(Almost) C*-algebras as sheaves with self-action

Flori, Cecilia; Fritz, Tobias

doi:10.4171/jncg/11-3-9

Cited by 4 publications

(2 citation statements)

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“…Our proposal for a notion of noncommutative spectrum bears structural similarity with related functional and order-theoretic constructions that represent noncommutative algebras as a commutative fragment augmented with a unitary group action [76,58,51,28,11]. Novel abstract frameworks for understanding noncommutative algebras in terms of commutative subalgebras have appeared since our starting to work on this line of research, notably [58,41], which include some results with a similar flavour to ours. Understanding the relationships between these approaches, and the question of whether some synthesis of them might better clarify noncommutative geometry is an important line of future work.…”

Section: Discussionmentioning

confidence: 72%

Contextuality and Noncommutative Geometry in Quantum Mechanics

Silva

Barbosa

2019

Commun. Math. Phys.

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Observable properties of a classical physical system can be modelled deterministically as functions from the space of pure states to outcomes; dually, states can be modelled as functions from the algebra of observables to outcomes. The probabilistic predictions of quantum physics are contextual in that they preclude this classical assumption of reality: noncommuting observables, which are not assumed to be comeasurable, cannot be consistently ascribed deterministic values even if one enriches the description of a quantum state.Here, we consider the geometrically dual objects of noncommutative operator algebras of observables as being generalisations of classical (deterministic) state spaces to the quantum setting and argue that these generalised geometric spaces represent the objects of study of noncommutative operator geometry. By adapting the spectral presheaf of Hamilton-Isham-Butterfield, a formulation of quantum state space that collates contextual data, we reconstruct tools of noncommutative geometry in an explicitly geometric fashion. In this way, we bridge the foundations of quantum mechanics with the foundations of noncommutative geometryà la Connes et al.To each unital C * -algebra A we associate a geometric object-a diagram of topological spaces collating quotient spaces of the noncommutative space underlying A-that performs the role of a generalised Gel'fand spectrum. We show how any functor F from compact Hausdorff spaces to a suitable target category can be applied directly to these geometric objects to automatically yield an extensionF acting on all unital C * -algebras.This procedure is used to give a novel formulation of the operator K 0 -functor via a finitary variantK f of the extensionK of the topological K-functor.We then delineate a C * -algebraic conjecture that the extension of the functor that assigns to a topological space its lattice of open sets assigns to a unital C * -algebra the Zariski topological lattice of its primitive ideal spectrum, i.e. its lattice of closed, two-sided ideals. We prove the von Neumann algebraic analogue of this conjecture.

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Section: Discussionmentioning

confidence: 72%

Contextuality and Noncommutative Geometry in Quantum Mechanics

Silva

Barbosa

2019

Commun. Math. Phys.

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“…• Density also appears in quantum contextuality. For example, the boolean algebras are dense in the effect algebras [42], and compact Hausdorff spaces are dense in piecewise C*-algebras [17, Thm. 4.5].…”

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confidence: 99%

Infinite-Dimensionality in Quantum Foundations: W*-algebras as Presheaves over Matrix Algebras

Rennela¹,

Staton

Furber

2017

Electron. Proc. Theor. Comput. Sci.

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In this paper, W*-algebras are presented as canonical colimits of diagrams of matrix algebras and completely positive maps. In other words, matrix algebras are dense in W*-algebras. IntroductionIn the foundations of quantum mechanics and quantum computing, there is often a split between research using infinite dimensional structures and research using finite dimensional structures. On the one hand, in axiomatic quantum foundations there is often a focus on finite dimensional spaces and matrix mechanics (e.g. [1,45,23,46,5,3,8,9,10,13,27]), and the same is true for circuit based quantum computing (e.g. [16,31]). On the other hand, infinite dimensional spaces arise naturally in subjects such as quantum field theory [47], and moreover the register space in a scalable quantum computer arguably has an infinite dimensional aspect (see e.g. [36]), which has led some researchers to use infinite dimensional spaces in the semantics of quantum programming languages [6,37,38,21]. The 'spaces' in quantum theory are really non-commutative, so we understand them as W*-algebras, by analogy to Gelfand duality [11, 1.4 ].A natural question, then, is whether foundational research frameworks that focus on finite dimensional structures can approximate their infinite-dimensional counterparts. In brief, the answer to this question is positive when one deals with W*-algebras. In detail, when we focus on completely positive maps, as is usual in quantum foundations and quantum computation, one can show that every infinite dimensional W*-algebra is a canonical colimit of matrix algebras. This characteristic is expressed in the following theorem, which constitutes our main result. Note that it is about colimits in the opposite category of W*-algebras and completely positive maps, and therefore about limits in the category of W*-algebras and completely positive maps. Theorem. Let W * -Alg CP be the category of W*-algebras together with completely positive maps. Let N CP be the category whose objects are natural numbers, with n considered as the algebra of n × n complex matrices, and completely positive maps between them. Let Set be the category of sets and functions.The hom-set functor W * -Alg CP (−, =) : (e.g. [26, III.7]). Recall too that a full-and-faithful functor is the same thing as a full subcategory, up to categorical equivalence. Thus we can say that every W*-algebra is a canonical colimit of matrix algebras. A category theorist would say that the matrix algebras are dense in the W*-algebras (following [25, Ch. 5]), making an analogy with topology (e.g. the rational numbers are dense among the reals). We phrase the result in terms of the dual category W * -Alg Infinite-dimensionality in Quantum FoundationsRelated ideas. Our theorem is novel (as far as we can tell) but the theme is related to various research directions.• Density theorems occur throughout category theory. Perhaps the most famous situation is simplicial sets, which are functors [∆ op , Set], where ∆ is a category whose objects are natural numbers, with n consider...

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Extending the Topos Quantum Theory Approach

Flori

2018

Lecture Notes in Physics

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(Almost) C*-algebras as sheaves with self-action

Cited by 4 publications

References 20 publications

Contextuality and Noncommutative Geometry in Quantum Mechanics

Contextuality and Noncommutative Geometry in Quantum Mechanics

Infinite-Dimensionality in Quantum Foundations: W*-algebras as Presheaves over Matrix Algebras

Extending the Topos Quantum Theory Approach

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