In the theory of C*-algebras, the universal representation of a C*-algebra is a faithful representation which is the direct sum of the GNS representations corresponding to the states of the C*-algebra. The various properties of the universal representation are used to obtain information about the ideals and quotients of the C*-algebra. The close relationship between an arbitrary representation of a C*-algebra and its universal representation can be exploited to obtain several criteria for determining whether a linear functional on the algebra is ultraweakly continuous. The method of using the properties of the universal representation as a tool to prove results about the C*-algebra and its representations is commonly referred to as universal representation techniques in the literature.
Formal definition and properties
As the universal representation is faithful, A is *-isomorphic to the C*-subalgebra Φ of B.
States of Φ(''A'')
With τ a state of A, let πτ denote the corresponding GNS representation on the Hilbert spaceHτ. Using the notation defined here, τ is ωx ∘ πτ for a suitable unit vectorx in Hτ. Thus τ is ωy ∘ Φ, where y is the unit vector ∑ρ∈S ⊕yρ in HΦ, defined by yτ=x, yρ=0. Since the mapping τ → τ ∘ Φ−1 takes the state space of A onto the state space of Φ, it follows that each state of Φ is a vector state.
Bounded functionals of Φ(''A'')
Let Φ− denote the weak-operator closure of Φ in B. Each bounded linear functional ρ on Φ is weak-operator continuous and extends uniquely preserving norm, to a weak-operator continuous linear functional on the von Neumann algebra Φ−. If ρ is hermitian, or positive, the same is true of. The mapping ρ → is an isometric isomorphism from the dual space Φ* onto the predual of Φ−. As the set of linear functionals determining the weak topologies coincide, the weak-operator topology on Φ− coincides with the ultraweak topology. Thus the weak-operator and ultraweak topologies on Φ both coincide with the weak topology of Φ obtained from its norm-dual as a Banach space.
Ideals of Φ(''A'')
If K is a convex subset of Φ, the ultraweak closure of K coincides with the strong-operator, weak-operator closures of K in B. The norm closure of K is Φ ∩ K−. One can give a description of norm-closed left ideals in Φ from the structure theory of ideals for von Neumann algebras, which is relatively much more simple. If K is a norm-closed left ideal in Φ, there is a projection E in Φ−such that If K is a norm-closed two-sided ideal in Φ, E lies in the center of Φ−.
Representations of ''A''
If π is a representation of A, there is a projection P in the center of Φ− and a *-isomorphism α from the von Neumann algebra Φ−P onto π− such that π = α for each a in A. This can be conveniently captured in the commutative diagram below : Here ψ is the map that sends a to aP, α0 denotes the restriction of α to ΦP, ι denotes the inclusion map. As α is ultraweakly bicontinuous, the same is true of α0. Moreover, ψ is ultraweakly continuous, and is a *-isomorphism if π is a faithful representation.
Ultraweakly continuous, and singular components
Let A be a C*-algebra acting on a Hilbert space H. For ρ in A* and S in Φ−, let Sρ in A* be defined by Sρ = for all a in A. If P is the projection in the above commutative diagram when π:A → B is the inclusion mapping, then ρ in A* is ultraweakly continuous if and only if ρ = Pρ. A functional ρ in A* is said to be singular if Pρ = 0. Each ρ in A* can be uniquely expressed in the form ρ=ρu+ρs, with ρu ultraweakly continuous and ρs singular. Moreover, ||ρ||=||ρu||+||ρs|| and if ρ is positive, or hermitian, the same is true of ρu, ρs.
Applications
Christensen–Haagerup principle
Let f and g be continuous, real-valued functions on C4m and C4n, respectively, σ1, σ2,..., σm be ultraweakly continuous, linear functionals on a von Neumann algebra R acting on the Hilbert space H, and ρ1, ρ2,..., ρn be bounded linear functionals on R such that, for each a in R, Then the above inequality holds if each ρj is replaced by its ultraweakly continuous component u.
