The C*-algebra generated by the CAR representation and the shell unitaries

MathlibAnnex.CStarAlgebra.CAR.AtomicCounterexampleAlgebra

abbrev

Defines the single algebra used in the irreducible and separable faithful representations.

Statement

Let be the completed CAR algebra and let be its distinguished pure state. Let be the set of unitary-equivalence classes of pure-state GNS representations. Choose a representative in each class, keeping the GNS representation of in its distinguished class, and put and . Fix the automorphisms and shell partial isometries supplied by CAR homogeneity, and choose the corresponding unitaries on by the cited shell construction. Define

Definition

is the norm-closed unital complex -subalgebra of generated by and the unitaries . The automorphisms, shell partial isometries and resulting unitaries are chosen once; the abbreviation does not choose another isomorphic copy of the algebra.

Assumptions

The CAR algebra, the pure state , the GNS representatives and the unitaries are those in the cited fixed construction. The index set is not assumed countable.

Conclusion

The notation refers to this particular generated C*-algebra. It does not denote a new isomorphic copy when different representations of are considered.

The abbreviation defines the algebra; separate theorems establish its simplicity, uniqueness of the nonzero irreducible representation up to unitary equivalence, and failure to be isomorphic to an algebra of compact operators. The fixed construction uses the proved CAR homogeneity theorem, without an additional KOS or CH hypothesis.

Supporting route explanation

abbrev AtomicCounterexampleAlgebra := ShellFamilyTarget homogeneityShellFamily

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