Operations on continuous bundles of C*-algebras

Kirchberg, Eberhard; Wassermann, Simon

doi:10.1007/bf01461011

Cited by 95 publications

(128 citation statements)

References 12 publications

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“…algebra of classical observables A 0 in (2.24), whereas for the N → ∞ limit of quantum statistical mechanics there are even two (relevant) possibilities: the (noncommutative) algebras B N in (2.34) can be glued continuously to either the algebra of (quasi-) local observables B l ∞ in (2.39), which is noncommutative, too (and hence is the more obvious choice), or to the commutative algebra of global observables B g ∞ in (2.40). The "glueing" is done using the formalism of continuous fields of C*-algebras (of observables); we will just look at the cases of interest for our three models, and refer the reader to Dixmier (1977) and Kirchberg & Wassermann (1995) for the abstract theory. 40 Continuity of the dynamics in the various limits at hand will be a corollary, provided time-evolution is expressed in terms of the one-parameter automorphism groups τ .…”

Section: Continuitymentioning

confidence: 99%

“…40 The idea of a continuous fields of C*-algebras goes back to Dixmier (1977), who gave a direct definition in terms of glueing conditions between the fibers, and was usefully reformulated by Kirchberg & Wassermann (1995), who stressed the role of the continuous sections of the field already in its definition. Both definitions are reviewed in Landsman (1998).…”

Section: Continuous Fields Of C*-algebrasmentioning

confidence: 99%

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Spontaneous symmetry breaking in quantum systems: Emergence or reduction?

Landsman¹

2013

Studies in History and Philosophy of Science Part B: Studies in

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Beginning with Anderson (1972), spontaneous symmetry breaking (ssb) in infinite quantum systems is often put forward as an example of (asymptotic) emergence in physics, since in theory no finite system should display it. Even the correspondence between theory and reality is at stake here, since numerous real materials show ssb in their ground states (or equilibrium states at low temperature), although they are finite. Thus against what is sometimes called 'Earman's Principle', a genuine physical effect (viz. ssb) seems theoretically recovered only in some idealization (namely the thermodynamic limit), disappearing as soon as the idealization is removed.We review the well-known arguments that (at first sight) no finite system can exhibit ssb, using the formalism of algebraic quantum theory in order to control the thermodynamic limit and unify the description of finite-and infinite-volume systems. Using the striking mathematical analogy between the thermodynamic limit and the classical limit, we show that a similar situation obtains in quantum mechanics (which typically forbids ssb) versus classical mechanics (which allows it).This discrepancy between formalism and reality is quite similar to the measurement problem (now regarded as an instance of asymptotic emergence), and hence we address it in the same way, adapting an argument of the author and Reuvers (2013) that was originally intended to explain the collapse of the wave-function within conventional quantum mechanics. Namely, exponential sensitivity to (asymmetric) perturbations of the (symmetric) dynamics as the system size increases causes symmetry breaking already in finite but very large quantum systems. This provides continuity between finite-and infinite-volume descriptions of quantum systems featuring ssb and hence restores Earman's Principle (at least in this particularly threatening case). Motto"The characteristic behaviour of the whole could not, even in theory, be deduced from the most complete knowledge of the behaviour of its components, taken separately or in other combinations, and of their proportions and arrangements in this whole. This is what I understand by the 'Theory of Emergence'. I cannot give a conclusive example of it, since it is a matter of controversy whether it actually applies to anything." (Broad, 1925, p. 59) * Final version, to be published in Studies in History and Philosophy of Modern Physics.

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Section: Continuitymentioning

confidence: 99%

Section: Continuous Fields Of C*-algebrasmentioning

confidence: 99%

Spontaneous symmetry breaking in quantum systems: Emergence or reduction?

Landsman¹

2013

Studies in History and Philosophy of Science Part B: Studies in

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“…Recall that a continuous field of C * -algebras (see [4,9]) is the natural notion of a bundle of C * -algebras. The fibers are all C * -algebras, the space of (continuous) sections is a C * -algebra, and for each point of the base space there is an evaluation map, a * -homomorphism of the algebra of sections onto the fiber algebra.…”

Section: Generalitiesmentioning

confidence: 99%

Geometric Quantization of Vector Bundles and the Correspondence with Deformation Quantization

Hawkins¹

2000

Communications in Mathematical Physics

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“…The above notion of continuous bundle has been given in [14]: it is a simplified version of the classical notion of continuous field (see [7, §10]). We refer to the last-cited reference for the notions of restriction ( [7, 10.1.7] and local triviality ([7, 10.1.8]), which are the analogues to well-known notions in the setting of topological bundles.…”

Section: Introductionmentioning

confidence: 99%

Bundles of C*-algebras and the KK(X;−,−)-bifunctor

Vasselli

Trends in Mathematics

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An overview about C*-algebra bundles with a Z -grading is presented, with particular emphasis on classification questions. In particular, we discuss the role of the representable KK(X; −, −) -bifunctor introduced by Kasparov. As an application, we consider Cuntz-Pimsner algebras associated with vector bundles, and give a classification in terms of Ktheoretical invariants in the case in which the base space is an n -sphere.

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Operations on continuous bundles of C*-algebras

Cited by 95 publications

References 12 publications

Spontaneous symmetry breaking in quantum systems: Emergence or reduction?

Spontaneous symmetry breaking in quantum systems: Emergence or reduction?

Geometric Quantization of Vector Bundles and the Correspondence with Deformation Quantization

Bundles of C*-algebras and the KK(X;−,−)-bifunctor

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