Unitary equivalence of irreducible representations

Scope

Selected GNS cyclic subspaces assemble into a surjective isometry and then a unitary equivalence of the target models. The nonunital case first proves unit preservation; shell sums are compared on their own Hilbert spaces without assuming strong continuity of an arbitrary representation.

9 direct Cards + 20 reused prerequisites = 29 unique Cards. This count is a selected Card closure, not a source-declaration count.

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Cards in this route

Key result: Unitary equivalence without an initial unitality assumption
Key result: The cyclic sum fills every irreducible target representation

Read this route with prerequisites

Reference index: direct Cards and reused prerequisites

Direct references: 42 — The CAR source map into the fixed shell-family target · 43 — The ambient inclusion of the fixed shell-family target · 44 — Assembling selected GNS cyclic subspaces into an isometric source representation · 45 — The cyclic sum fills every irreducible target representation · 49 — The shell-generated algebra is not an algebra of all compact operators · 46 — Every irreducible representation is unitarily equivalent to the inclusion · 47 — Unitary equivalence without an initial unitality assumption · 48 — Closed-ideal simplicity of the shell-generated algebra · 50 — Structural properties of one shell-generated C*-algebra

Reused prerequisites: 38 — The atomic common range is one embedded GNS line · 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 · 36 — The matching GNS fiber retains exactly its cyclic line · 28 — Distinct GNS classes have no unitary intertwiner · 29 — The atomic direct sum of the selected CAR representations · 33 — Realizing a shell family by a faithful irreducible operator algebra · 9 — The decreasing root projections in the CAR completion · 4 — Purity of the completed root state · 30 — An irreducible operator algebra constructed from projection shells · 2 — The CAR algebra as a completion of finite matrix stages · 37 — An inequivalent GNS fiber has no residual common range · 40 — An irreducible target representation has a surviving fixed space · 39 — Reconstructing a represented unitary from its shells and residual corner · 25 — A fixed representative family of CAR shell data · 31 — The atomic-shell construction on selected pure-GNS fibers · 32 — The closed operator algebra generated by a representation and extra operators · 5 — The completed CAR algebra is infinite-dimensional · 41 — A common root vector realizes every selected state through the generators

Detailed route conditions

For a fixed supplied family and its once-chosen links, put . The maps have distinct domains:

The selected vector states give a cyclic-sum isometry . In the generated-algebra setting, root normalization and irreducibility show that the chosen isometry is surjective. The full equivalence proof uses that same underlying map after bundling it as a unitary, and additionally uses the prescribed cyclic-vector action of the links. Shell sums are compared separately on the two Hilbert spaces; strong continuity of an arbitrary representation is never assumed.

The nonunital version first proves unit preservation for the same nonzero irreducible representation. Simplicity and compact-operator exclusion are subsequent consequences. The existential unitary in a separately applied theorem need not equal a witness selected in another application.

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.

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Level 0 (4 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 (4 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

Level 1 (5 Cards)

The product-vector state on the completed CAR algebra

The completed CAR algebra is infinite-dimensional

An irreducible operator algebra constructed from projection shells

Distinct GNS classes have no unitary intertwiner

The decreasing root projections in the CAR completion

Level 2 (3 Cards)

Root compression becomes scalar in norm

Purity of the completed root state

The atomic-shell construction on selected pure-GNS fibers

Level 3 (3 Cards)

The atomic direct sum of the selected CAR representations

The matching GNS fiber retains exactly its cyclic line

An inequivalent GNS fiber has no residual common range

Level 4 (3 Cards)

Assembling selected GNS cyclic subspaces into an isometric source representation

The atomic common range is one embedded GNS line

Reconstructing a represented unitary from its shells and residual corner

Level 5 (2 Cards)

Realizing a shell family by a faithful irreducible operator algebra

An irreducible target representation has a surviving fixed space

Level 6 (3 Cards)

The CAR source map into the fixed shell-family target

The ambient inclusion of the fixed shell-family target

A common root vector realizes every selected state through the generators

Level 7 (2 Cards)

The shell-generated algebra is not an algebra of all compact operators

The cyclic sum fills every irreducible target representation

Level 8 (1 Card)

Every irreducible representation is unitarily equivalent to the inclusion

Level 9 (2 Cards)

Unitary equivalence without an initial unitality assumption

Closed-ideal simplicity of the shell-generated algebra

Level 10 (1 Card)

Structural properties of one shell-generated C*-algebra