Rosenberg Compact-Operator Model

Let A be a complex C*-algebra, not assumed unital, H a complex Hilbert space, and e a nonunital complex star-algebra homomorphism from A to the bounded operators on H. The model predicate consists of all three clauses: e is injective; every represented e(a) is compact; and each compact bounded operator T on H equals e(a) for some a in A.

Let D be a unital star subalgebra of a unital complex C*-algebra A. D is commutative, and any commutative unital star subalgebra E containing D is contained in D; thus D is maximal for inclusion among such subalgebras.

Let A be a unital complex C*-algebra and let φ : A → ℂ be a continuous complex-linear functional. φ is pure exactly when it belongs to the real extreme points of the normalized positive state space stateSpace A. Membership in that extreme-point set includes membership in the state space.

Let A be a complex C*-algebra, not assumed unital, and H a complex Hilbert space. A representation of A on H is a complex-linear nonunital star-algebra homomorphism into the bounded complex-linear operators on H. It preserves addition, multiplication, complex scalar multiplication and involution; neither preservation of a unit nor faithfulness is built in.

Let A be a complex C*-algebra, not assumed unital, and form its minimal unitization Unitization ℂ A. The infinity character is the character induced by projection onto the scalar coordinate: it evaluates a unitization element at its ℂ component. It is a character of Unitization ℂ A, not of A itself.

Let A be a unital complex C*-algebra, H a complex Hilbert space, and π a unital representation of A on H. The representation π is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A: π is irreducible, and for every nonzero irreducible representation ρ presented through the nonunital representation interface, π is unitarily equivalent to the associated unital representation ρ.toUnital. Faithfulness and separability are not part of the definition.

Let A be a unital complex C*-algebra, H a nonzero finite-dimensional complex Hilbert space, and π an irreducible unital representation of A on H. No singleton hypothesis or separability assumption on A is made. π is surjective onto all bounded complex-linear operators on H.

Let D be a commutative unital complex C*-algebra, J an ideal whose carrier is norm-closed, and d∈D with d∉J. There is a character χ of D that vanishes on every x∈J while χ(d)≠0. Properness follows from d∉J and is not separately assumed.

Let A be a complex C*-algebra, not assumed unital, H a complex Hilbert space, and ρ a representation of A on H. compactPreimageIdeal ρ is the two-sided algebraic ideal of elements whose represented bounded operator is compact. Norm-closedness is a separate theorem and is not a field of this returned TwoSidedIdeal.

Let X be nonempty, countable, T1 and Baire. At least one singleton {x} is open in X.

Let A be a complex C*-algebra, not assumed unital, H a complex Hilbert space, and π a representation of A on H. The predicate says that π acts nontrivially and every closed complex submodule L invariant under both π(a) and its adjoint for every a is either {0} or H.

Let A be a complex C*-algebra, not assumed unital; let π and ρ be representations of A on complex Hilbert spaces H and K, respectively. They are unitarily equivalent when an onto complex-linear isometry U from H to K intertwines every represented operator at every vector.

Let A be a complex C*-algebra, not assumed unital. Let D be a closed commutative unital star subalgebra of Unitization ℂ A with countable character space, and let 0 ≠ d ∈ D have zero scalar coordinate. There exists a character χ distinct from the restricted infinity character for which {χ} is open in the character space. This theorem itself does not assume that any representation represents a unique irreducible-representation class.

Let x be self-adjoint in a unital complex C*-algebra A. There exists a maximal abelian unital star subalgebra D of A containing x.

Let A be a nonzero unital complex C*-algebra, D a closed unital star subalgebra of A, and χ a character of D. There is a continuous complex-linear functional φ on A in stateSpace A that is pure and restricts to χ: for every d in D, φ(d)=χ(d).

Let A be a unital complex C*-algebra and let φ be a pure state of A. The canonical GNS star-algebra homomorphism built from the positive map associated to φ is irreducible.

Let A be a complex C*-algebra, not assumed unital, H a complex Hilbert space, and π a representation of A on H. The representation π is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A: π is nonzero and irreducible, and every nonzero irreducible representation of A on any complex Hilbert space is unitarily equivalent to π. The definition does not assume faithfulness or separability of A or H.

Let A be a nonzero unital complex C*-algebra, H a complex Hilbert space, and π a representation of A on H that is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A. Assume additionally that π is injective. No separability assumption on H is required. A is simple in the exact closed-two-sided-ideal sense IsSimpleCStarAlgebra A.

Let A be a nonzero unital complex C*-algebra and let a ∈ A be nonzero. No separability assumption on A is required. Some pure state φ in stateSpace A satisfies φ(a* a)≠0.

Let A be a nonzero complex C*-algebra, not assumed unital; let H be a complex Hilbert space; let π be a representation of A on H that is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A; let D be a closed unital star subalgebra of Unitization ℂ A; and let χ be a character of D distinct from infinityCharacterOn D. No separability assumption on H is required. There is a unit vector η in H such that π.unitization(d)η=χ(d)η for every d in D.

Let A be a nonzero complex C*-algebra, not assumed unital, H a complex Hilbert space, and π a representation of A on H that is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A. No separability assumption on H is required. Then π is injective.

Let A be a nonzero unital complex C*-algebra, H a separable complex Hilbert space, and π a representation of A on H that is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A. H is finite-dimensional over ℂ. The conclusion is conditional on the unique-class hypothesis, not a statement about all unital irreducible representations.

Let A be a nonzero complex C*-algebra, not assumed unital; let H be a separable complex Hilbert space; let π be a representation of A on H that is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A; and let D be a closed unital star subalgebra of Unitization ℂ A. The entire character space of D is countable, including the distinguished scalar character.

Let A be a nonzero unital complex C*-algebra, H a separable complex Hilbert space, and π a unital representation of A on H that is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A. π is injective; its underlying nonunital star homomorphism is injective, maps every a∈A to a compact operator, and represents every compact operator on H.

Let A be a nonzero unital complex C*-algebra, H a separable complex Hilbert space, and π a unital representation of A on H that is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A. Then A is finite-dimensional over ℂ; finite dimension of A is the conclusion, not an initial hypothesis.

Let A be a nonzero unital complex C*-algebra, H a separable complex Hilbert space, and π a unital representation of A on H that is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A. Then H is finite-dimensional over ℂ.

Let A be a nonzero complex C*-algebra, not assumed unital, H a separable complex Hilbert space, and π a representation of A on H that is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A. There is a nonzero star projection p in A whose every corner p a p is a complex scalar multiple of p.

Let A be a nonzero infinite-dimensional unital complex C*-algebra, H a separable complex Hilbert space, and π a unital representation of A on H. The representation π is not a representative of the unique unitary-equivalence class of nonzero irreducible representations of A. Equivalently, π is not both irreducible and unitarily equivalent to every nonzero irreducible representation of A. This does not rule out separable irreducible representations in general.

Let A be a nonzero complex C*-algebra, not assumed unital, H a separable complex Hilbert space, and π a representation of A on H that is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A. There are a nonzero star projection p∈A and a unit vector e∈H with π(p)=rankOne e e; the represented operator π(p) is compact. Minimality is not an explicit clause of this theorem type.

Let A be a nonzero complex C*-algebra, not assumed unital; let H be a separable complex Hilbert space; and let π be a representation of A on H that is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A. For every pair x,y∈H, including zero vectors, there exists a∈A with π(a)=rankOne ℂ x y; this operator has rank at most one.

Let A be a nonzero complex C*-algebra, not assumed unital; let H be a separable complex Hilbert space; let π be a representation of A on H that is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A; and let T be a compact bounded operator on H. There exists a∈A with π(a)=T. This is the inclusion K(H)⊆π(A); compactness of each π(a) is proved separately.

Let A be a nonzero complex C*-algebra, not assumed unital, H a separable complex Hilbert space, and π a representation of A on H that is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A. No simplicity assumption is made. Every operator π(a), a∈A, is compact.

Let A be a nonzero complex C*-algebra, not assumed unital, H a separable complex Hilbert space, and π a representation of A on H that is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A. No separability of A, prior faithfulness, or simplicity is assumed. π is injective and satisfies the full unbundled compact-operator model: every π(a) is compact and every compact T on H has an exact preimage in A.