MathlibAnnex.Analysis.CStarAlgebra.irreducibleSpectrumHomeomorphOfGeometricStructure
def
Transports the irreducible spectrum along a homeomorphism of geometric structure spaces.
Statement
Let be complex C*-algebras, possibly without identities. Suppose
is a homeomorphism of their geometric structure closure spaces. Both and its inverse are continuous for the specified closure operators, with continuity tested on every subset.
The map determines an ordinary homeomorphism
between the independent irreducible spectra with their kernel-induced ordinary topologies.
Definition
For , denotes kernels of extreme points of the closed unit ball of the Banach dual, with real convexity, and . The extreme-functional kernel characterization gives its geometric identification. Let be the component space and the specified spectral homeomorphism.
The induced component homeomorphism is , with and inverse . Full closure continuity gives ordinary continuity for the specified topologies.
The defining map and its inverse, with composition read from right to left, are
Thus the three stages go from to , then , then ; the last stage uses the inverse of .
Assumptions
The source requires nonzero complex C*-algebras , each with its compatible order, and the specified full closure-space homeomorphism . The underlying Hilbert spaces of irreducible representations are not fixed as additional inputs; the independent spectra use their pure-state GNS presentation. The map has domain , whose points are subsets of ; it is not a map and no linear or multiplicative property on algebra elements is assumed.
Conclusion
The output is this specific ordinary homeomorphism between the independent irreducible spectra. It uses the given full closure-space homeomorphism and the two fixed spectral comparisons. The input acts on geometric points, which are subsets of algebras, not on algebra elements or representation operators.
Main citations
Exact full definition:
irreducibleSpectrumHomeomorphOfGeometricStructure.The ordinary spectral comparison: the two endpoint homeomorphisms and their exact topologies.
Normed/GeometricStructure/SpectrumMap.lean,
geometricSpectrumHomeomorph,geometricSpectrumHomeomorph_val,geometricSpectrumHomeomorph_toGeometricSpectrum, andimage_geometricSpectrumClosure: the induced component map and full closure transport.The concrete geometric-spectrum closure: the component-intersection closure used by this construction.
Ryotaro Tanaka, arXiv:2504.05551v1, Corollary 5.13: the spectral consequence of a geometric-structure homeomorphism.
Lean source signature (exact)
The full definition below is copied from the pinned source.
def irreducibleSpectrumHomeomorphOfGeometricStructure : IrreducibleSpectrum A ≃ₜ IrreducibleSpectrum B :=
(irreducibleSpectrumHomeomorph A).trans
(((NormedSpace.geometricSpectrumHomeomorph e).toTopologicalHomeomorph
(isTopologizable_geometricSpectrumClosure A) (isTopologizable_geometricSpectrumClosure B)).trans
(irreducibleSpectrumHomeomorph B).symm)
| In the source | Mathematical meaning |
|---|---|
| The two ambient algebras and independent universesExact declaration | The independently specified nonzero complex C*-algebras with compatible orders; no units are required. |
Exact declaratione in the surrounding scope |
The supplied full closure-space homeomorphism of geometric points, not a map on algebra elements. |
Exact declaration(irreducibleSpectrumHomeomorph A).trans |
The first stage is . |
Exact declarationNormedSpace.geometricSpectrumHomeomorph e |
The middle stage is the component map with its inverse. |
Exact declaration.toTopologicalHomeomorph |
The existing topologizability results give ordinary continuity of that same component map. |
Exact declaration(irreducibleSpectrumHomeomorph B).symm |
The final stage is , giving the displayed composite . |
Exact surrounding source context (separate excerpts)
Source lines 27–27:
variable {A : Type u} [NonUnitalCStarAlgebra A] [PartialOrder A] [StarOrderedRing A] [Nontrivial A]
Source lines 28–28:
variable {B : Type v} [NonUnitalCStarAlgebra B] [PartialOrder B] [StarOrderedRing B] [Nontrivial B]
Source lines 29–30:
variable (e : ClosureSpace.Homeomorph (NormedSpace.geometricStructureClosure ℂ A)
(NormedSpace.geometricStructureClosure ℂ B))
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