Now we use the results on analytic subordination from the previous section to get the reverse radius bounds that we could not prove before. In the proposition below, we conjecture that the constants 3 + 2√
2 and 2 can be replaced by 1, but we were not able to prove this in all cases. However, the values of the constants do not make any qualitative difference in our results, either here or in the rest of the paper.
Proposition 6.3.1. Let ind ∈ {bool, free, mono, mono †}, and let µ1 and µ2 be B-valued laws. Then
rad(µj) ≤ (3 + 2√ 2)
max(kµ1(X)k, kµ2(X)k) + 2 rad(µ1indµ2) .
Proof. First, consider the boolean case. Let µ = µ1 ] µ2. Let (σ, b), (σ1, b1), and (σ2, b2) correspond to µ, µ1, and µ2 as in Theorem 4.5.3. Then σ = σ1+ σ2. Hence,
rad(σ1) ≤ rad(σ) ≤ rad(µ) and
kσ1(1)k1/2 ≤ kσ(1)k1/2≤ kµ(X2)k1/2 ≤ rad(µ).
Therefore,
rad(µ1) ≤ kb1k + rad(σ) + kσ(1)k1/2 ≤ kb1k + 2 rad(µ).
Of course, the analogous bound holds for µ2. This is already a better estimate than what we asserted above.
Next, consider the (anti-)monotone case. Let µ = µ1 B µ2. Note that µ = µ2 ] ν where ν is the law given by Proposition 6.2.1. Therefore, rad(µ2) ≤ kb2k + 2 rad(µ), where b2 = µ2(X). In order to get an estimate for rad(µ1), observe that ˜Gµ1 = ˜Gµ◦ ˜G−1µ2. As we remarked in the proof of Lemma 4.5.9, it follows from the inverse function theorem that if R ≤ 1/ rad(µ2), then ˜G−1µ maps B(0, (3 − 2√
2)R) to B(0, (1 − 1/√
2)R). This implies that G˜µ1 is fully matricial and bounded on B(0, (3 − 2√
2)R) provided that R < 1
kb2k + 2 rad(µ),
1 − 1
√2
R < 1 rad(µ). The first condition is strictly stronger than the second. Therefore,
rad(µ1) ≤ 1 3 − 2√
2(kb2k + 2 rad(µ))
= (3 + 2√
2) (kb2k + 2 rad(µ)) . This finishes the (anti-)monotone case.
The free case follows because µ1 µ2 = µ1B ν2 = µ2B ν1 for some laws ν1, ν2 given by Proposition 6.2.2.
CHAPTER 7
Results: Evolution equations for subordination families
7.1 Introduction
Our main goal in this chapter is to describe how the F -transforms FXt evolve over time when (Xt)t∈[0,T ]is a process with independent increments for each of the four types of independence described in §5. The main result will be roughly speaking that if µt is the law of Xt, then Fµt satisfies the equation
∂tFµ(n)t (z) =
−[b(t) + Gσ(·,t)(z)], boolean case,
−DFµ(n)t (z)[b(t) + G(n)σ(·,t)(Fµ(n)t (z))], free case,
−DFµ(n)t (z)[b(t) + G(n)σ(·,t)(z)], monotone case,
−[b(t) + G(n)σ(·,t)(Fµ(n)t (z))], anti-monotone case,
(7.1)
where Gσ(·,t) is the Cauchy-Stieltjes transform of a generalized law σ(·, t) depending on t, and where DFµt(z) = ∆Fµt(z, z).
The evolution of FXt (or equivalently of GXt) has been studied in many previous papers in special cases. The first case to be worked out for each type of independence was when (Xt)t∈[0,T ] has independent and stationary increments (that is, Xt− Xs ∼ Xt−s in law), or equivalently the laws (µt)t∈[0,T ] form a convolution semigroup. Prior work on the differential equations associated to such semigroups is summarized in Table 7.1. See also the discussion of the L´evy-Hinˇcin formula in §1.3 and the subsequent discussion of semigroups in §9.1.
To address the differential equations for processes with non-stationary increments in the operator-valued setting, we must deal with the technicalities of differentiation for Banach-valued functions in order to even make sense of the equation. These difficulties do not arise in the scalar-valued setting because scalar-valued absolutely continuous functions are
boolean free (anti-)monotone
scalar-valued [SW97, Thm. 3.6] [Voi86, Thm. 4.3] [Has10a] [Has10b]
[Bia98] [HS14] [AW14]
operator-valued / [BN08] [BPV12] [Spe98, §4.5 - 4.7] [AW16]
multivariable [PV13, §2] [BPV12] [PV13, §3] [Jek20]
Table 7.1: References on non-commutative convolution semigroups.
always differentiable almost everywhere. And in the operator-valued setting, if we have the additional symmetry of stationary increments, the differentiability of FXt in t can be estab-lished by direct arguments with inverse function theorems and/or iteration, as for instance in [AW16, Proposition 3.3]. However, in the non-stationary operator-valued setting (even with the assumption of bounded support), we inevitably run into the issue that not every absolutely continuous function from [0, T ] into a Banach space can be differentiated almost everywhere, even in the weak or weak-∗-topology. But we will circumvent this problem by instead treating the time-derivatives as operator-valued distributions on [0, T ].
The present author studied the monotone case of B-valued Lipschitz subordination fam-ilies with bounded support in [Jek20] (the free and boolean cases being easier to understand by previously existing techniques), and this chapter uses many of the same material as in that paper. We first discuss some preliminary definitions and observations about processes with independent increments. Then we describe the properties of the derivatives of Lips-chitz functions from [0, T ] into Banach spaces, and of LipsLips-chitz families of fully matricial functions. Finally, using these tools, we show that the transforms FXt for a process Xt with independent increments (and some Lipschitz conditions in time) will satisfy the equation above for some σ.
7.1.1 Processes and subordination families
Definition 7.1.1. Let ind ∈ {bool, free, mono, mono †}. A process with B-valued ind-independent increments on [0, T ] is a collection of non-commutative self-adjoint operators (Xt)t∈[0,T ] in B-valued probability space (A, E) such that for every 0 = t0 < t1 < · · · < tN = T , the operators Xt0, Xt1 − Xt0, . . . , XtN − XtN −1 are ind-independent over B.
Another viewpoint on the same idea is to look at the non-commutative law µt of Xt rather than the operator itself. This leads to the following definition.
Definition 7.1.2. Let ind ∈ {bool, free, mono, mono †}. An ind-subordination family on [0, T ] is a collection of non-commutative laws (µt)t∈[0,T ] such that for each 0 ≤ s ≤ t ≤ T , there exists another non-commutative law µs,t such that µt= µsindµs,t.
We make the following observations:
• If (Xt)t∈[0,T ] is a process with independent increments, and if µt is the law of Xt, then (µt)t∈[0,T ] is a subordination family because we can take µs,t to be the law of Xt− Xs.
• Suppose that (µt)t∈[0,T ] is an ind-subordination family. Then there is only one possible choice of µs,t satisfying µt= µsindµs,t. This is because the analytic transforms of µs,t
must satisfy the equations
Kµt= Kµs + Kµs,t, boolean case, Φµt= Φµs + Φµs,t, free case, Fµt = Fµs ◦ Fµs,t, monotone case, Fµt = Fµs,t◦ Fµs, anti-monotone case,
and µs,t is uniquely determined by knowing Kµs,t, Φµs,t, or Fµs,t in a neighborhood of
∞.
• Again, let (µt)t∈[0,T ] be a subordination family. Using associativity of convolution, we have for s ≤ t ≤ u that µu = µs ind(µs,t indµt,u), and therefore it follows that µs,u = µs,tindµt,u by the previous claim about uniqueness of µs,u.
• A desirable property for a subordination chain would be that the rad(µt) is uniformly bounded for t ∈ [0, T ]. However, this is automatic; it follows from Proposition 6.3.1 that
sup
t∈[0,T ]
rad(µt) ≤ (3 + 2√
2) 2 rad(µT) + sup
t∈[0,T ]
kµt(X)k
! .
Remark 7.1.3. It is not difficult to show that any subordination family arises from a process with independent increments. Indeed, if we consider finitely many times 0 = t0 < · · · < tN, then we can construct independent variables Yt0, Yt0,t1, . . . , YtN −1,tN and set Ytj = Yt0 + Yt0,t1 + · · · + YtN −1,tN. Then Yt0, . . . , YtN are a family of variables indexed by {t0, . . . , tN} with independent increments. We can do this for any finite family of times. It remains to show that all the finite-time marginals can be realized simultaneously by the same process.
One can reduce this claim to the case where B is a von Neumann algebra. Then by using compactness in the pointwise WOT of the space of laws of processes satisfying kXtk ≤ C, one can conclude that there is a family (Xt)t∈[0,T ] that realizes each of these finite-time marginals simultaneously. In the last argument, the only challenge is to get a uniform bound on the operator norm Yt over all partitions {t0, . . . , tN} that contain t, in order to obtain the existence of bounded operators (Xt)t∈[0,T ]. This can be done by a careful use of our operator-norm bounds Lemma 6.1.1. However, we will not carry out this argument in detail because we will discuss a more enlightening systematic construction of processes with independent increments in the next chapter, under some continuity assumptions.
7.1.2 Setup and conditions for differentiation
Consider a subordination family (µt)t∈[0,T ]. Under what reasonably general conditions can we expect to be able to differentiate Fµt with respect to t? We know that suptrad(µt) < +∞, so it would be natural for ˜Gµt to also be differentiable with respect to t in a neighborhood of zero, which implies that each of the moments of µt should be differentiable. Thus, we
should at least guarantee that the mean µt(X) and the variance at 1 given by Var(µt)[1] = µt((X − µt(X))2) are differentiable with respect to t, and in fact this will turn out to be sufficient.
Next, under what conditions can we differentiate the maps t 7→ µt(X) and t 7→ µt(X2)?
We should require that they are absolutely continuous as maps from [0, T ] into the Banach space B. That is, for every > 0, there exists δ > 0 such that if {[ai, bi)}Ni=1 are disjoint intervals in [0, T ] with P
i(bi − ai) < δ, then P
ikµbi(X) − µai(X)k < , and the same for µt((X − µt(X))2) rather than µt(X). Now if we let φ(t) and ψ(t) be the total variation of t 7→ µt(X) and t 7→ µt((X − µt(X))2) respectively, then we can reparametrize time using the inverse function of f (t) = φ(t) + ψ(t) + t. Letting νf−1(t), then we have kνt(X) − νs(X)k ≤ C|t − s| for some constant C, and the same is true for νt((X − νt(X))2).
Therefore, if we want to study subordination families where µt(X) and µt((X − µt(X))) are absolutely continuous, then without loss of generality, we can restrict our attention to the case where they are Lipschitz in t. Of course, for many concrete examples of subordination families, the mean and variance might be of the form b0 + tb for some b0, b ∈ B, and the Lipschitz assumption obviously holds in such cases. Thus, we make the following definition.
Definition 7.1.4. Let (µt)t∈[0,T ] be a subordination family with respect to boolean, free, monotone, or anti-monotone independence. We say that (µt)t∈[0,T ] is Lipschitz if t 7→ µt(X) and t 7→ µt((X − µt(X))2) are Lipschitz on [0, T ].
However, even in the Lipschitz case, we run into technical issues with differentiation. Our solution will ultimately be to avoid pointwise differentiation altogether using a distributional theory presented in the next section. As motivation, we will first explain why pointwise differentiation is hopeless in the level of generality we are aiming for, where we do not assume Fµt is C1 in t and where B is allowed to be a general C∗-algebra.
A Lipschitz function from [0, T ] into a Banach space X may not be differentiable almost everywhere with respect to the norm on X or even with respect to the weak topology, or the weak-∗ topology if X happens to be a dual space. Known results about differentiating Banach valued functions (see e.g. [Kom67, Appendix]) rely on either separability or reflex-itvity of X , which is something we cannot assume in an operator algebras setting. Indeed, infinite-dimensional C∗-algebras are never reflexive, and furthermore, infinite-dimensional von Neumann algebras are never separable in the norm topology.
Pointwise differentiation will certainly not be possible in the norm topology. If A is a von Neumann algebra acting on a separable Hilbert space, then differentiation in the strong operator topology (SOT) may be possible (thanks to the theory of differentiation of Hilbert-valued functions). However, in order to use the chain rule and similar manipulations for SOT differentiation (which we will have to do here since we must consider composition of time-dependent functions), we would have to make the additional assumption that the Fr´echet derivatives of the maps we are composing are SOT-continuous, which means making additional SOT continuity assumptions about the laws µt that are not automatic.
Furthermore, suppose that we can for a fixed z, differentiate F (z, t) almost everywhere with respect to t; then it would still be problematic to carry out such differentiation with the same exceptional null set of times for all values of z ranging over an open set in a non-separable Banach space. One might try to solve this problem by assuming that A is separable in SOT and that our analytic functions are continuous in SOT. However, even this is not sufficient because we cannot enforce SOT equicontinuity of (F (z, t + δ) − F (z, t))/δ as δ → 0.
Another possible idea would be to assume that A is tracial von Neumann algebra and that for each function F (z) = z − Gσ(z) that we are dealing with, the state τ ◦ σ is tracial on AhXi. The problems with the SOT approach sketched above would be solved by using explicit estimates in L2 norm to guarantee SOT-equicontinuity of (F (z, t + δ) − F (z, t))/δ for different values of δ, as well as SOT equicontinuity of a 7→ DFt(z)[a] for different values of t.
However, traciality of σ seems like an artificial and restrictive condition. If Fµ(z) = z−Gσ(z), it is unclear (at least to the author) whether traciality of σ and traciality of µ are related.
For free independence, we at least know that if µ and ν can be realized by variables in a tracial von Neumann algebra, then so can µ ν. And perhaps tracial von Neumann algebras are a good place to start developing the theory of non-commutative laws for unbounded operator-valued random variables. However, that is not the goal of the present work.