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

MathlibAnnex.CStarAlgebra.CAR.not_isCompactOperatorModel_shellFamilyTarget

theorem

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

Statement

For the fixed shell-generated algebra below and any complete complex Hilbert space , there is no injective complex-linear star representation whose image is exactly the compact operators on . The map is not assumed to preserve the unit.

Assumptions

Let be the completed CAR algebra with matrix stages embedded by . Write for the image of the first diagonal matrix unit, , and for the root state characterized by on each stage, where is its canonical embedding.

Index chosen pure-state Gelfand-Naimark-Segal (GNS) representations by their unitary-equivalence classes , with root index and representative states .

A fixed shell family consists of complex-linear unital star automorphisms and elements satisfying , and .

At the root, is the identity and .

Let be the Hilbert direct sum of these GNS spaces, their representation of , and its embedded unit cyclic vector, where is the selected unit cyclic vector and is coordinate inclusion.

Choose once a family of unitary operators from the construction of the shell unitaries: , , and . Here denotes the bounded complex-linear operators on .

Put , the norm-closed unital star algebra generated by these operators. Let be with its codomain restricted to , and let be inclusion.

The comparison space may have any dimension and belongs to an arbitrary independent universe. The exact range condition says both that each is compact and that every compact operator on equals some . Inner products in the rank-one formula are linear in the second argument.

Conclusion

Thus is not isomorphic to the C*-algebra of all compact operators on any complete complex Hilbert space , including the zero Hilbert space. No preservation of the identity is imposed on the candidate representation.

The obstruction uses that is unital and infinite-dimensional, with infinite dimension established in the target-dimension theorem using its injective copy of the completed CAR algebra. It is distinct from results about the intersection of one represented image with the compact operators.

Proof route

Assume such a map exists. Injectivity and first force . Since every rank-one operator is in the image, acts as the identity on every vector: apply to a rank-one operator sending a fixed unit vector to the desired vector. Thus . But is compact, forcing to be finite-dimensional. Then and its complex-linear subspace are finite-dimensional, contradicting the infinite dimension of .

theorem not_isCompactOperatorModel_shellFamilyTarget
    (family : RepresentativeShellFamily)
    {K : Type v} [NormedAddCommGroup K] [InnerProductSpace ℂ K]
    [CompleteSpace K]
    (e : ShellFamilyTarget family →⋆ₙₐ[ℂ] (K →L[ℂ] K)) :
    ¬ IsCompactOperatorModel e

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