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Universal representation (C*-algebra)

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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
Φ := ∑ ρ ∈ S ⊕ π ρ {\displaystyle \Phi :=\sum _{\rho \in S}\oplus \;\pi _{\rho }}
on the Hilbert space H Φ {\displaystyle 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 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) 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 ρ on the von Neumann algebra Φ(A). If ρ is hermitian, or positive, the same is true of ρ. The mapping ρ → ρ is an isometric isomorphism from the dual space Φ(A) onto the predual of Φ(A). As the set of linear functionals determining the weak topologies coincide, the weak-operator topology on Φ(A) 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)coincides with the strong-operator, weak-operator closures of K in B(HΦ). The norm closure of K is Φ(A) ∩ K. 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) such that

K = Φ ( A ) ∩ Φ ( A ) − E , K − = Φ ( A ) − E {\displaystyle 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).

Representations of A

If π is a representation of A, there is a projection P in the center of Φ(A) and a *-isomorphism α from the von Neumann algebra Φ(A)P onto π(A) such that π(a) = α(Φ(a)P) 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 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), let Sρ in A be defined by Sρ(a) = ρ∘Φ(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.

Applications

Christensen–Haagerup principle

Let f and g be continuous, real-valued functions on C and C, 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 ( σ 1 ( a ) , σ 1 ( a ∗ ) , σ 1 ( a a ∗ ) , σ 1 ( a ∗ a ) , ⋯ , σ m ( a ) , σ m ( a ∗ ) , σ m ( a a ∗ ) , σ m ( a ∗ a ) ) {\displaystyle 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))}
≤ g ( ρ 1 ( a ) , ρ 1 ( a ∗ ) , ρ 1 ( a a ∗ ) , ρ 1 ( a ∗ a ) , ⋯ , ρ n ( a ) , ρ n ( a ∗ ) , ρ n ( a a ∗ ) , ρ n ( a ∗ a ) ) . {\displaystyle \leq 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.

References

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