Akcoglu's Dilation Theory in L1-Spaces

TL;DR

This paper discusses Akcoglu's dilation theory within L1-spaces, outlining its proof and application in embedding probabilistic processes into larger deterministic systems.

Contribution

It provides a detailed outline of Akcoglu's dilation proof, clarifying its use in probabilistic to deterministic system embedding.

Findings

Clarifies Akcoglu's dilation proof

Demonstrates embedding of probabilistic processes into deterministic systems

Enhances understanding of dilation theory in L1-spaces

Abstract

We provide an outline for the proof of Akcoglu's dilation theory used by Gustafson at el (1997) in investigating the embedding of a probabilistic process into a larger deterministic dynamical system.

Equations

$$A^n=P{B}^n|H,$$

$$(J,\mathfrak{B}{)}^n=(J^n,{\mathfrak{B}}^n)\text{mbox}and(J,\mathfrak{B}{)}^{\infty }=(J^{\infty },{\mathfrak{B}}^{\infty }).$$

$$\varsigma (A^{\prime })=\vartheta ({\rho }^{-1}(A^{\prime })),A^{\prime }\in {{\mathcal{S}}^{\prime }}^{{}^{\prime }},$$

$$\varpi (A\times M)={\int }_A\eta (M,s)\vartheta (ds),$$

$$(Ef)(s)={\int }_{Y}f(s,y)\eta (dy,s),f\in L_1(Z,\Im ,\varpi ),s\in S.$$

$$(Tf)(x_0)=(T1)(x_0){\int }_{J_1}f({\phi }_0^{-1}(x_0,x_1))d{x}_1$$

$${\tau }_{i}x$$

$${\tau }_{0}x$$

$${\tau }_{1}x$$

$${\mu }_{-\infty }^{\infty }$$

$${\nu }_{-\infty }^{\infty }$$

$$\frac{d{\nu }_{-\infty }^{\infty }}{d{\mu }_{-\infty }^{\infty }}(\cdots ,x_{-1},x_0,x_1,\cdots )=\frac{d\nu }{d\mu }(x_0)=(T1)(x_0),$$

$${\int }_{h^{-1}A}fd\mu ={\int }_{A}Qfd\mu$$

$$(Qf)(x)=\frac{d\nu }{d\mu }(x)f(h^{-1}x).$$

$$(Qf)(\cdots ,x_{-1},x_0,x_1,\cdots )$$

$$=(T1)(x_0)f({\tau }^{-1}(\cdots ,x_{-1},x_0,x_1,\cdots )),$$

Full text

1. Introduction

The motivation for studying dilations in operator theory was to gain insight into the properties of an operator on a smaller space by using the properties of the operator on a larger space.

Over the years, researchers have applied results from dilation theory to mathematical physics. One example is the study conducted by Gustafson et al. in investigating the embedding of a probability process into a larger deterministic dynamical system [8, 9]. 111R.J. Reynolds Tobacco Company requested a reprint of [8] shortly after it was published [11]. The physical significance of their research is related to the work on irreversible processes by Proggigne, a Nobel Prize chemist [18].

In their work [8, 9], Gustafson et al. used two dilation theories: one from Rokhlin [7, 23, 21], the other from Akcoglu [1, 2, 3, 4]. A general result was proved in [9] using Akcoglu’s dilation theorem.

With deep and interesting results, Akcoglu’s dilation theorem was first proved in $L_{1}$ and subsequently generalized to $L_{p}$. The original proof in Akcoglu’s dilation theorem was rather complicated and simpler versions of the proof were presented [14, 19]. Nonetheless, it may still be worthwhile for those who are new to the field to go through Akcoglu’s proof. With this in mind, the purpose of this article is to provide an outline for the proof in $L_{1}$ [1], delineating the interplay of several fundamental ideas.

The organization of this article is as follows. In Section 2, we present the definition of a dilation, the most general result of Gustafson et al. in [9] and Akcoglu’s dilation theorem in $L_{1}$. In section 3, we outline the main ideas of Akcgolu’s proof. In the last section, we state the open problem discussed in [10] and present a dilation result from Rota [22].

2. Preliminaries

We first introduce the definition of a dilation. An operator $B$ on a Hilbert space $K$ is said to be a “dilation” of an operator $A$ on a Hilbert space $H$ provided $H$ is a subspace of $K$ and for all positive integers $n$,

[TABLE]

where P is the orthogonal projection of $K$ onto $H$ with range $H$. (The original definition introduced by Paul Halmos was a weaker form than the above as it did not have the power $n$ [12]).

A landmark theorem on dilations was proven by Sz-Nagy, which states that very contraction has a power dilation that is unitary [24]. For a historical development on dilation theory, see [6, 13, 15].

The dilation theories used to determine the types of Markov semigroup which could be embedded in a deterministic dynamical system without approximations in [8, 9] include that of Rokhlin and Akcoglu. Roughly speaking, Sz-Nagy’s dilation theory can be viewed as dilating Hilbert spaces, Rohlin’s as dilating the underlying dynamical systems and Akcoglu’s the measure spaces.

Using Akcoglu’s dilation theorem, Gustafson et at. proved that an arbitrary Markov semigroup satisfying certain basic properties can be dilated into a deterministic dynamical system.

Theorem 1**.**

Every Markov semigroup $M_{t}$ acting on the Hilbert space $K$ arises as a projection of a dynamical system in a larger Hilbert space $H$.

The Markov semigroup family $\{M_{t}\}_{t\geq 0}$ is defined in [9] as a semigroup of Markov operators, which are the positive integral-preserving contractions $T$ in Akcoglu’s Theorem (see Theorem 2 below). Under certain assumptions, a Markov operator is induced by transition probabilities, which in turn, determines a stochastic process called the Markov process (see Proposition V.4.4 in [20] and [5]).

In the next section, we state Akcoglu’s dilation theorem and outline its proof.

3. Akcoglu’s Theorem

Akcoglu’s dilation result (see Theorem 2 below) was first proved in $L_{1}$ [1] and then extended to $L_{p}$ [2, 3, 4]. Since the ideas in the $L_{p}$ proof are similar to those in $L_{1}$, for the rest of the paper, we outline the $L_{1}$ proof in [1].

Here is the dilation result in $L_{1}$ [1]:

Theorem 2**.**

Let $(X,\mho,\mu)$ be a Borel Space and let $T$ be a positive contraction on $L_{1}(X,\mho,\mu)$. Then there exists another Borel Space $(Y,\Im,\nu)$ and a non-singular invertible transformation $\tau:Y\to Y$ so that the positive isometry $Q$ on $L_{1}(Y,\Im,\nu)$ by $\tau$ is a dilation of $T$.

The setup of the proof is as follows:

Let $J=[0,1]$, $\mathfrak{B}$ be the $\sigma$-algebra of the Borel subsets of $J$ and $\mu$ is the (finite) measure on $(J,\mathfrak{B})$. The cartesian product of finitely or countably many copies of $(J,\mathfrak{B})$ is denoted by:

[TABLE]

Also, let $(J_{i},\mathfrak{B}_{i})$ be the copies of $(J,\mathfrak{B})$.

Let $f\in L_{1}(J^{\infty}_{-\infty},\mathfrak{B}^{\infty}_{-\infty},\mu^{\infty}_{-\infty})$ be a function depending only on $x_{0}$ and consider $f$ being a member of $L_{1}(J,\mathfrak{B},\mu)$ also. Then Theorem 2 was proved by showing that $EQ^{n}f=T^{n}f$ where

• •

$T$ is a positive contraction on $L_{1}(J_{0},\mathfrak{B}_{0},\mu)$,

• •

$Q$ is the positive contraction on $L_{1}(J^{\infty}_{-\infty},\mathfrak{B}^{\infty}_{-\infty},\mu^{\infty}_{-\infty})$, and

• •

$E$ is the conditional expectation operator with respect to $\mathfrak{B}_{0}$ i.e. $E:L_{1}(J^{\infty}_{-\infty},\mathfrak{B}^{\infty}_{-\infty},\mu^{\infty}_{-\infty})\to L_{1}(J,\mathfrak{B}_{0},\mu)$.

The two main questions that the proof has to address are:

1. (1)

From the original measure space $X$, how do we obtain a larger space $Y$? 2. (2)

How do we obtain the dilation $Q$ from the larger space $Y$?

A short answer to the first question is to ”dilate” the measure space $X$ via conditioned measures (see Section 4.1.3). As for the second question, once we obtain a larger space $Y$, we can then construct a non-singular invertible transformation $\tau$, which in turn invokes a Frobenius-Perron operator serving as the dilation of $T$ (see Section 4.5). We will give a more detailed discussion in the next section.

4. Outline of Proof

4.1. Definitions

4.1.1. Positive Contractions

A linear operator $T:L_{1}(S,\mathcal{S},\vartheta)\to L_{1}(S,\mathcal{S},\vartheta)$ is called a positive operator if $Tf\geq 0$ for every $f\geq 0$ in $L_{1}(S,\mathcal{S},\vartheta)$. It is a contraction if its norm is less than 1, that is, $||T||\leq 1$.

4.1.2. Transporting Measures

Suppose that $(S,\mathcal{S},\vartheta)$ is a finite measure space. Let $\rho$ be a measurable map from $(S,\mathcal{S})$ to another measurable space $(S^{\prime},\mathcal{S}^{{}^{\prime}})$. If $\rho$ is invertible, that is, it has a measurable inverse, then the set function $\varsigma$ on $\mathcal{S}^{{}^{\prime}}$ defined by

[TABLE]

is a measure in $(S^{\prime},\mathcal{S}^{{}^{\prime}})$ and is called the transport of the measure $\vartheta$ via $\rho$ i.e. $\rho$ transports measure $\vartheta$ to $\varsigma$. (If $\vartheta$ is a probability measure and $(S^{\prime},\mathcal{S}^{{}^{\prime}})=(\mathbb{R}^{n},\mathcal{R}^{n})$, then $\varsigma$ is called the distribution of $\rho$.)

4.1.3. Conditioned Measures

Let $(S,\mathcal{S})$ and $(\Xi,\mathcal{M})$ be two measurable spaces. A family of normalized measures $\{\eta\}$ on $(\Xi,\mathcal{M})$ is said to be conditioned by $(S,\mathcal{S})$ if this family is indexed by $s\in S$ and denoted by $\{\eta\}_{S}$. Let $(Z,\Im)=(S,\mathcal{S})\times(\Xi,\mathcal{M})$. With the conditioned family $\{\eta\}_{S}$ on $(\Xi,\mathcal{M})$ and a measure $\vartheta$ on $(S,\mathcal{S})$, there exists a measure $\varpi=\vartheta\times\{\eta\}$ on $(Z,\Im)$ such that

[TABLE]

for each $A\in\mathcal{S}$ and $M\in\mathcal{M}$.

With $\varpi=\varphi\times\{\eta\}$, we can define the conditional expectation operator $E:L_{1}(Z,\Im,\varpi)\to L_{1}(Z,\mathcal{S},\varpi)$ with respect to $\mathcal{S}$ as:

[TABLE]

4.1.4. Equivalence

A mapping between two measurable spaces is called an equivalence if it is measurable and invertible in both directions.

4.2. Obtaining a Larger Space

Conditioned families of measures were investigated by both Rokhlin [21] and Maharam [17]. The Rokhlin-Maharam theorem, stated in the following form in [1] and [3], provides us with tools to dilate spaces.

Theorem 3** (Rokhlin-Maharam Theorem).**

Let $(\Omega,\Sigma,\sigma)$ be a Borel space and let $f:\Omega\to J$ be a measurable function transporting $\sigma$ to a measure $\nu$ on $(J,\mathfrak{B})$. Then there exists an isomorphism $\Phi:J^{2}\to\Omega$ between $(J^{2},\mathfrak{B}^{2},\nu\times\{\eta\})$ and $(\Omega,\Sigma,\sigma)$, for some choice of the conditioned family $\{\eta\}_{J}$, so that $f\Phi:J^{2}\to J$ is the projection of $J^{2}$ to its first component $J$, $\nu\times{\eta}-$ a.e. (If $(\Omega,\Sigma)$ is equivalent to $(J,\mathfrak{B})$, then $\Phi$ can be chosen as an equivalence.)

The Rokhlin-Maharam Theorem shows how we can embed $J$ (the first component) in $J^{2}$ and then recover it from $J^{2}$ via an isomorphism $\Phi$ and a conditioned family $\{\eta\}_{J}$:

\Omega$$J$$J\times J$$f$$\Phi$$f\circ\Phi

We will briefly show how the Rokhlin-Maharam Theorem is used in the following theorem, which is key to proving Akcoglu’s dilation result in Theorem 2.

Theorem 4**.**

Let $T$ be an integral-preserving positive contraction on $L_{1}(J_{0},\mathfrak{B}_{0},\mu)$. Then there exists a conditioned family $\{\alpha\}=\{\alpha\}_{J_{0}}$ on $(J_{-1},\mathfrak{B}_{-1})$ and an equivalence $\varphi:J_{-1}\times J_{0}\to J_{0}\times J_{1}$ so that $\varphi$ transports $\{\alpha\}\times\mu$ on $J_{-1}\times J_{0}$ to $\nu\times\lambda$ on $J_{0}\times J_{-1}$ where $d\nu=(T1)d\mu$, so that

[TABLE]

for each $f\in L_{1}(J_{0},\mathfrak{B}_{0},\mu)$ and for $\mu$-a.a. $x_{0}\in J_{0}$. Here the integration is with respect to the standard Borel measure $\lambda$ on $(J_{1},\mathfrak{B}_{1})$ and $\varphi^{-1}_{0}(x_{0},x_{1})$ denotes the $J_{0}$-coordinate of $\varphi^{-1}(x_{0},x_{1})\in J_{-1}\times J_{0}$, where $(x_{0},x_{1})\in J_{0}\times J_{1}$.

Here is how the Rokhlin-Maharam Theorem is used. We first start with a measurable function $g:J_{0}\times J_{1}\to J$. Then apply Theorem 3 to $g$ and obtain an equivalence $\varphi:J_{-1}\times J_{0}\to J_{0}\times J_{1}$ on $(J_{-1},\mathfrak{B_{-1}})$ so that $g\varphi:J_{-1}\times J_{0}\to J$ is the identification of $J_{0}$-component in $J_{-1}\times J_{0}$ via the conditioned family $\{\alpha\}_{J_{0}}$ on $(J_{-1},\mathfrak{B}_{-1})$ as illustrated below:

J_{0}\times J_{1}$$J$$J_{-1}\times J_{0}$$g$$\varphi$$g\circ\varphi

4.3. Integral Preserving

Note that the positive contraction $T$ in Theorem 4 is integral-preserving, that is, $\int fd\mu=\int Tfd\mu$ for all $f\in L_{1}$. With the following theorem [1], in proving Akcoglu’s dilation result, we can reduce the case of a general positive contraction to that of an integral preserving positive contraction.

Theorem 5**.**

Every positive contraction on the $L_{1}$ space of a Borel space has a dilation to an operator of the same type which is also integral preserving.

4.4. Construct $\tau$

Recall our ultimate goal is to obtain a dilation defined on $L_{1}(J^{\infty}_{-\infty},\mathfrak{B}^{\infty}_{-\infty},\mu^{\infty}_{-\infty})$. To do that, we first need to use $\varphi$ in Theorem 4 to construct an equivalence $\tau:J^{\infty}_{-\infty}\to J^{\infty}_{-\infty}$ on the measurable space $(J^{\infty}_{-\infty},\mathfrak{B}^{\infty}_{-\infty})$ as follows: Recall that $\varphi$ is defined as $\varphi:J_{-1}\times J_{0}\to J_{0}\times J_{1}$. For each $x\in J^{\infty}_{-\infty}$, we define $\tau$ as:

[TABLE]

Then via the conditioned measures $\{\alpha_{n}\}_{J^{0}_{-n+1}}$ on $(J_{-n},\mathfrak{B}_{-n})$, it can be shown that $\tau$ transports the measure

[TABLE]

on $(J^{\infty}_{-\infty},\mathfrak{B}^{\infty}_{-\infty},\mu^{\infty}_{-\infty})$ to

[TABLE]

on the same space. It follows then that

[TABLE]

which is a key result to use in obtaining a dilation $Q$ as shown in the next section.

4.5. Obtaining a Dilation

So far, we still have not shown how an integral-preserving positive contraction $T$ can invoke a dilation operator $Q$. The key to that issue lies with the transformation $\tau$: as a nonsingular transformation, $\tau$ induces a positive contraction called the Froebenius-Perron operator, which becomes the dilation operator $Q$.

Here is the definition of a Frobenius-Perron operator.

4.5.1. Frobenius-Perron Operator

If $h:X\to X$ is a nonsingular transformation, then $h$ induces a positive contraction $Q$ of $L_{1}(X,\mathcal{A},\mu)$ defined uniquely by:

[TABLE]

for each $f\in L_{1}(X,\mathcal{A},\mu)$ and for each $A\in\mathcal{A}$ and we call $Q$ the Frobenius-Perron operator.

If $h$ is invertible and we let $\nu(A)=\mu(h^{-1}A)$, that is, $h$ transports measure $\mu$ to $\nu$, then

[TABLE]

(See Lasota (1994), Ch 1 & 3 in [16].)

4.5.2. $\tau$ induces Q

Recall that $\tau$ is defined on $(J^{\infty}_{-\infty},\mathfrak{B}^{\infty}_{-\infty},\mu^{\infty}_{-\infty})$. Since $\tau$ is invertible and non-singular (because $\nu^{\infty}_{-\infty}$ is absolutely continuous with respect to $\mu^{\infty}_{-\infty}$), it invokes a positive contraction $Q$ of $L_{1}(J^{\infty}_{-\infty},\mathfrak{B}^{\infty}_{-\infty},\mu^{\infty}_{-\infty})$ as in (4.3). Then by (4.4) and (4.2), we obtain

[TABLE]

which links the dilation $Q$ with the original operator $T$. Then it can be shown that $EQf=EQEf$ and eventually $EQ^{n}f=T^{n}f$.

5. Open Question

An open question was raised in [10] as to whether the semigroup property of $M_{t}$ in Theorem 1 can be relaxed to some wider stochastic structures which are of more martingale-type or which permit memory effects.

While the open problem still remains unsolved, we state without proof an interesting result from Rota relating dilations and reverse martingales [22].

Here is Rota’s dilation theorem.

Theorem 6**.**

Let $P$ be a linear positive contraction in $L_{2}(S,\sum,\mu)$ which is self-adjoint and maps the constant function of value 1 to 1 i.e. $P1=1$. Then

1. (1)

there is a dilation of the sequence of operators $P^{2n}$ into a (reversed) martingale $E_{n}$ and 2. (2)

for $f$ in $L_{p}(S,\mathcal{F},\mu)$, $p>1$, $lim_{n\to\infty}P^{2n}f$ exists almost everywhere.

Here, the dilation is in the sense that $P^{2n}=\hat{E}\circ E_{n}$ where $E_{n}$ are conditional expectations of a decreasing filtration and $\hat{E}$ is a conditional expectation projecting onto $L_{2}(S,\sum,\mu)$.