MathlibAnnex.Analysis.CStarAlgebra.MinimalClosedStarIdeal.CompactModelSpace
abbrev
Names the cyclic Hilbert space chosen for a minimal closed ideal.
Statement
Let be a complex C*-algebra with compatible order, and let be a faithful representation with every value compact, on a complete complex Hilbert space. For a minimal nonzero closed -ideal of , this abbreviation names the chosen cyclic Hilbert subspace .
Definition
The minimal-ideal model construction chooses a pair with the following properties: is a minimal projection, the closed ideal generated by is the displayed ideal , , , and the same cyclic restriction is an exact compact-operator model: it is injective, every value is compact, and every compact operator on this same is for some . Faithfulness and compact-valuedness alone do not include the last condition. These properties are guaranteed for the chosen pair; they are not additional assumptions.
Write for the closed ideal generated by . The chosen model subspace is
for that same pair , with its inherited norm and complex inner product. The closedness and Hilbert-space results ensure that is complete and nonzero.
Assumptions
The algebra carries a partial order and compatible order. The input ideal is minimal among nonzero closed -ideals. The representation is injective and every value is compact. There is no additional choice of Hilbert structure, no countability assumption and no global nonzero-algebra hypothesis.
Conclusion
The resulting space depends on the ideal and the chosen representation . The compact-model isomorphism uses this same Hilbert space; different ideals need not give the same space.
Main citations
- Exact complete definition.
- MinimalClosedStarIdeal: the indexing type.
- compactModelWitness: one generator/vector pair and its full specification.
- IsCompactOperatorModel: the three conditions on one action.
- compactModelSubspace: the exact cyclic subspace.
- isClosed_compactModelSubspace and the completeness/nontriviality instances.
Lean source signature (exact)
The following is the complete defining right-hand side, copied from the fixed source.
abbrev CompactModelSpace (i : MinimalClosedStarIdeal A)
(pi : NonUnitalCStarRepresentation A H) (hpi : pi.IsCompactlyRepresented) : Type v :=
i.compactModelSubspace pi hpi
| In the source | Mathematical meaning |
|---|---|
Exact declarationi : MinimalClosedStarIdeal A |
A minimal nonzero closed -ideal. |
Exact declarationpi and hpi |
The fixed representation and its faithful compact-valued condition, used to choose the projection and cyclic vector. |
Exact declaration: Type v |
The resulting space is a subspace of the given Hilbert space . |
Exact declarationi.compactModelSubspace pi hpi |
The closed cyclic subspace associated with the chosen pair , with norm and inner product inherited from . |
Exact surrounding source context (separate excerpts)
Source lines 128–129:
variable {A : Type u} [NonUnitalCStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
variable {H : Type v} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
Source lines 140–140:
namespace MinimalClosedStarIdeal
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