Closure spaces and norm geometry
Begin with hull–kernel closure on the kernel families obtained from faces. Closure need not be topological; finite continuity tests finite subsets, whereas continuity tests every subset.
From unit-ball geometry to C*-algebras
The geometric structure space connects faces and supporting functionals of the norm unit ball with irreducible representations and primitive ideals. The project follows those connections to normal parts, compact representations, decomposition and rigidity.
Read the mathematical overview · Research project · Reader Project PDF
Assumptions and limits
The geometric construction begins with real or complex normed spaces; the C*-algebra comparisons retain their complex, possibly nonunital hypotheses. The zero space has empty geometric structure space. Individual results retain their nonzero, closedness, irreducibility and continuity assumptions.
- For the elementary-model rigidity result, at least one algebra is *-isomorphic to the whole K(H) of a complex Hilbert space.
- For finite-dimensional rigidity, at least one algebra is finite-dimensional over the complex numbers.
- For bounded-operator rigidity, at least one algebra is *-isomorphic to the whole B(H) of a complex Hilbert space.
These conclusions equate the existence of a *-isomorphism with the existence of a GSS homeomorphism. They do not assert that a specified nonlinear homeomorphism is induced by that isomorphism. The exact manuscripts govern zero-algebra and zero-model cases. An abstract type-I factor is not substituted for the concrete B(H) model assumption.
GSS rigidity when either algebra has an elementary model · GSS rigidity when either algebra is finite-dimensional · GSS rigidity from a full bounded-operator model
Mathematical routes
These six groups follow the accepted reading guide. Choosing a group includes its prerequisite closure. The displayed levels always retain the full selected scope.
Faces, pure states and spectra
Faces and extreme dual functionals lead to pure states and representation components. Unitary-equivalence classes of irreducible representations and their kernels remain different objects; the two spectral comparisons use their respective hull–kernel topologies.
Normal parts and semilinear structure
Pass from a representation component to its coefficient-kernel normal part with relative closure. Projective parametrization is a bijection. The same map, its inverse, the scalar automorphism and global orientation retain the hypotheses specified in each manuscript; full continuity requires extra conditions.
The central characterization
The faithful, fully normal representation condition and continuous finite conjugation meet in the elementary-algebra characterization. Retain the same GNS model, equality of the compact-operator image, singleton condition, and two-sided conjugate continuity.
From components to classification
Use compact comparisons to read representation-theoretic conditions. Compact containment differs from equality; prescribed dimension differs from an upper bound. Transport through a specified homeomorphism retains that specified map.
Compact representations, decomposition and rigidity
Faithful compact representation leads to the canonical c₀-decomposition indexed by minimal nonzero closed ideals. Its compact-operator summands are distinct from compact operators on a Hilbert direct sum. The final rigidity results assert existence of an isomorphism, with the different model hypotheses below.
Boundary inputs
The mathematical background below follows reading A. Links identify fixed source entry points using these tools; they are not new graph edges or claims of new independent proofs.
- Hahn–Banach — Norming and extension functionals supply the connection from norm geometry to states. Fixed source entry
- Krein–Milman and weak* compactness — The weak* support family and extreme dual ball connect maximal faces to their supports. Fixed source entry
- Pure states and GNS representations — The pure-state construction and irreducibility comparison give representation classes, distinct from kernels. Fixed source entry
- Simultaneous interpolation — Simultaneous control of representation vectors enters the compact-representation results. Fixed source entry
- Quotient by the kernel — Representation images are compared through the quotient by the representation kernel. Fixed source entry
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Formal dependency graph
Arrows go from prerequisites to their users. Search and focus dim the same excluded nodes; graph links locate the matching tile safely. Levels and edges are unchanged from the exact saved declaration graph.
Dependency-first reading route
Each tile opens the accepted manuscript presentation. Focus includes its target and all selected prerequisites; Used by is separate. Selection keys below are editorial locators, not assigned native Card IDs.
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Level 0 (9 Cards)
Closure spaces before topology
Editorial locator 1 · level 0
Prerequisites and uses
Prerequisites: None in this selected scope
Used by: Finite-homeomorphisms of closure spaces, The hull–kernel closure construction, Linked points in a closure space
Closed hyperplane images force semilinear regularity
Derives scalar regularity and bounded invertibility from closed images of closed hyperplanes.
Editorial locator 38 · level 0
Prerequisites and uses
Prerequisites: None in this selected scope
Used by: Forward continuity of normal parts preserves Hilbert-basis cardinality, The semilinear factors and their inverses are continuous, Each selected scalar automorphism is identity or conjugation
The dependent c₀-direct sum
Names the algebra of bounded families that vanish at infinity.
Editorial locator c0-direct-sum · level 0
Prerequisites and uses
Prerequisites: None in this selected scope
Used by: The specified equivalence from a dense orthogonal ideal sum, The dependent c₀-sum of compact-operator algebras
The irreducible spectrum as pure-GNS representation classes
Defines the irreducible spectrum using unitary equivalence classes of pure-state GNS representations.
Editorial locator irreducible-spectrum · level 0
Prerequisites and uses
Prerequisites: None in this selected scope
Used by: The irreducible spectrum and the geometric spectrum
A compact-operator model for a representation
Packages faithfulness together with equality of the represented image and the compact operators.
Editorial locator is-compact-operator-model · level 0
Prerequisites and uses
Prerequisites: None in this selected scope
Used by: The Hilbert space of a minimal-ideal compact model, A specified GSS homeomorphism preserves compact-operator equality, Elementary C*-algebras from singleton spectrum and finite conjugation, Compact inclusion in every irreducible image, Irreducible dimension equal to a fixed number, Irreducible dimension at most a fixed number, Compact equality in every irreducible image, GSS rigidity from a full bounded-operator model
The primitive spectrum as representation kernels
Defines the primitive spectrum as the set of pure-state GNS representation kernels.
Editorial locator primitive-spectrum · level 0
Prerequisites and uses
Prerequisites: None in this selected scope
Used by: Existence of a faithful compact representation, The same reduced atomic representation and its exact diagonal range, The irreducible spectrum and the geometric spectrum, The primitive spectrum and the primitive geometric spectrum
An extreme dual functional has a pure-state unitary translate
Expresses each extreme dual functional as a unitary translate of a pure state.
Editorial locator 15 · level 0
Prerequisites and uses
Prerequisites: None in this selected scope
Used by: Maximal faces of the C*-algebra unit ball and their unique support
Unions of kernels of supporting functionals
Editorial locator 3 · level 0
Prerequisites and uses
Prerequisites: None in this selected scope
Used by: The geometric structure family is the family of extreme-functional kernels, Points of the geometric structure space, The primitive geometric spectrum
Finite projective incidence determines a semilinear equivalence
Realizes a finite-incidence-preserving projective bijection by a semilinear equivalence.
Editorial locator 35 · level 0
Prerequisites and uses
Prerequisites: None in this selected scope
Used by: Forward continuity of normal parts preserves Hilbert-basis cardinality, The semilinear factors and their inverses are continuous, Semilinear factors recover all coefficient kernels, Each selected scalar automorphism is identity or conjugation
Level 1 (8 Cards)
Finite-homeomorphisms of closure spaces
Editorial locator 12 · level 1
Prerequisites and uses
Prerequisites: Closure spaces before topology
Used by: A continuous finite comparison with a compact-operator GSS, Continuous finite conjugation relative to a class of source spaces, Faithful normal-part transport through a specified GSS homeomorphism, Finite-homeomorphisms detect finite dimension, A geometric-structure homeomorphism transports the irreducible spectrum, Semilinear vector changes preserve finite normal-part closures, Semilinear factors recover all coefficient kernels
The hull–kernel closure construction
Editorial locator 2 · level 1
Prerequisites and uses
Prerequisites: Closure spaces before topology
Used by: The primitive geometric spectrum, The geometric structure closure
Linked points in a closure space
Editorial locator 7 · level 1
Prerequisites and uses
Prerequisites: Closure spaces before topology
Used by: Pairwise linked subsets of a closure space, Components generated by linked pairs, Existence of a faithful compact representation, Components of coefficient kernels encode unitary equivalence, The same reduced atomic representation and its exact diagonal range, Linked normal kernels share a projective coordinate
The specified equivalence from a dense orthogonal ideal sum
Identifies an algebra with the c₀-sum of a dense orthogonal family of closed ideals.
Editorial locator ideal-c0-sum-equivalence · level 1
Prerequisites and uses
Prerequisites: The dependent c₀-direct sum
The dependent c₀-sum of compact-operator algebras
Names the c₀-sum of a family of compact-operator algebras.
Editorial locator compact-operator-c0-sum · level 1
Prerequisites and uses
Prerequisites: The dependent c₀-direct sum
The Hilbert space of a minimal-ideal compact model
Names the cyclic Hilbert space chosen for a minimal closed ideal.
Editorial locator compact-model-space · level 1
Prerequisites and uses
Prerequisites: A compact-operator model for a representation
Maximal faces of the C*-algebra unit ball and their unique support
Characterizes maximal proper faces of a C*-algebra unit ball by extreme dual functionals.
Editorial locator 14 · level 1
Prerequisites and uses
Prerequisites: An extreme dual functional has a pure-state unitary translate
Used by: The geometric structure family is the family of extreme-functional kernels
Points of the geometric structure space
Editorial locator 4 · level 1
Prerequisites and uses
Prerequisites: Unions of kernels of supporting functionals
Used by: The coefficient-kernel normal part, The geometric structure closure
Level 2 (6 Cards)
Pairwise linked subsets of a closure space
Editorial locator 13 · level 2
Prerequisites and uses
Prerequisites: Linked points in a closure space
Components generated by linked pairs
Editorial locator 8 · level 2
Compact coordinates from the inverse ideal-sum map
Expresses an algebra in compact-operator coordinates through its orthogonal ideal decomposition.
Editorial locator compact-c0-sum-equivalence · level 2
Prerequisites and uses
Prerequisites: The specified equivalence from a dense orthogonal ideal sum, The dependent c₀-sum of compact-operator algebras
The geometric structure family is the family of extreme-functional kernels
Identifies the geometric structure family with kernels of extreme dual functionals.
Editorial locator 16 · level 2
Prerequisites and uses
Prerequisites: Maximal faces of the C*-algebra unit ball and their unique support, Unions of kernels of supporting functionals
Used by: A representation component is the full coefficient-kernel family, Existence of a faithful compact representation, The geometric structure of a representation image as a component closure, Components of coefficient kernels encode unitary equivalence, The same reduced atomic representation and its exact diagonal range, Two-sided continuity is exact operator equality, Continuity is an operator-inclusion condition, Every geometric-spectrum component contains a pure-state kernel, The geometric structure family is the family of pure-state translate kernels, Linked normal kernels share a projective coordinate, The normal part as a product of projective spaces, Semilinear vector changes preserve finite normal-part closures
The coefficient-kernel normal part
Defines the normal part by vector coefficient kernels that are geometric points.
Editorial locator 27 · level 2
Prerequisites and uses
Prerequisites: Points of the geometric structure space
Used by: A coefficient-kernel component equals the normal part, Two-sided continuity is exact operator equality, Continuity is an operator-inclusion condition, A singleton irreducible spectrum is witnessed by a faithful, fully normal representation, Linked normal kernels share a projective coordinate, The normal part as a product of projective spaces, Semilinear vector changes preserve finite normal-part closures
The geometric structure closure
Editorial locator 5 · level 2
Prerequisites and uses
Prerequisites: The hull–kernel closure construction, Points of the geometric structure space
Used by: A full homeomorphism to a compact-operator GSS, A continuous finite comparison with a compact-operator GSS, A finite-dimensional compact model with a dimension condition, A representation component is the full coefficient-kernel family, Existence of a faithful compact representation, Components of coefficient kernels encode unitary equivalence, The same reduced atomic representation and its exact diagonal range, Two-sided continuity is exact operator equality, Continuity is an operator-inclusion condition, Linked normal kernels share a projective coordinate, Semilinear vector changes preserve finite normal-part closures, The geometric spectrum, The closure-space homeomorphism induced by an orthogonality equivalence
Level 3 (16 Cards)
A full homeomorphism to a compact-operator GSS
Requires a full homeomorphism to the geometric structure space of a compact-operator algebra.
Editorial locator 47 · level 3
A continuous finite comparison with a compact-operator GSS
Requires a finite homeomorphism to a compact-operator geometric space whose forward map is fully continuous.
Editorial locator 46 · level 3
Prerequisites and uses
Prerequisites: Finite-homeomorphisms of closure spaces, The geometric structure closure
Continuous finite conjugation relative to a class of source spaces
Defines when finite conjugation is continuous in both directions.
Editorial locator 44 · level 3
Prerequisites and uses
Prerequisites: Finite-homeomorphisms of closure spaces, Components generated by linked pairs
Used by: Elementary C*-algebras from singleton spectrum and finite conjugation
A finite-dimensional compact model with a dimension condition
Requires a finite-dimensional compact model satisfying a specified condition on its dimension.
Editorial locator 48 · level 3
Prerequisites and uses
Prerequisites: The geometric structure closure
Used by: Irreducible dimension equal to a fixed number, Irreducible dimension at most a fixed number
The canonical compact c₀-decomposition
Defines the compact-operator c₀-decomposition associated with a faithful compact-valued representation.
Editorial locator 57 · level 3
Prerequisites and uses
Prerequisites: Compact coordinates from the inverse ideal-sum map, The Hilbert space of a minimal-ideal compact model
Used by: Existence of a faithful compact representation, The same reduced atomic representation and its exact diagonal range, GSS rigidity when either algebra has an elementary model, One-sided compact representability gives GSS rigidity, GSS rigidity when either algebra is finite-dimensional
A representation component is the full coefficient-kernel family
Describes each irreducible representation component by all of its nonzero vector coefficients.
Editorial locator 19 · level 3
Prerequisites and uses
Prerequisites: Components generated by linked pairs, The geometric structure family is the family of extreme-functional kernels, The geometric structure closure
Used by: A coefficient-kernel component equals the normal part, The intersection of a pure-state component is its GNS representation kernel
Components of coefficient kernels encode unitary equivalence
Characterizes equality of coefficient-kernel components by unitary equivalence of representations.
Editorial locator 18 · level 3
Prerequisites and uses
Prerequisites: Linked points in a closure space, The geometric structure family is the family of extreme-functional kernels, The geometric structure closure
Used by: The irreducible spectrum and the geometric spectrum
Two-sided continuity is exact operator equality
Characterizes two-sided continuity of a normal-part equivalence by equality of operator sets.
Editorial locator 32 · level 3
Prerequisites and uses
Prerequisites: The geometric structure family is the family of extreme-functional kernels, The coefficient-kernel normal part, The geometric structure closure
Used by: A specified GSS homeomorphism preserves compact-operator equality, Classifying the specified normal-part equivalence with the selected bounded lifts, Elementary C*-algebras from singleton spectrum and finite conjugation, Compact inclusion in every irreducible image, Compact equality in every irreducible image, GSS rigidity from a full bounded-operator model, GSS rigidity when either algebra has an elementary model, One-sided compact representability gives GSS rigidity, GSS rigidity when either algebra is finite-dimensional
Continuity is an operator-inclusion condition
Characterizes continuity of a normal-part map by an inclusion of operator sets.
Editorial locator 31 · level 3
Prerequisites and uses
Prerequisites: The geometric structure family is the family of extreme-functional kernels, The coefficient-kernel normal part, The geometric structure closure
Used by: A specified GSS homeomorphism preserves compact-operator equality, Classifying the specified normal-part equivalence with the selected bounded lifts, Elementary C*-algebras from singleton spectrum and finite conjugation, Compact inclusion in every irreducible image, Compact equality in every irreducible image, GSS rigidity from a full bounded-operator model, GSS rigidity when either algebra has an elementary model, One-sided compact representability gives GSS rigidity, GSS rigidity when either algebra is finite-dimensional
The geometric structure family is the family of pure-state translate kernels
Identifies geometric points with kernels of unitary translates of pure states.
Editorial locator 17 · level 3
Prerequisites and uses
Prerequisites: The geometric structure family is the family of extreme-functional kernels
Used by: None in this selected scope
Linked normal kernels share a projective coordinate
Characterizes linked normal kernels by agreement of one projective coordinate.
Editorial locator 33 · level 3
Prerequisites and uses
Prerequisites: Linked points in a closure space, The geometric structure family is the family of extreme-functional kernels, The coefficient-kernel normal part, The geometric structure closure
Used by: Finite-homeomorphisms detect finite dimension, Maximal linked families are projective fibers, Semilinear factors recover all coefficient kernels
The normal part as a product of projective spaces
Parametrizes the normal part by an ordered pair of complex projective lines.
Editorial locator 29 · level 3
Prerequisites and uses
Prerequisites: The geometric structure family is the family of extreme-functional kernels, The coefficient-kernel normal part
Used by: Finite-homeomorphisms detect finite dimension, Maximal linked families are projective fibers, Semilinear factors recover all coefficient kernels
Semilinear vector changes preserve finite normal-part closures
Defines a finite homeomorphism of normal parts from two semilinear vector isomorphisms.
Editorial locator 30 · level 3
Prerequisites and uses
Prerequisites: Finite-homeomorphisms of closure spaces, The geometric structure family is the family of extreme-functional kernels, The coefficient-kernel normal part, The geometric structure closure
Used by: Elementary C*-algebras from singleton spectrum and finite conjugation, Compact inclusion in every irreducible image, GSS rigidity from a full bounded-operator model
The geometric spectrum
Editorial locator 9 · level 3
The primitive geometric spectrum
Editorial locator 10 · level 3
Prerequisites and uses
Prerequisites: The hull–kernel closure construction, Components generated by linked pairs, Unions of kernels of supporting functionals
The closure-space homeomorphism induced by an orthogonality equivalence
Editorial locator 6 · level 3
Prerequisites and uses
Prerequisites: The geometric structure closure
Used by: Finite-homeomorphisms detect finite dimension, Semilinear factors recover all coefficient kernels
Level 4 (9 Cards)
Existence of a faithful compact representation
Characterizes existence of a faithful compact action by compact irreducible images and discrete primitive spectrum.
Editorial locator 55 · level 4
Prerequisites and uses
Prerequisites: Linked points in a closure space, The canonical compact c₀-decomposition, The primitive spectrum as representation kernels, The geometric structure family is the family of extreme-functional kernels, The geometric structure closure
Used by: None in this selected scope
The geometric structure of a representation image as a component closure
Identifies the geometric structure of a representation image with a component closure.
Editorial locator 25 · level 4
Prerequisites and uses
Prerequisites: The geometric structure family is the family of extreme-functional kernels, The geometric spectrum
Used by: Faithful normal-part transport through a specified GSS homeomorphism, A singleton irreducible spectrum is witnessed by a faithful, fully normal representation, Elementary C*-algebras from singleton spectrum and finite conjugation, Compact inclusion in every irreducible image, Irreducible dimension equal to a fixed number, Irreducible dimension at most a fixed number, Compact equality in every irreducible image, GSS rigidity when either algebra has an elementary model, One-sided compact representability gives GSS rigidity, GSS rigidity when either algebra is finite-dimensional
The same reduced atomic representation and its exact diagonal range
Collects the simultaneous properties and exact diagonal range of the same reduced atomic representation.
Editorial locator 56 · level 4
Prerequisites and uses
Prerequisites: Linked points in a closure space, The canonical compact c₀-decomposition, The primitive spectrum as representation kernels, The geometric structure family is the family of extreme-functional kernels, The geometric structure closure
Used by: None in this selected scope
A coefficient-kernel component equals the normal part
Identifies the normal part with one component of the ambient geometric structure space.
Editorial locator 28 · level 4
Prerequisites and uses
Prerequisites: A representation component is the full coefficient-kernel family, The coefficient-kernel normal part
Used by: Faithful normal-part transport through a specified GSS homeomorphism
Every geometric-spectrum component contains a pure-state kernel
Places the kernel of a pure state in every geometric-spectrum component.
Editorial locator 20 · level 4
Prerequisites and uses
Prerequisites: The geometric structure family is the family of extreme-functional kernels, The geometric spectrum
Used by: None in this selected scope
Finite-homeomorphisms detect finite dimension
Recovers finite dimensionality and equal dimensions from a finite-homeomorphism of normal parts.
Editorial locator 36 · level 4
Prerequisites and uses
Prerequisites: Finite-homeomorphisms of closure spaces, Linked normal kernels share a projective coordinate, The normal part as a product of projective spaces, The closure-space homeomorphism induced by an orthogonality equivalence
Used by: Irreducible dimension equal to a fixed number, Irreducible dimension at most a fixed number, Forward continuity of normal parts preserves Hilbert-basis cardinality, The semilinear factors and their inverses are continuous, Each selected scalar automorphism is identity or conjugation
Maximal linked families are projective fibers
Characterizes maximal linked subsets of the normal part as projective fibers.
Editorial locator 34 · level 4
Prerequisites and uses
Prerequisites: Pairwise linked subsets of a closure space, Linked normal kernels share a projective coordinate, The normal part as a product of projective spaces
Used by: None in this selected scope
Semilinear factors recover all coefficient kernels
Evaluates a normal-part finite-homeomorphism and its inverse using the same semilinear factors.
Editorial locator 37 · level 4
Prerequisites and uses
Prerequisites: Finite-homeomorphisms of closure spaces, Linked normal kernels share a projective coordinate, The normal part as a product of projective spaces, The closure-space homeomorphism induced by an orthogonality equivalence, Finite projective incidence determines a semilinear equivalence
Used by: A specified GSS homeomorphism preserves compact-operator equality, Classifying the specified normal-part equivalence with the selected bounded lifts, Elementary C*-algebras from singleton spectrum and finite conjugation, Compact inclusion in every irreducible image, Compact equality in every irreducible image, GSS rigidity from a full bounded-operator model, GSS rigidity when either algebra has an elementary model, One-sided compact representability gives GSS rigidity, GSS rigidity when either algebra is finite-dimensional
From components to their common kernel
Editorial locator 11 · level 4
Prerequisites and uses
Prerequisites: The geometric spectrum, The primitive geometric spectrum
Used by: The intersection of a pure-state component is its GNS representation kernel
Level 5 (5 Cards)
Faithful normal-part transport through a specified GSS homeomorphism
Transports the normal part through one faithful pure-GNS representation and the same preimage map.
Editorial locator 53 · level 5
Prerequisites and uses
Prerequisites: Finite-homeomorphisms of closure spaces, The geometric structure of a representation image as a component closure, A coefficient-kernel component equals the normal part
Used by: A specified GSS homeomorphism preserves compact-operator equality, GSS rigidity from a full bounded-operator model
The intersection of a pure-state component is its GNS representation kernel
Identifies a pure-state component intersection with the kernel of its GNS representation.
Editorial locator 21 · level 5
Prerequisites and uses
Prerequisites: A representation component is the full coefficient-kernel family, From components to their common kernel
Used by: Primitive representation kernels are exactly geometric component intersections
Forward continuity of normal parts preserves Hilbert-basis cardinality
Preserves Hilbert-basis cardinality under forward full continuity of a finite normal-part homeomorphism.
Editorial locator 42 · level 5
Prerequisites and uses
Prerequisites: Closed hyperplane images force semilinear regularity, Finite-homeomorphisms detect finite dimension, Finite projective incidence determines a semilinear equivalence
Used by: GSS rigidity from a full bounded-operator model, GSS rigidity when either algebra has an elementary model, One-sided compact representability gives GSS rigidity, GSS rigidity when either algebra is finite-dimensional
The semilinear factors and their inverses are continuous
Establishes continuity of the fixed semilinear factors and their inverses.
Editorial locator 40 · level 5
Prerequisites and uses
Prerequisites: Closed hyperplane images force semilinear regularity, Finite-homeomorphisms detect finite dimension, Finite projective incidence determines a semilinear equivalence
Used by: A specified GSS homeomorphism preserves compact-operator equality, Classifying the specified normal-part equivalence with the selected bounded lifts, Elementary C*-algebras from singleton spectrum and finite conjugation, Compact inclusion in every irreducible image, Compact equality in every irreducible image, GSS rigidity from a full bounded-operator model, GSS rigidity when either algebra has an elementary model, One-sided compact representability gives GSS rigidity, GSS rigidity when either algebra is finite-dimensional
Each selected scalar automorphism is identity or conjugation
Identifies each scalar automorphism of the fixed semilinear factors as identity or conjugation.
Editorial locator 39 · level 5
Prerequisites and uses
Prerequisites: Closed hyperplane images force semilinear regularity, Finite-homeomorphisms detect finite dimension, Finite projective incidence determines a semilinear equivalence
Used by: A specified GSS homeomorphism preserves compact-operator equality, Classifying the specified normal-part equivalence with the selected bounded lifts, Elementary C*-algebras from singleton spectrum and finite conjugation, Compact inclusion in every irreducible image, Compact equality in every irreducible image, GSS rigidity from a full bounded-operator model, GSS rigidity when either algebra has an elementary model, One-sided compact representability gives GSS rigidity, GSS rigidity when either algebra is finite-dimensional
Level 6 (2 Cards)
Classifying the specified normal-part equivalence with the selected bounded lifts
Classifies the given normal-part homeomorphism using its fixed factors and an operator image equality.
Editorial locator 41 · level 6
Prerequisites and uses
Prerequisites: Two-sided continuity is exact operator equality, Continuity is an operator-inclusion condition, The semilinear factors and their inverses are continuous, Semilinear factors recover all coefficient kernels, Each selected scalar automorphism is identity or conjugation
Used by: None in this selected scope
Primitive representation kernels are exactly geometric component intersections
Identifies primitive representation kernels with intersections of geometric components.
Editorial locator 22 · level 6
Prerequisites and uses
Prerequisites: The intersection of a pure-state component is its GNS representation kernel
Used by: The irreducible spectrum and the geometric spectrum, The primitive spectrum and the primitive geometric spectrum
Level 7 (2 Cards)
The irreducible spectrum and the geometric spectrum
Identifies irreducible representation classes with components of the geometric structure space.
Editorial locator 24 · level 7
Prerequisites and uses
Prerequisites: The irreducible spectrum as pure-GNS representation classes, Components of coefficient kernels encode unitary equivalence, The primitive spectrum as representation kernels, Primitive representation kernels are exactly geometric component intersections
Used by: A specified GSS homeomorphism preserves compact-operator equality, A singleton irreducible spectrum is witnessed by a faithful, fully normal representation, Elementary C*-algebras from singleton spectrum and finite conjugation, Compact inclusion in every irreducible image, Irreducible dimension equal to a fixed number, Irreducible dimension at most a fixed number, Compact equality in every irreducible image, A geometric-structure homeomorphism transports the irreducible spectrum, GSS rigidity from a full bounded-operator model
The primitive spectrum and the primitive geometric spectrum
Identifies the primitive ideal spectrum with the primitive geometric spectrum.
Editorial locator 23 · level 7
Prerequisites and uses
Prerequisites: The primitive spectrum as representation kernels, Primitive representation kernels are exactly geometric component intersections
Used by: GSS rigidity when either algebra has an elementary model, One-sided compact representability gives GSS rigidity, GSS rigidity when either algebra is finite-dimensional
Level 8 (9 Cards)
A specified GSS homeomorphism preserves compact-operator equality
Preserves equality with the compact operators under the specified full geometric homeomorphism.
Editorial locator 54 · level 8
Prerequisites and uses
Prerequisites: A compact-operator model for a representation, Two-sided continuity is exact operator equality, Continuity is an operator-inclusion condition, Faithful normal-part transport through a specified GSS homeomorphism, The irreducible spectrum and the geometric spectrum, The semilinear factors and their inverses are continuous, Semilinear factors recover all coefficient kernels, Each selected scalar automorphism is identity or conjugation
Used by: None in this selected scope
A singleton irreducible spectrum is witnessed by a faithful, fully normal representation
Characterizes a singleton irreducible spectrum by one faithful representation with a full normal part.
Editorial locator 43 · level 8
Prerequisites and uses
Prerequisites: The geometric structure of a representation image as a component closure, The irreducible spectrum and the geometric spectrum, The coefficient-kernel normal part
Used by: None in this selected scope
Elementary C*-algebras from singleton spectrum and finite conjugation
Characterizes a pure-GNS compact model by singleton spectrum and continuous finite conjugation.
Editorial locator 45 · level 8
Prerequisites and uses
Prerequisites: Continuous finite conjugation relative to a class of source spaces, A compact-operator model for a representation, The geometric structure of a representation image as a component closure, Two-sided continuity is exact operator equality, Continuity is an operator-inclusion condition, The irreducible spectrum and the geometric spectrum, Semilinear vector changes preserve finite normal-part closures, The semilinear factors and their inverses are continuous, Semilinear factors recover all coefficient kernels, Each selected scalar automorphism is identity or conjugation
Used by: None in this selected scope
Compact inclusion in every irreducible image
Characterizes compact containment in every irreducible image by forward-continuous component comparisons.
Editorial locator 49 · level 8
Prerequisites and uses
Prerequisites: A continuous finite comparison with a compact-operator GSS, A compact-operator model for a representation, The geometric structure of a representation image as a component closure, Two-sided continuity is exact operator equality, Continuity is an operator-inclusion condition, The irreducible spectrum and the geometric spectrum, Semilinear vector changes preserve finite normal-part closures, The semilinear factors and their inverses are continuous, Semilinear factors recover all coefficient kernels, Each selected scalar automorphism is identity or conjugation
Used by: None in this selected scope
Irreducible dimension equal to a fixed number
Characterizes one fixed irreducible dimension by finite-dimensional compact component models.
Editorial locator 51 · level 8
Prerequisites and uses
Prerequisites: A finite-dimensional compact model with a dimension condition, A compact-operator model for a representation, The geometric structure of a representation image as a component closure, Finite-homeomorphisms detect finite dimension, The irreducible spectrum and the geometric spectrum
Used by: None in this selected scope
Irreducible dimension at most a fixed number
Characterizes a uniform finite dimension bound by full compact component models.
Editorial locator 52 · level 8
Prerequisites and uses
Prerequisites: A finite-dimensional compact model with a dimension condition, A compact-operator model for a representation, The geometric structure of a representation image as a component closure, Finite-homeomorphisms detect finite dimension, The irreducible spectrum and the geometric spectrum
Used by: None in this selected scope
Compact equality in every irreducible image
Characterizes equality with the compact operators by full component homeomorphisms.
Editorial locator 50 · level 8
Prerequisites and uses
Prerequisites: A full homeomorphism to a compact-operator GSS, A compact-operator model for a representation, The geometric structure of a representation image as a component closure, Two-sided continuity is exact operator equality, Continuity is an operator-inclusion condition, The irreducible spectrum and the geometric spectrum, The semilinear factors and their inverses are continuous, Semilinear factors recover all coefficient kernels, Each selected scalar automorphism is identity or conjugation
Used by: None in this selected scope
A geometric-structure homeomorphism transports the irreducible spectrum
Transports the irreducible spectrum along a homeomorphism of geometric structure spaces.
Editorial locator 26 · level 8
Prerequisites and uses
Prerequisites: Finite-homeomorphisms of closure spaces, The irreducible spectrum and the geometric spectrum
Used by: GSS rigidity when either algebra has an elementary model, One-sided compact representability gives GSS rigidity, GSS rigidity when either algebra is finite-dimensional
GSS rigidity from a full bounded-operator model
Obtains algebraic rigidity from an actual full bounded-operator model on either side.
Editorial locator 61 · level 8
Prerequisites and uses
Prerequisites: A compact-operator model for a representation, Two-sided continuity is exact operator equality, Continuity is an operator-inclusion condition, Faithful normal-part transport through a specified GSS homeomorphism, Forward continuity of normal parts preserves Hilbert-basis cardinality, The irreducible spectrum and the geometric spectrum, Semilinear vector changes preserve finite normal-part closures, The semilinear factors and their inverses are continuous, Semilinear factors recover all coefficient kernels, Each selected scalar automorphism is identity or conjugation
Used by: None in this selected scope
Level 9 (3 Cards)
GSS rigidity when either algebra has an elementary model
Obtains algebraic rigidity from an elementary compact-operator model on either side.
Editorial locator 59 · level 9
Prerequisites and uses
Prerequisites: A full homeomorphism to a compact-operator GSS, The canonical compact c₀-decomposition, The geometric structure of a representation image as a component closure, Two-sided continuity is exact operator equality, Continuity is an operator-inclusion condition, Forward continuity of normal parts preserves Hilbert-basis cardinality, A geometric-structure homeomorphism transports the irreducible spectrum, The semilinear factors and their inverses are continuous, Semilinear factors recover all coefficient kernels, Each selected scalar automorphism is identity or conjugation, The primitive spectrum and the primitive geometric spectrum
Used by: None in this selected scope
One-sided compact representability gives GSS rigidity
Obtains algebraic rigidity when either side admits a faithful compact action.
Editorial locator 58 · level 9
Prerequisites and uses
Prerequisites: A full homeomorphism to a compact-operator GSS, The canonical compact c₀-decomposition, The geometric structure of a representation image as a component closure, Two-sided continuity is exact operator equality, Continuity is an operator-inclusion condition, Forward continuity of normal parts preserves Hilbert-basis cardinality, A geometric-structure homeomorphism transports the irreducible spectrum, The semilinear factors and their inverses are continuous, Semilinear factors recover all coefficient kernels, Each selected scalar automorphism is identity or conjugation, The primitive spectrum and the primitive geometric spectrum
Used by: None in this selected scope
GSS rigidity when either algebra is finite-dimensional
Obtains algebraic rigidity when either algebra is finite-dimensional.
Editorial locator 60 · level 9
Prerequisites and uses
Prerequisites: A full homeomorphism to a compact-operator GSS, The canonical compact c₀-decomposition, The geometric structure of a representation image as a component closure, Two-sided continuity is exact operator equality, Continuity is an operator-inclusion condition, Forward continuity of normal parts preserves Hilbert-basis cardinality, A geometric-structure homeomorphism transports the irreducible spectrum, The semilinear factors and their inverses are continuous, Semilinear factors recover all coefficient kernels, Each selected scalar automorphism is identity or conjugation, The primitive spectrum and the primitive geometric spectrum
Used by: None in this selected scope
Source and qualification
Source GSS v0.5.0 is fixed at ac405356bb98afed954da9b9c795d23cc8d5d149. The legacy published LFH v0.4.0 / R09 binding remains separate. Checkpoint-assisted qualification is inherited from run 38063627833, attempt 1. Independent clean single-runner qualification is NOT_RUN.
This edition contains 69 exact Card references, their expositions and reading guides. Website edition, source releases and Card revisions are separate; whole-manuscript correspondence is not asserted.
Original paper, fixed v1 · Original v1 PDF · Saved graph and witnesses · Version record