Unital representative of the unique irreducible-representation class

MathlibAnnex.Analysis.CStarAlgebra.Representation.IsSingletonIrreducibleModelAmongNonUnital

def

Defines the corresponding condition for a unital representation, with competing nonzero irreducible representations quantified through the nonunital representation interface.

Statement

Let A be a unital complex C*-algebra, H a complex Hilbert space, and π a unital representation of A on H. The representation π is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A: π is irreducible, and for every nonzero irreducible representation ρ presented through the nonunital representation interface, π is unitarily equivalent to the associated unital representation ρ.toUnital. Faithfulness and separability are not part of the definition.

Definition

IsSingletonIrreducibleModelAmongNonUnital π ↔ IsIrreducible π ∧ ∀ K ρ, IsIrreducible ρ → UnitaryEquivalent π (ρ.toUnital ...).

Assumptions

Let A be a unital complex C*-algebra, H a complex Hilbert space, and π a unital representation of A on H. Competing nonzero irreducible representations are quantified through the nonunital representation interface; irreducibility supplies the associated unital representations used in the comparison.

Conclusion

The representation π is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A, with competitors quantified through the nonunital representation interface. Faithfulness and separability are not part of the definition.

Main citations

Lean source signature (exact)

def IsSingletonIrreducibleModelAmongNonUnital
    (pi : Representation A H) : Prop

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