Characterisation of the Berkovich spectrum of the Banach algebra of bounded continuous functions

Mihara, Tomoki

doi:10.4171/dm/463

Cited by 6 publications

(1 citation statement)

References 4 publications

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“…Since every ultrafilter on R(X) converges to a unique point in X, one obtains a continuous open surjection from the Stone space of R(X), denoted by X ε , onto X. Note that this construction is different from the 'Stone space associated with X' as introduced in [23], where the collection of clopen subsets instead of regular open subsets were considered.…”

Section: Ortho-sets and Their Relation To Ortholatticesmentioning

confidence: 99%

Ortho-sets and Gelfand spectra

Ding¹,

Ng²

2021

J. Phys. A: Math. Theor.

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Motivated by the relation on quantum states induced by zero transition probability, we introduce the notion of ortho-set, which is a set equipped with a relation ≠q satisfying: x ≠q y implies both x ≠ y and y ≠q x. For an ortho-set, a canonical complete ortholattice is constructed. Conversely, every complete ortholattice comes from an ortho-set in this way. Hence, the theory of ortho-sets captures almost everything about quantum logics. For a quantum system modeled by the self-adjoint part B sa of a C*-algebra B, we also introduce a ‘semi-classical object’ called the Gelfand spectrum. It is the ortho-set, P(B), of pure states of B equipped with an ‘ortho-topology’, which is a collection of subsets of P(B), defined via a hull-kernel construction with respects to closed left ideals of B. We establish a generalization of the Gelfand theorem by showing that a bijection between the Gelfand spectra of two quantum systems that preserves the respective ortho-topologies is induced by a Jordan isomorphism between the self-adjoint parts of the underlying C*-algebras (i.e. an isomorphism of the quantum systems), when the underlying C*-algebras satisfy a mild condition.

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Section: Ortho-sets and Their Relation To Ortholatticesmentioning

confidence: 99%

Ortho-sets and Gelfand spectra

Ding¹,

Ng²

2021

J. Phys. A: Math. Theor.

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Le lemme d’Abhyankar perfectoide

André

2017

Publ.math.IHES

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ABSTRACT. Nousétendons le théorème de presque-pureté de Faltings-Scholze-KedlayaLiu sur les extensionsétales finies d'algèbres perfectoïdes au cas des extensions ramifiées, sans restriction sur le discriminant. Le point clé est une version perfectoïde du théorème d'extension de Riemann. Au préalable, nous revenons sur les aspects catégoriques des algèbres de Banach uniformes et des algèbres perfectoïdes.ABSTRACT. We extend Faltings's "almost purity theorem" on finite etale extensions of perfectoid algebras (as generalized by Scholze and Kedlaya-Liu) to the ramified case, without restriction on the discriminant. The key point is a perfectoid version of Riemann's extension theorem. Categorical aspects of uniform Banach algebras and perfectoid algebras are revisited beforehand.

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On Tate’s Acyclicity and uniformity of Berkovich spectra and adic spectra

Mihara¹

2016

Isr. J. Math.

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We construct a non-sheafy uniform Banach algebra such that a rational localisation of the Berkovich spectrum does not preserve the uniformity. We also construct uniform affinoid rings in the sense of Roland Huber such that rational localisations of the adic spectra do not preserve the uniformity. One of them is an example of a non-sheafy uniform affinoid ring. We introduce the notion of local uniformity instead, and prove that the local uniformity implies the sheaf condition.

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Characterisation of the Berkovich spectrum of the Banach algebra of bounded continuous functions

Cited by 6 publications

References 4 publications

Ortho-sets and Gelfand spectra

Ortho-sets and Gelfand spectra

Le lemme d’Abhyankar perfectoide

On Tate’s Acyclicity and uniformity of Berkovich spectra and adic spectra

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