Fixed subspaces and reconstruction of the generators

Scope

Decreasing transported projections determine common fixed subspaces by strong vectorwise limits. Shell sums and residual corners reconstruct the generators in arbitrary unital target representations. For a nonzero irreducible target representation, a common fixed space survives and supplies compatible selected vectors.

8 direct Cards + 12 reused prerequisites = 20 unique Cards. This count is a selected Card closure, not a source-declaration count.

Route reading PDF · Preserved source exploration

Cards in this route

Key result: Reconstructing a represented unitary from its shells and residual corner
Key result: An irreducible target representation has a surviving fixed space

Read this route with prerequisites

Reference index: direct Cards and reused prerequisites

Direct references: 34 — Shell matching fixes the trace of transported flags · 36 — The matching GNS fiber retains exactly its cyclic line · 37 — An inequivalent GNS fiber has no residual common range · 39 — Reconstructing a represented unitary from its shells and residual corner · 35 — Transported flags vanish on vectors generated by a trace vector · 38 — The atomic common range is one embedded GNS line · 40 — An irreducible target representation has a surviving fixed space · 41 — A common root vector realizes every selected state through the generators

Reused prerequisites: 3 — The product-vector state on the completed CAR algebra · 10 — Root compression becomes scalar in norm · 27 — The selected GNS representation of a pure-state class · 28 — Distinct GNS classes have no unitary intertwiner · 29 — The atomic direct sum of the selected CAR representations · 1 — The normalized trace on a finite CAR stage · 9 — The decreasing root projections in the CAR completion · 4 — Purity of the completed root state · 7 — Constructing the normalized trace on the completed CAR algebra · 2 — The CAR algebra as a completion of finite matrix stages · 25 — A fixed representative family of CAR shell data · 32 — The closed operator algebra generated by a representation and extra operators

Detailed route conditions

For the supplied family, write . On the matching fiber its common fixed projection is the projection onto ; on an inequivalent fiber it is zero. Coordinatewise reasoning therefore identifies the common range in with . These are strong, vectorwise consequences of decreasing projections, not operator-norm limits.

In an arbitrary representation of the generated algebra, shell sums and their adjoints are reconstructed on that representation’s own Hilbert space. The generator splits into a strong shell sum and its residual corner. Nonzero irreducibility prevents every transported fixed space from vanishing. A surviving index may be different from the root class ; its vector is transported to a common root vector , then pulled back to the compatible vectors . No equality is imposed until the additional root-generator normalization is available.

Reference index: exact Card references

Exact Card references

Dependency-first reading route

Read selected Card prerequisites before their uses. Levels are recomputed from the selected reachability-preserving projection; omitted source helpers remain traceable in the source exploration.

Search Cards Route 20 Cards

Level 0 (5 Cards)

Level 0

The selected GNS representation of a pure-state class

A choice of representative pure states, fixed literally at a root, gives one concrete GNS representation per equivalence class.

MathlibAnnex.CStarAlgebra.PureState.selectedRepresentation

Immediate Card prerequisites: None in this selected scope

Used by in this scope: The matching GNS fiber retains exactly its cyclic line · Distinct GNS classes have no unitary intertwiner · The atomic direct sum of the selected CAR representations

Scope

Cards in this route

Detailed route conditions

Exact Card references

Dependency-first reading route

Level 0 (5 Cards)

The selected GNS representation of a pure-state class

The CAR algebra as a completion of finite matrix stages

A fixed representative family of CAR shell data

The closed operator algebra generated by a representation and extra operators

The normalized trace on a finite CAR stage

Level 1 (4 Cards)

Constructing the normalized trace on the completed CAR algebra

The product-vector state on the completed CAR algebra

Distinct GNS classes have no unitary intertwiner

The decreasing root projections in the CAR completion

Level 2 (3 Cards)

Shell matching fixes the trace of transported flags

Root compression becomes scalar in norm

Purity of the completed root state

Level 3 (4 Cards)

The matching GNS fiber retains exactly its cyclic line

Transported flags vanish on vectors generated by a trace vector

An inequivalent GNS fiber has no residual common range

The atomic direct sum of the selected CAR representations

Level 4 (2 Cards)

The atomic common range is one embedded GNS line

Reconstructing a represented unitary from its shells and residual corner

Level 5 (1 Card)

An irreducible target representation has a surviving fixed space

Level 6 (1 Card)

A common root vector realizes every selected state through the generators