generated by its projections, and any atom of C (A) is a C * -subalgebra of A that is generated by single proper projection in A. For details, we refer to Heunen & Lindenhovius (2015).…”

Section: Projectionsmentioning

confidence: 99%

Symmetries in Exact Bohrification

Landsman¹,

Lindenhovius

2018

Springer Proceedings in Mathematics &Amp; Statistics

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The 'Bohrification" program in the foundations of quantum mechanics implements Bohr's doctrine of classical concepts through an interplay between commutative and non-commutative operator algebras. Following a brief conceptual and mathematical review of this program, we focus on one half of it, called "exact" Bohrification, where a (typically noncommutative) unital C * -algebra A is studied through its commutative unital C * -subalgebras C ⊂ A, organized into a poset C (A). This poset turns out to be a rich invariant of A. To set the stage, we first give a general review of symmetries in elementary quantum mechanics (i.e., on Hilbert space) as well as in algebraic quantum theory, incorporating C (A) as a new kid in town. We then give a detailed proof of a deep result due to Hamhalter (2011), according to which C (A) determines A as a Jordan algebra (at least for a large class of C * -algebras). As a corollary, we prove a new Wignertype theorem to the effect that order isomorphisms of C (B(H)) are (anti) unitarily implemented. We also show how C (A) is related to the orthomodular poset P(A) of projections in A. These results indicate that C (A) is a serious player in C * -algebras and quantum theory.

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Domains of Commutative C*-Subalgebras

Cited by 4 publications

References 43 publications

Domains of commutative C*-subalgebras

Domains of commutative C*-subalgebras

The Many Classical Faces of Quantum Structures

Symmetries in Exact Bohrification

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