Universal representation (C*-algebra)

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

 * Definition. Let A be a C*-algebra with state space S. The representation
 * $$\Phi := \sum_{\rho \in S} \oplus \; \pi_\rho$$
 * on the Hilbert space $$H_{\Phi}$$ is known as the universal representation of A.

As the universal representation is faithful, A is *-isomorphic to the C*-subalgebra Φ(A) of B(HΦ).

States of Φ(A)
With τ a state of A, let πτ denote the corresponding GNS representation on the Hilbert space Hτ. Using the notation defined here, τ is ωx ∘ πτ for a suitable unit vector x(=xτ) in Hτ. Thus τ is ωy ∘ Φ, where y is the unit vector Σρ∈S ⊕yρ in HΦ, defined by yτ=x, yρ=0(ρ≠τ). Since the mapping τ → τ ∘ Φ−1 takes the state space of A onto the state space of Φ(A), it follows that each state of Φ(A) is a vector state.

Bounded functionals of Φ(A)
Let Φ(A)&minus; denote the weak-operator closure of Φ(A) in B(HΦ). Each bounded linear functional ρ on Φ(A) is weak-operator continuous and extends uniquely preserving norm, to a weak-operator continuous linear functional $\overline{ρ}$ on the von Neumann algebra Φ(A)&minus;. If ρ is hermitian, or positive, the same is true of $\overline{ρ}$. The mapping ρ → $\overline{ρ}$ is an isometric isomorphism from the dual space Φ(A)* onto the predual of Φ(A)&minus;. As the set of linear functionals determining the weak topologies coincide, the weak-operator topology on Φ(A)&minus; coincides with the ultraweak topology. Thus the weak-operator and ultraweak topologies on Φ(A) both coincide with the weak topology of Φ(A) obtained from its norm-dual as a Banach space.

Ideals of Φ(A)
If K is a convex subset of Φ(A), the ultraweak closure of K (denoted by K&minus;)coincides with the strong-operator, weak-operator closures of K in B(HΦ). The norm closure of K is Φ(A) ∩ K&minus;. One can give a description of norm-closed left ideals in Φ(A) 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(A), there is a projection E in Φ(A)&minus; such that
 * $$K = \Phi(A) \cap \Phi(A)^{-}E, K^{-} = \Phi(A)^{-}E$$

If K is a norm-closed two-sided ideal in Φ(A), E lies in the center of Φ(A)&minus;.

Representations of A
If π is a representation of A, there is a projection P in the center of Φ(A)&minus; and a *-isomorphism α from the von Neumann algebra Φ(A)&minus;P onto π(A)&minus; such that π(a) = α(Φ(a)P) for each a in A. This can be conveniently captured in the commutative diagram below :


 * Univ rep diag.png

Here ψ is the map that sends a to aP, α0 denotes the restriction of α to Φ(A)P, 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(A)&minus;, let Sρ in A* be defined by Sρ(a) = $\overline{ρ∘Φ^{−1}}$(Φ(a)S) for all a in A. If P is the projection in the above commutative diagram when π:A → B(H) 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.

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,
 * $$f(\sigma_1(a), \sigma_1(a^*), \sigma_1(aa^*), \sigma_1(a^*a), \cdots, \sigma_m(a), \sigma_m(a^*), \sigma_m(aa^*), \sigma_m(a^*a))$$
 * $$ \le g(\rho_1(a), \rho_1(a^*), \rho_1(aa^*), \rho_1(a^*a), \cdots, \rho_n(a), \rho_n(a^*), \rho_n(aa^*), \rho_n(a^*a)).$$

Then the above inequality holds if each ρj is replaced by its ultraweakly continuous component (ρj)u.