The primitive spectrum as representation kernels

MathlibAnnex.Analysis.CStarAlgebra.PrimitiveSpectrum

def

Defines the primitive spectrum as the set of pure-state GNS representation kernels.

Statement

Let be a complex C*-algebra with compatible order, not necessarily unital or nonzero. The primitive spectrum here is the set of subsets of occurring as kernels of pure-state GNS representations.

Definition

A pure state is a bounded complex-linear functional that is positive, has norm one, and is extreme among positive norm-one functionals, for real convexity. Its canonical state extension to the forced unitization has a GNS representation; denotes the restriction of that representation to .

A point is the representation kernel itself, not a preferred pure state. The ideal identification regards it as a two-sided ideal of . This definition assigns no topology; the spectrum topology is defined separately.

Assumptions

The source requires a nonunital complex C*-algebra, a partial order and compatible order. No nontriviality, separability, faithfulness, finite dimension or chosen representation is an additional input.

Conclusion

Points are representation kernels, regarded as subsets of . Pure states with the same GNS kernel determine the same point. This is not the quotient by unitary equivalence, nor is it the family of functional kernels.

def PrimitiveSpectrum := {P : Set A // P ∈ primitiveKernelFamily A}

namespace MathlibAnnex.Analysis.CStarAlgebra

universe u v
variable (A : Type u) [NonUnitalCStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
open NonUnitalCStarRepresentation

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Technical documentation

def PrimitiveSpectrum := {P : Set A // P ∈ primitiveKernelFamily A}

def PrimitiveSpectrum := {P : Set A // P ∈ primitiveKernelFamily A}

def PrimitiveSpectrum := {P : Set A // P ∈ primitiveKernelFamily A}

def PrimitiveSpectrum := {P : Set A // P ∈ primitiveKernelFamily A}