We now turn to the definition of B-valued laws (and generalized laws). The results of this section are based on [Voi95, PV13, AW16].
Recall that if X is a real random variable on a probability space (Ω, P ), then the law of X is the measure µX on R given by R f dµX = E[f (X)]. Similarly, if φ : A → C is a state and X ∈ A is self-adjoint, then the law of X with respect to φ is the measure µX given by R f dµX = φ[f (X)]. In either case, if the measure µX is compactly supported, then it is uniquely specified by its moments R tndµX(t) = φ(Xn), that is, it suffices to consider polynomial test functions. In the B-valued setting, there is no clear way to express these moments in terms of a measure, we will simply define the law of X by its action on polynomial test functions.
Definition 2.6.1. We denote by BhXi the algebra of non-commutative polynomials in a formal variable X with coefficients in B, that is, the universal (non-commutative) algebra generated by B and an indeterminate X. (As a vector space, BhXi is the linear span of terms of the form b0Xb1X . . . bk−1Xbk.) We endow AhXi with the ∗-operation determined by
(b0Xb1X . . . bk−1Xbk)∗ = b∗kXb∗k−1. . . Xb∗1Xb∗0.
Definition 2.6.2. Let (A, E) be a B-valued probability space and x a self-adjoint element of A. The law of x is the map µx : BhXi → B given by p(X) 7→ E[p(x)].
In probability theory, it is a standard fact that every probability measure on R is the law of some random variable. Indeed, the random variable given by the identity function on the probability space (R, µ) will have the law µ. Thus, laws which arise from random variables are characterized abstractly as measures. In operator-valued non-commutative probability, there is also an abstract characterization of laws, and a way to explicitly construct a random variable which realizes a given law, which is a version of the GNS construction.
Definition 2.6.3. An B-valued law is a linear map µ : BhXi → B such that
(1) µ is completely positive: For any P (X) ∈ Mn(BhXi) we have µ(n)(P (X)∗P (X)) ≥ 0 in Mn(B).
(2) µ is exponentially bounded: There exist some M > 0 and R > 0 such that kµ(b0Xb1X . . . bk−1Xbk)k ≤ M Rkkb0k . . . kbkk for all b0, . . . , bk∈ B.
(3) µ is unital: µ(1) = 1.
(4) µ is a B-B-bimodule map: µ(b1p(X)b2) = b1µ(p(X))b2 for b1, b2 ∈ B.
Definition 2.6.4. Let µ : BhXi → B. If kµ(b0Xb1X . . . bk−1Xbk)k ≤ M Rkkb0k . . . kbkk, then we say that R is an exponential bound for µ. Finally, we define the radius of µ as
rad(µ) := inf{R : R is an exponential bound for µ}.
The characterization of B-valued laws is proved in [PV13, Proposition 1.2], and it is similar to earlier results such as [AGZ09, Proposition 5.2.14].
Theorem 2.6.5. A linear map µ : BhXi → B is a B-valued law if and only if there exists a B-valued probability space (A, E) and a self-adjoint x ∈ A with µ = µx. Moreover, for each µ, we can choose x such that kxk = rad(µ).
The most substantial part of the proof will work in greater generality, and we will need the more general result later when we work with analytic transforms associated to B-valued laws.
Theorem 2.6.6. Let B and C be C∗-algebras and σ : BhXi → C a linear map. Then the following are equivalent:
(1) σ is completely positive and exponentially bounded.
(2) There exists a B-C-correspondence H, a vector ξ ∈ H, and a self-adjoint operator x ∈ B(H) with σ(f (X)) = hξ, f (x)ξi for all f ∈ BhXi.
(3) There exists a C∗-algebra A and a ∗-homomorphism ρ : BhXi → A, and a completely positive map σ : A → C such that σ = σ ◦ ρ.
Moreover, for a completely positive and exponentially bounded σ, the operator x can be chosen such that kxk = rad(σ), and we have
kσ(b0Xb1. . . Xbk)k ≤ kσ(1)k rad(σ)kkb0k . . . kbkk. (2.1) Proof. (1) =⇒ (2). We define a C-valued pre-inner-product on BhXi ⊗alg C by
hf1(X) ⊗ c1, f2(X) ⊗ c2iµ= c∗1µ(f1(X)∗g2(X))c2.
As we saw earlier with tensor products and the GNS construction, the complete positivity of σ implies that the pre-inner-product is nonnegative. Therefore, the Cauchy-Schwarz inequality holds and we can define the separation-completion with respect to this pre-inner-product, which we denote by BhXi ⊗σ C.
The space H := BhXi ⊗σ C is a B-C-correspondence with the left action of B defined in the natural way. Indeed, to show that the left multiplication by b ∈ B extends to the separation completion, it suffices to show that kbξk ≤ kbkkξk for ξ ∈ BhXi ⊗alg C. This is done by writing kbk2− b∗b = y∗y for some y ∈ B as before.
Next, we claim that the linear operator given by f (X) ⊗ c 7→ Xf (X) ⊗ c on the algebraic tensor product passes to a well-defined and bounded operator x on the separation-completion BhXi ⊗σC. Let R be an exponential bound for σ and let T > R. Unfortunately, we cannot claim that T2− X2 is a positive element of BhXi, or that it can be written as g(X)∗g(X) for some g(X) ∈ BhXi, since BhXi does not have the same completeness properties as a C∗-algebra. However, we can fix this problem by looking at a certain power-series completion of BhXi and defining g(X) as the power series for √
T2− X2. For a monomial b0Xb1. . . Xbk, we denote
p(b0Xb1. . . Xbk) = Rkkb0k . . . kbkk.
Then for f (X) ∈ BhXi, we define
kf (X)kR= inf ( n
X
j=1
p(fj) : fj monomials and f =
n
X
j=1
fj )
.
Let BhhXiiR be the completion of BhXi in this norm. One checks easily that kf (X)g(X)kR≤ kf (X)kRkg(X)kR,
and this inequality extends to the completion, which makes BhhXiiR a Banach algebra.
Similarly, the ∗-operation on BhXi extends to the completion. By standard results from complex analysis, the function g(t) =√
T2− t2 has a power series expansion g(t) =
∞
X
j=0
αjtj
which converges for |t| < T . In particular, the series converges absolutely for t = R, which implies that
g(X) =
∞
X
j=0
αjXj
converges absolutely in BhhXiiR. Moreover, because of the absolute convergence and the Banach algebra properties, we can compute ψ(X)2 by multiplying the series term by term and hence conclude that g(X)2 = T2− X2. Because R is an exponential bound for σ, we know that kσ(f (X))k ≤ M kf (X)kR, where M is a constant such that kb0Xb1. . . Xbkk ≤ M Rkkb0k . . . kbkk. Hence, σ extends to a linear map BhhXiiR→ C, which is still completely positive, and hence the pre-inner-product h·, ·i extends to BhhXiiR ⊗alg C. Then for each vector ζ ∈ BhhXiiR⊗algC, we have
hζ, (T2− X2)ξi = hg(X)ζ, g(X)ζi ≥ 0,
which implies that kXζk ≤ T kζk, and in particular, this holds for ζ ∈ BhXi ⊗alg C. By taking T & R, we have kXζk ≤ Rkζk, which means that the multiplication operator by X is bounded with respect to the pre-inner-product and hence extends to the separation-completion.
The operator x thus defined is clearly self-adjoint. Moreover, letting ξ = 1 ⊗ 1 ∈ BhXi ⊗σ C, we have
hξ, f (x)ξi = σ(f (X)) for f ∈ BhXi.
Since R was an arbitrary exponential bound, we have kxk ≤ rad(σ), and thus,
kσ(b0Xb1. . . Xbk)k = khξ, b0xb1. . . xbkξik ≤ kξk2kxkkkb0k . . . kbkk ≤ kσ(1)k rad(σ)kkb0k . . . kbkk.
This proves (1) =⇒ (2) as well as the last claim of the theorem.
(2) =⇒ (3). Fix H, ξ, and x as in (2), and let A = B(H). Let π be the ∗-homomorphism B → B(H) given by the left B-module structure. Then there is a unique ∗-homomorphism ρ : BhXi → B(H) satisfying ρ|B = π and ρ(X) = x. Moreover, the map σ : B(H) → C given by a 7→ hξ, aξi is completely positive and satisfies σ ◦ ρ = σ.
(3) =⇒ (1). If (3) holds, then σ is completely positive because it is the composition of the two completely positive maps ρ and σ. Moreover, it is exponentially bounded because
kσ(b0Xb1. . . Xbk)k = kσ(b0ρ(X)b1. . . ρ(X)bk)k ≤ kσ(1)kkρ(X)kkkb0k . . . kbkk by Lemma 2.4.5.
Proof of Theorem 2.6.5. Suppose that µ is a B-valued law. Let H, ξ, and x be as in Theorem 2.6.6 (2) for σ = µ. Since hξ, bξi = µ(b) = b, Lemma 2.5.5 implies that ξ is a B-central unit vector, hence A = B(H) and E = hξ, (·)ξi form a B-valued probability space. And clearly µx = µ. Conversely, if µ = µx for some x in a B-valued probability space (A, E), then by Theorem 2.6.5, µ is completely positive and exponentially bounded. Since E(1) = 1 and E is a B-B-bimodule map, the same holds for µ.
From the use of Lemma 2.5.5 above, the following corollary is obvious.
Corollary 2.6.7. Let σ : BhXi → B be completely positive and exponentially bounded. Then σ is a law if and only if σ|B = idB.
Lemma 2.6.8. If σ and τ are generalized laws, then σ + τ is also a generalized law, and we have rad(σ + τ ) = max(rad(σ), rad(τ )).
Proof. It is straightforward to check from the definition that σ +τ is a generalized law. Next, if p(X) = b0Xb1. . . Xbk, then
k(σ + τ )(p(X))k ≤ kσ(p(X))k + kτ (p(X))k
≤ kσ(1)k rad(σ)kkb0k . . . kbkk + kτ (1)k rad(τ )kkb0k . . . kbjk
≤ (kσ(1)k + kτ (1)k) max(rad(σ), rad(τ ))kkb0k . . . kbkk, which shows that rad(σ + τ ) ≤ max(rad(σ), rad(τ )). On the other hand,
kσ(p(X))k ≤ kσ(1)k1/2kσ(p(X)∗p(X))k1/2
≤ k(σ + τ )(1)k1/2k(σ + τ )(p(X)∗p(X))k1/2
≤ k(σ + τ )(1)k1/2 k(σ + τ )(1)k rad(σ + τ )2kkb0k2. . . kbkk21/2
= k(σ + τ )(1)k rad(σ + τ )kkb0k . . . kbkk,
where we have used the fact that 0 ≤ σ(p(X)∗p(X)) ≤ (σ + τ )(p(X)∗p(X)). But the above estimate implies that rad(σ) ≤ rad(σ + τ ), and of course rad(τ ) ≤ rad(σ + τ ) by symmetry.