- Source: 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
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−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)− 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−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.
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,
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
Kadison, Richard, Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory, American Mathematical Society. ISBN 978-0821808191.
Kadison, Richard, Fundamentals of the Theory of Operator Algebras, Vol. II : Advanced Theory, American Mathematical Society. ISBN 978-0821808207.
Kadison, Richard V. (1993), "On an inequality of Haagerup–Pisier", Journal of Operator Theory, 29 (1): 57–67, MR 1277964.
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