Naimark’s problem

The normalized matrix trace provides a positive tracial functional compatible with the CAR inclusions.

Names the completed CAR algebra in which finite matrix calculations can be extended by norm approximation.

A point-norm approximation followed by close-projection conjugacy gives exact support identities for an arbitrary projection family.

One family records a transporting automorphism and exact shell links for each selected pure-state class, with an explicitly fixed root component.

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

Places a represented algebra and a chosen family of bounded operators in one concrete closed algebra.

Compatible evaluation at the distinguished matrix coordinate extends continuously to the CAR completion.

Rules out finite dimension by retaining matrix subalgebras of unbounded dimension.

Turns an arbitrary element of the completed CAR algebra into one commuting with a prescribed finite matrix stage.

Carries compatible bounded matrix traces through the algebraic limit, normed union and completion.

The distinguished stage corners give a nested sequence of projections supporting the root state.

The homogeneity property asks for exact state transport by an automorphism that is locally approximable by inner automorphisms.

An inner sequence records both forward and actual inverse convergence data before any limiting automorphism is constructed.

The class index makes the selected pure-GNS family pairwise unitarily inequivalent.

Shell partial isometries with rank-one limiting defects can be completed to unitary links that join inequivalent irreducible fibers.

Passes extremality of the root vector states from every finite matrix stage to the completed CAR algebra.

Proves uniqueness without assuming positivity of the competing continuous functional.

Extends an exact rank-one compression identity on finite stages to a pointwise norm limit on the completed algebra.

The forward and inverse limits close the homogeneity argument once an intertwining sequence is supplied for every pure-state pair.

Pure-state GNS data supplies the irreducible and inequivalent fibers required by the generic shell construction.

Recovers exact trace values from the initial and final supports of shell links.

Every nonzero closed two-sided ideal of the CAR algebra contains the identity.

All selected pure-GNS fibers of the completed CAR algebra act together on one Hilbert direct sum.

Turns decay of projection traces into norm convergence on each source-orbit vector.

Identifies the residual projection of a transported CAR flag in its matching pure-state representation.

Uses compression and cyclic transport to exclude fixed vectors in every other chosen pure-state class.

Simplicity and infinite dimensionality rule out compact operators coming from nonzero CAR elements.

Assembles the matching and inequivalent fiber calculations in an arbitrary Hilbert direct sum.

Separates a represented generator into a strong shell sum and the exact operator between its limiting fixed spaces.

Joins mutually orthogonal pure-state cyclic copies and controls the limiting flag projections on their sum.

The infinite-dimensional root corner contains an isometric copy of any finite family of vectors.

A prescribed unitary involution on a represented root corner can be realized on finitely many vectors by one lifted corner exponential.

Constructs unitary links between pure-state summands and an irreducible algebra containing a faithful copy of CAR.

Rules out simultaneous disappearance of all limiting CAR flags by passing reduction through the reconstructed generators.

Every finite-stage restriction of a unit vector state can be realized by a unit vector in any irreducible CAR representation.

Finite matrix-state tests ensure exact transport while protecting a prescribed finite set throughout the path.

Builds a compatible family of unit vectors from a surviving residual space and identifies their CAR vector states.

Restricting the codomain of the selected atomic representation gives the source map into one fixed generated target.

The same concrete target has a literal representation on the selected atomic Hilbert space.

Finite entrywise tests allow new state approximation while an entire unitary path nearly fixes earlier data.

A fixed irreducible CAR representation can approximate any vector state on finitely many elements by moving a given unit vector.

Uses shell reconstruction and residual lines to turn an isometric CAR intertwiner into a unitary equivalence on the source.

Excludes an injective representation of the infinite-dimensional unital algebra onto all compact operators.

The source trace first becomes a state on the existing target algebra.

Source cyclicity and strong shell sums extend the canonical CAR orbit isometry to the whole target.

Finite-set vector-state transport supplies explicit corrections; their summable conjugation bounds and state estimates yield the required two-sided sequence.

Extends the unitary equivalence from the CAR source to all additional generators and the norm-closed target.

The chosen state on the existing target supplies its cyclic GNS representation.

The constructed inner intertwining sequences discharge the supplier hypothesis and give an approximately inner transporting automorphism.

Shows why allowing nonunital input representations does not enlarge the irreducible representation class of the shell-generated algebra.

Uses a faithful irreducible representation in the unique unitary-equivalence class to rule out proper nonzero closed two-sided ideals.

Pointed transport identifies arbitrary state extensions before traciality is proved.

Unitary invariance, strong shell sums, and a closed centralizer extend the trace identity beyond the source.

A fixed pointed unitary transports the target action onto the Hilbert space constructed from CAR alone.

CAR homogeneity and exact shell matching supply one family whose distinguished component is fixed by an explicit identity choice.

Collects the proved properties of a single generated algebra, keeping the shell-family input separate from its consequences.

Every tracial state is forced back to the CAR trace and hence to its unique extension.

The ideal dichotomy of the same target forces the constructed nonzero representation to be injective.

Unitary conjugation preserves injectivity of the target-state representation.

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

Establishes the properties that make the fixed algebra a counterexample to Naimark's problem.

Defines the inclusion representation .

Defines a representation of the same algebra on the GNS Hilbert space of the CAR trace.

Proves that the representation is injective.

Rules out irreducible representations of on separable Hilbert spaces by the finite-dimensionality theorem for a single irreducible representation class.

The same algebra has a faithful representation on a separable Hilbert space, but no nonzero irreducible representation on any separable Hilbert space.

Combines a cardinality bound from a faithful representation on a separable Hilbert space with a lower bound for every norm-dense subset.

Gives a C*-algebra of norm density whose nonzero irreducible representations form one unitary-equivalence class, without an identification with the compact operators.