New York Journal of Mathematics

New York J. Math. 14 (2008) 539–576.

On automorphisms of type II Arveson systems (probabilistic approach)

Boris Tsirelson

Abstract. We give a counterexample to the conjecture that the automorphisms of an arbitrary Arveson system act transitively on its normalized units.

Contents

Introduction 539 1. Definitions, basic observations, and the result formulated 540 2. Extensions of automorphisms and isomorphisms of extensions 544 3. Toy models: Hilbert spaces 547 4. Toy models: probability spaces 548 5. Binary extensions: probability spaces 550 6. Binary extensions: Hilbert spaces 553 7. Products of binary extensions 555 8. Some necessary conditions of isomorphism 557 9. A binary extension of Brownian motion 561 10. A new noise extending the white noise 564 11. The binary extension inside the new noise 569 References 575

Introduction

We do not know how to calculate the gauge group in this generality…

W. Arveson [1, Section 2.8]

539^{ISSN 1076-9803/08}

Received April 25, 2008.

Mathematics Subject Classification. Primary 46L55, secondary 60G99.

Key words and phrases. product system, automorphism.

This research was supported by the israel science foundation (grant No. 683/05).

At the moment, most important seems to us to answer the question whether the automorphisms of an arbitrary product system act transitively on the normalized units.

V. Liebscher [5, Section 11]

By an Arveson system I mean a product system as defined by Arveson [1, 3.1.1]. Roughly, it consists of Hilbert spaces H_{t} (for 0 < t < ∞) satisfying H_{s} ⊗ H_{t} = H_{s+t}. Classical examples are given by Fock spaces; these are type I systems, see [1, 3.3 and Part 2]. Their automorphisms are described explicitly, see [1, 3.8.4]. The group of automorphisms, called the gauge group of the Arveson system, for type I is basically the group of motions of the N-dimensional Hilbert space. The parameter N ∈ {0, 1, 2, … } ∪ {∞} is the so-called (numerical) index; accordingly, the system is said to be of type I_{0}, I_{1}, I_{2}, … or I_{∞}. All Hilbert spaces are complex (that is, over ℂ).

Some Arveson systems contain no type I subsystems; these are type III systems, see [1, Part 5]. An Arveson system is of type II, if it is not of type I, but contains a type I subsystem. (See [9, 6g and 10a] for examples.) In this case the greatest type I subsystem exists and will be called the classical part of the type II system. The latter is of type II_{N} where N is the index of its classical part.

Little is known about the gauge group of a type II system and its natural homomorphism into the gauge group of the classical part. In general, the homomorphism is not one-to-one, and its range is a proper subgroup. The corresponding subgroup of motions need not be transitive, which is the main result of this work (Theorem 1.10); it answers a question asked by Liebscher [5, Notes 3.6, 5.8 and Section 11 (question 1)] and (implicitly) Bhat [2, Definition 8.2]; see also [9], Question 9d3 and the paragraph after it. A partial answer is obtained by Markiewicz and Powers [6] using a different approach.

Elaborate constructions (especially, counterexamples) in a Hilbert space often use a coordinate system (orthonormal basis). In other words, the sequence space l_{2} is used rather than an abstract Hilbert space. An Arveson system consists of Hilbert spaces, but we cannot choose their bases without sacrificing the given tensor product structure. Instead, we can choose maximal commutative operator algebras, which leads to the probabilistic approach. Especially, the white noise (or Brownian motion) will be used rather than an abstract type I_{1} Arveson system.

1. Definitions, basic observations, and the result formulated

I do not reproduce here the definition of an Arveson system [1, 3.1.1], since we only need the special case

(1.1)  H_{t} = L_{2}(Ω, F_{0,t}, P)

corresponding to a noise.

Definition 1.2. A noise consists of a probability space (Ω, F, P), subσ-fields F_{s,t} ⊂ F given for all s, t ∈ ℝ, s < t, and a measurable action (T_{h})_{h} of ℝ on Ω, having the following properties:

(a) F_{r,s} ⊗ F_{s,t} = F_{r,t} whenever r < s < t ,

(b) T_{h} sends F_{s,t} to F_{s+h,t+h} whenever s < t and h ∈ ℝ ,

(c) F is generated by the union of all F_{s,t} .

See [9, 3d1] for details. As usual, all probability spaces are standard, and everything is treated mod 0. Item (a) means that F_{r,s} and F_{s,t} are (statistically) independent and generate F_{r,t}. Invertible maps T_{h} : Ω → Ω preserve the measure P.

The white noise is a classical example; we denote it (Ω^{white}, F^{white}, P^{white}), (F_{s,t}^{white})_{s<t}, (T_{h}^{white})_{h}. It is generated by the increments of the one-dimen-sional Brownian motion (B_{t})_{−∞<t<∞}, B_{t} : Ω → ℝ.

Given a noise, we construct Hilbert spaces H_{t} consisting of F_{0,t}-measur-able complex-valued random variables, see (1.1). The relation

H_{s} ⊗ H_{t} = H_{s+t},

or rather a unitary operator H_{s} ⊗ H_{t} → H_{s+t}, emerges naturally:

H_{s+t} = L_{2}(F_{0,s+t}) = L_{2}(F_{0,s} ⊗ F_{s,s+t}) = L_{2}(F_{0,s}) ⊗ L_{2}(F_{s,s+t})
= L_{2}(F_{0,s}) ⊗ L_{2}(F_{0,t}) = H_{s} ⊗ H_{t};

the time shift T_{s} is used for turning F_{s,s+t} to F_{0,t}. Thus, (H_{t})_{t>0} is an Arveson system. Especially, the white noise leads to an Arveson system (H_{t}^{white})_{t>0} (of type I_{1}, as will be explained).

For X ∈ H_{s}, Y ∈ H_{t} the image of X ⊗ Y in H_{s+t} will be denoted simply XY (within this section).

We specialize the definition of a unit [1, 3.6.1] to systems of the form (1.1).

Definition 1.3. A unit (of the system (1.1)) is a family (u_{t})_{t>0} of nonzero vectors u_{t} ∈ H_{t} = L_{2}(F_{0,t}) ⊂ L_{2}(F) such that t ↦ u_{t} is a Borel measurable map (0, ∞) → L_{2}(F), and

u_{s}u_{t} = u_{s+t} for all s, t > 0 .

(In other words, the given unitary operator H_{s} ⊗H_{t} → H_{s+t} maps u_{s} ⊗u_{t} to u_{s+t}.) The unit is normalized, if ∥u_{t}∥ = 1 for all t. (In general,

∥u_{t}∥ = exp(ct)

for some c ∈ ℝ.)

Here is the general form of a unit in (H_{t}^{white})_{t}:

u_{t} = exp(zB_{t} + z_{1}t) ; z, z_{1} ∈ ℂ;

it is normalized iff (Re z)^{2} + Re z_{1} = 0. The units generate (H_{t}^{white})_{t} in the following sense: for every t > 0, H_{t}^{white} is the closed linear span of vectors of the form (u_{1})_{\frac{t}{n}} (u_{2})_{\frac{t}{n}} … (u_{n})_{\frac{t}{n}}, where u_{1}, …, u_{n} are units, n = 1, 2, . . .. Indeed, L_{2}(F_{0,t}) is spanned by random variables of the form exp(i ∫_{0}^{t} f(s) dB_{s}) where f runs over step functions (0, t) → ℝ constant on (0, \frac{1}{n}t), …, (\frac{n−1}{n}t, t).

We specialize two notions, ‘type I’ and ‘automorphism’, to systems of the form (1.1).

Definition 1.4. A system of the form (1.1) is of type I if it is generated by its units.

We see that (H_{t}^{white})_{t} is of type I.

Definition 1.5. An automorphism (of the system (1.1)) is a family (Θ_{t})_{t>0} of unitary operators Θ_{t} : H_{t} → H_{t} such that Θ_{s+t}(XY) = (Θ_{s}X)(Θ_{t}Y) for all X ∈ H_{s}, Y ∈ H_{t}, s > 0, t > 0, and the function t ↦ ⟨Θ_{t}X_{t}, Y_{t}⟩ is Borel measurable whenever t ↦ X_{t} and t ↦ Y_{t} are Borel measurable maps (0, ∞) → L_{2}(F) such that X_{t}, Y_{t} ∈ L_{2}(F_{0,t}) ⊂ L_{2}(F).

Basically, Θ_{s}⊗Θ_{t} = Θ_{s+t}. The group G of all automorphisms is called the gauge group. Clearly, G acts on the set of normalized units, (u_{t})_{t} ↦ (Θ_{t}u_{t})_{t}.

Automorphisms Θ_{t} = Θ_{t}^{trivial(λ)} = e^{iλt} (for λ ∈ ℝ), consisting of scalar operators, will be called trivial; these commute with all automorphisms, and are a one-parameter subgroup G^{trivial} ⊂ G. Normalized units (u_{t})_{t} and (e^{iλt}u_{t})_{t} will be called equilavent. The factor group G/G^{trivial} acts on the set of all equivalence classes of normalized units.

We turn to the gauge group G^{white} of the classical system (H_{t}^{white})_{t}. Equivalence classes of normalized units of (H_{t}^{white})_{t} are parametrized by numbers z ∈ ℂ, since each class contains exactly one unit of the form

u_{t} = exp(zB_{t} − (Re z)^{2}t) .

The scalar product corresponds to the distance:

|⟨u_{t}^{(1)}, u_{t}^{(2)}⟩| = exp(−\frac{1}{2}|z_{1} − z_{2}|^{2}t)

for u_{t}^{(k)} = exp(z_{k}B_{t} − (Re z_{k})^{2}t), k = 1, 2. The action of G^{white}/G^{trivial} on equivalence classes boils down to its action on ℂ by isometries. The orientation of ℂ is preserved, since

\frac{⟨u_{t}^{(1)}, u_{t}^{(2)}⟩⟨u_{t}^{(2)}, u_{t}^{(3)}⟩⟨u_{t}^{(3)}, u_{t}^{(1)}⟩}{|⟨u_{t}^{(1)}, u_{t}^{(2)}⟩⟨u_{t}^{(2)}, u_{t}^{(3)}⟩⟨u_{t}^{(3)}, u_{t}^{(1)}⟩|} = exp(itS(z_{1}, z_{2}, z_{3})),

where S(z_{1}, z_{2}, z_{3}) = Im((z_{2} − z_{1})\overline{(z_{3} − z_{1})}) is twice the signed area of the triangle. So, G^{white}/G^{trivial} acts on ℂ by motions (see [1, 3.8.4]).

Shifts of ℂ along the imaginary axis, z ↦ z + iλ (for λ ∈ ℝ) emerge from automorphisms

Θ_{t} = Θ_{t}^{shift(iλ)} = exp(iλB_{t});

here the random variable exp(iλB_{t}) ∈ L_{∞}(F_{0,t}^{white}) is treated as the multiplication operator, X ↦ X exp(iλB_{t}) for X ∈ L_{2}(F_{0,t}^{white}).

Shifts of ℂ along the real axis, z ↦ z + λ (for λ ∈ ℝ) emerge from less evident automorphisms

(1.6)  Θ_{t}^{shift(λ)}X = D_{t}^{1/2} · (X ∘ θ_{t}^{λ});

here θ_{t}^{λ} : C[0, t] → C[0, t] is the drift transformation (θ_{t}^{λ}b)(s) = b(s) − 2λs (for s ∈ [0, t]), D_{t} is the Radon–Nikodym derivative of the Wiener measure shifted by θ_{t}^{λ} w.r.t. the Wiener measure itself,

(1.7)  D_{t} = exp(2λB_{t} − 2λ^{2}t),

and X ∈ L_{2}(F_{0,t}^{white}) is treated as a function on C[0, t] (measurable w.r.t. the Wiener measure). Thus,

(Θ_{t}^{shift(λ)}X)(b) = exp(λb(t) − λ^{2}t)X(θ_{t}^{λ}b).

By the way, these two one-parameter subgroups of G^{white} satisfy Weyl relations

Θ_{t}^{shift(λ)}Θ_{t}^{shift(iμ)} = e^{−2iλμt}Θ_{t}^{shift(iμ)}Θ_{t}^{shift(λ)};

that is, Θ^{shift(λ)}Θ^{shift(iμ)} = Θ^{trivial(−2λμ)}Θ^{shift(iμ)}Θ^{shift(λ)}.

Rotations of ℂ around the origin, z ↦ e^{iλ}z (for λ ∈ ℝ) emerge from automorphisms Θ^{rotat(λ)}. These will not be used, but are briefly described anyway. They preserve Wiener chaos spaces H_{n},

Θ_{t}^{rotat(λ)}X = e^{inλ}X for X ∈ H_{n} ∩ L_{2}(F_{0,t}^{white}) ;

the n-th chaos space H_{n} ⊂ L_{2}(F^{white}) consists of stochastic integrals

X = ∫ ⋯ ∫_{−∞<s_{1}<···<s_{n}<∞} f(s_{1}, …, s_{n}) dB_{s_{1}} … dB_{s_{n}}

where f ∈ L_{2}(ℝ^{n}) (or rather, the relevant part of ℝ^{n}). One may say that Θ^{rotat(λ)} just multiplies each dB_{s} by e^{iλ}.

Combining shifts and rotations we get all motions of ℂ. Accordingly, all automorphisms of (H_{t}^{white})_{t} are combinations of Θ^{shift(iλ)}, Θ^{shift(λ)}, Θ^{rotat(λ)} and Θ^{trivial(λ)}. More generally, the N-dimensional Brownian motion leads to the (unique up to isomorphism) Arveson system of type I_{N} and motions of ℂ^{N}. We need N = 1 only; (H_{t}^{white})_{t} is the Arveson system of type I_{1}.

Some noises are constructed as extensions of the white noise,

(1.8)  F_{s,t} ⊃ F_{s,t}^{white}

(also T_{h} conforms to T_{h}^{white}). More exactly, it means that B_{t} ∈ L_{2}(F) are given such that B_{t} − B_{s} is F_{s,t}-measurable for −∞ < s < t < ∞, and B_{t}−B_{s} ∼ N(0, t−s) (that is, the random variable (t−s)^{−1/2}(B_{t}−B_{s}) has the standard normal distribution), and B_{0} = 0, and (B_{t}−B_{s})∘T_{h} = B_{t+h}−B_{s+h}. Such (B_{t})_{t} may be called a Brownian motion adapted to the given noise. Then, of course, by F_{s,t}^{white} we mean the sub-σ-field generated by B_{u} − B_{s} for all u ∈ (s, t). The Arveson system (H_{t})_{t}, H_{t} = L_{2}(F_{0,t}), is an extension of the type I_{1} system (H_{t}^{white})_{t}, H_{t}^{white} = L_{2}(F_{0,t}^{white}),

(1.9)  H_{t} ⊃ H_{t}^{white}.

All units of (H_{t}^{white})_{t} are also units of (H_{t})_{t}. It may happen that (H_{t})_{t} admits no other units even though F_{s,t} ≠ F_{s,t}^{white}, H_{t} ≠ H_{t}^{white}. Then (H_{t})_{t} is of type II (units generate a nontrivial, proper subsystem), namely, of type II_{1}; (H_{t}^{white})_{t} is the classical part of (H_{t})_{t}, and the white noise is the classical part of the given noise. The automorphisms Θ^{trivial}(λ) and Θ^{shift(iλ)} for λ ∈ ℝ can be extended naturally from the classical part to the whole system (which does not exclude other possible extensions). For Θ^{shift(λ)} and Θ^{rotat(λ)} we have no evident extension. Moreover, these automorphisms need not have any extensions, as will be proved.

Two examples found by Warren [11], [12] are ‘the noise of splitting’ and ‘the noise of stickiness’; see also [13] and [9, Section 2]. For the noise of splitting the gauge group restricted to the classical part covers all shifts of ℂ (but only trivial rotations [10]), thus, it acts transitively on ℂ, therefore, on normalized units as well.

A new (third) example is introduced in Section 10 for proving the main result formulated as follows.

Theorem 1.10. There exists an Arveson system of type II_{1} such that the action of the group of automorphisms on the set of normalized units is not transitive.

The proof is given in Section 11, after the formulation of Proposition 11.1. The first version [8] of this paper raised some doubts [3, p. 6]. Hopefully

they will be dispelled by the present version.

First of all, in Section 2 we reformulate the problem as a problem of isomorphism. Isomorphism of some models simpler than Arveson systems are investigated in Sections 3–9. In Section 11 we reduce the problem for Arveson systems to the problem for the simpler models. In combination with the new noise of Section 10 it proves Theorem 1.10.

2. Extensions of automorphisms and isomorphisms of extensions

Assume that a given noise ((Ω, F, P), (F_{s,t}), (T_{h})) is an extension of the white noise (see (1.8) and the explanation after it) generated by a given Brownian motion (B_{t})_{t} adapted to the given noise. Assume that another noise ((Ω′, F′, P′), (F′_{s,t}), (T′_{h})) is also an extension of the white noise, according to a given adapted Brownian motion (B′_{t})_{t}. On the level of Arveson systems we have two extensions of the type I_{1} system:

H_{t} ⊃ H_{t}^{white} , H′_{t} ⊃ H_{t}^{′ white} ;

here H_{t} = L_{2}(Ω, F_{0,t}, P), H_{t}^{white} = L_{2}(Ω, F_{0,t}^{white}, P), F_{0,t}^{white} being generated by the restriction of B to [0, t]. (H′_{t} and H_{t}^{′ white} are defined similarly.)

An isomorphism between the two Arveson systems (H_{t})_{t}, (H′_{t})_{t} is defined similarly to 1.5 (Θ_{t} : H_{t} → H′_{t}, Θ_{s+t} = Θ_{s} ⊗ Θ_{t}, and the Borel measurability). If it exists, it is nonunique. In contrast, the subsystems (H_{t}^{white})_{t} and (H_{t}^{′ white})_{t} are naturally isomorphic:

Θ_{t}^{transfer}(X(B|_{[0,t]})) = X(B′|_{[0,t]}) for all X;

here B|_{[0,t]} is treated as a C[0, t]-valued random variable on Ω, distributed W_{t} (the Wiener measure); similarly, B′|_{[0,t]} is a C[0, t]-valued random variable on Ω′, distributed W_{t}; and X runs over L_{2}(C[0, t], W_{t}).

We define an isomorphism between extensions as an isomorphism (Θ_{t})_{t} between Arveson systems that extends Θ^{transfer}, that is,

Θ_{t}|_{H_{t}^{white}} = Θ_{t}^{transfer} for all t.

Adding a drift to the Brownian motion (B_{t})_{t} we get a random process (B_{t} + λt)_{t} locally equivalent, but globally singular to the Brownian motion. In terms of noises this idea may be formalized as follows.

Let (Ω, F, P̃) be a probability space, F_{s,t} ⊂ F sub-σ-fields, and (T_{h})_{h} a measurable action of ℝ on Ω, satisfying Conditions (b) and (c) of Definition 1.2 (but not (a)). Let P, P′ be (T_{h})-invariant probability measures on (Ω, F) such that P + P′ = 2P̃, and 1.2(a) holds for each of the two measures P, P′. Then we have two noises ((Ω, F, P), (F_{s,t}), (T_{h})), ((Ω, F, P′), (F_{s,t}), (T_{h})). Assume also that the restrictions P |_{F_{s,t}} and P′|_{F_{s,t}} are equivalent (that is, mutually absolutely continuous) whenever s < t. This relation between two noises may be called a change of measure. The corresponding Arveson systems are naturally isomorphic (via multiplication by the Radon–Nikodym derivative):

Θ_{t}^{change} : H_{t} → H′_{t}, Θ_{t}^{change}ψ = D_{t}^{−1/2}ψ, D_{t} = \frac{dP′|_{F_{0,t}}}{dP |_{F_{0,t}}} .

We are especially interested in a change of measure such that (recall (1.7))

D_{t} = exp(2λB_{t} − 2λ^{2}t) for t ∈ (0, ∞),

where (B_{t})_{t} is a Brownian motion adapted to the first noise, and λ ∈ ℝ a given number. In this case (B_{t} − 2λt)_{t} is a Brownian motion adapted to the second noise. We take B′_{t} = B_{t} − 2λt and get two extensions of the white noise. In such a situation we say that the second extension results from the first one by the drift 2λ, denote Θ_{t}^{change} by Θ_{t}^{change(λ)} and Θ_{t}^{transfer} by Θ_{t}^{transfer(λ)}.

Note that

Θ_{t}^{transfer(λ)}(X(B|_{[0,t]})) = (X ∘ θ_{t}^{λ})(B|_{[0,t]})

for X ∈ L_{2}(C[0, t], W_{t}); as before, θ_{t}^{λ} : C[0, t] → C[0, t] is the drift transformation, (θ_{t}^{λ}b)(s) = b(s) − 2λs for s ∈ [0, t], it sends the measure D_{t} · W_{t} to W_{t}.

The isomorphism Θ^{change(λ)} between the two Arveson systems (H_{t})_{t}, (H′_{t})_{t} is not an isomorphism of extensions (unless λ = 0), since its restiction to (H_{t}^{white})_{t} is not equal to Θ^{transfer(λ)}. Instead, by the lemma below, they are related via the automorphism Θ^{shift(λ)} of (H_{t}^{white})_{t} introduced in Section 1.

Lemma 2.1.

Θ^{change(λ)}Θ^{shift(λ)} = Θ^{transfer(λ)},

that is,

Θ_{t}^{change(λ)}Θ_{t}^{shift(λ)}ψ = Θ_{t}^{transfer(λ)}ψ

for all ψ ∈ H_{t}^{white} and all t ∈ (0, ∞).

Proof. We take X ∈ L_{2}(C[0, t], W_{t}) such that ψ = X(B|_{[0,t]}), then

Θ_{t}^{change(λ)}Θ_{t}^{shift(λ)}ψ = D_{t}^{−1/2} · D_{t}^{1/2} · (X ∘ θ_{t}^{λ})(B|_{[0,t]})

= (X ∘ θ_{t}^{λ})(B|_{[0,t]}) = Θ_{t}^{transfer(λ)}ψ. □

The situation is shown on the diagram

\xymatrix{(H_{t})_{t} \ar@{.>}[r]_{Θ} \ar@{.>}[rr]^{Θ′} & (H_{t})_{t} \ar[r]_{Θ^{change(λ)}} & (H′_{t})_{t} \\ (H_{t}^{white})_{t} \ar@{^{(}->}[u] \ar[r]^{Θ^{shift(λ)}} \ar[rr]_{Θ^{transfer(λ)}} & (H_{t}^{white})_{t} \ar@{^{(}->}[u] \ar[r]^{Θ^{change(λ)}} & (H_{t}^{′ white})_{t} \ar@{^{(}->}[u]}

and we see that the following conditions are equivalent:

• There exists an automorphism Θ of (H_{t})_{t} that extends Θ^{shift(λ)}.

• There exists an isomorphism Θ′ between (H_{t})_{t} and (H′_{t})_{t} that extends

Θ^{transfer(λ)}. In other words, Θ^{shift(λ)} can be extended to (H_{t})_{t} if and only if the two extensions of the type I_{1} system are isomorphic.

Corollary 2.2. In order to prove Theorem 1.10 it is sufficient to construct a noise, extending the white noise, such that for every λ ∈ ℝ \ {0} the extension obtained by the drift λ is nonisomorphic to the original extension on the level of Arveson systems (that is, the corresponding extensions of the type I_{1} Arveson system are nonisomorphic).

Proof. In the group of all motions of the complex plane we consider the subgroup G of motions that correspond to automorphisms of (H_{t}^{white})_{t} extendable to (H_{t})_{t}. Real shifts z ↦ z + λ (for λ ∈ ℝ \ {0}) do not belong to G, as explained above. Imaginary shifts z ↦ z + iλ (for λ ∈ ℝ) belong to G, since the operators Θ_{t}^{shift(iλ)} of multiplication by exp(iλB_{t}) act naturally on H_{t}. It follows that G contains no rotations (except for the rotation by π) and therefore is not transitive. □

Thus, we need a drift sensitive extension. Such extension is constructed in Section 10 and its drift sensitivity is proved in Section 11.

3. Toy models: Hilbert spaces

Definitions and statements of Sections 3 and 4 will not be used formally, but probably help to understand the idea.

The phenomenon of a nonextendable isomorphism (as well as nonisomorphic extensions) is demonstrated in this section by a toy model, — a kind of product system of Hilbert spaces, simpler than Arveson system.

Definition 3.1. A toy product system of Hilbert spaces is a triple

(H_{1}, H_{∞}, U),

where H_{1}, H_{∞} are Hilbert spaces (over ℂ, separable), and

U : H_{1} ⊗ H_{∞} → H_{∞}

is a unitary operator.

We treat it as a kind of product system, since

H_{∞} ∼ H_{1} ⊗ H_{∞} ∼ H_{1} ⊗ H_{1} ⊗ H_{∞} ∼ ⋯

where ‘∼’ means: may be identified naturally (using U).

An evident example: H_{∞} = (H_{1}, ψ_{1})^{⊗∞} is the infinite tensor product of (an infinite sequence of) copies of H_{1} relatively to (the copies of) a given vector ψ_{1} ∈ H_{1}, ∥ψ_{1}∥ = 1. The equation U(ψ ⊗ ξ) = ξ has exactly one solution: ψ = ψ_{1}, ξ = ψ_{1}^{⊗∞}.

An uninteresting modification: H_{∞} = (H_{1}, ψ_{1})^{⊗∞} ⊗ H_{0} for some Hilbert space H_{0}.

A more interesting example: H_{∞} = (H_{1}, ψ_{1})^{⊗∞}⊕(H_{1}, ψ_{2})^{⊗∞} is the direct sum of two such infinite tensor products, one relative to ψ_{1}, the other relative to another vector ψ_{2} ∈ H_{1}, ∥ψ_{2}∥ = 1, ψ_{2} ≠ ψ_{1}. The equation U(ψ ⊗ ξ) = ξ has exactly two solutions: ψ = ψ_{1}, ξ = ψ_{1}^{⊗∞} and ψ = ψ_{2}, ξ = ψ_{2}^{⊗∞}.

Definition 3.2. Let (H_{1}, H_{∞}, U) and (H′_{1}, H′_{∞}, U′) be toy product systems of Hilbert spaces. An isomorphism between them is a pair Θ = (Θ_{1}, Θ_{∞}) of unitary operators Θ_{1} : H_{1} → H′_{1}, Θ_{∞} : H_{∞} → H′_{∞} such that the diagram

\xymatrix{H_{1} ⊗ H_{∞} \ar[d]^{Θ_{1}⊗Θ_{∞}} \ar[r]^{U} & H_{∞} \ar[d]^{Θ_{∞}} \\ H′_{1} ⊗ H′_{∞} \ar[r]^{U′} & H′_{∞}}

is commutative.

Thus,

Θ_{∞} ∼ Θ_{1} ⊗ Θ_{∞} ∼ Θ_{1} ⊗ Θ_{1} ⊗ Θ_{∞} ∼ ⋯

A unitary operator Θ_{1} : H_{1} → H_{1} leads to an automorphism of

(H_{1}, ψ_{1})^{⊗∞}

(that is, of the corresponding toy product system) if and only if Θ_{1}ψ_{1} = ψ_{1}. Similarly, Θ_{1} leads to an automorphism of (H_{1}, ψ_{1})^{⊗∞} ⊕ (H_{1}, ψ_{2})^{⊗∞} if and only if either Θ_{1}ψ_{1} = ψ_{1} and Θ_{1}ψ_{2} = ψ_{2}, or Θ_{1}ψ_{1} = ψ_{2} and Θ_{1}ψ_{2} = ψ_{1}.

Taking Θ_{1} such that Θ_{1}ψ_{1} = ψ_{1} but Θ_{1}ψ_{2} ≠ ψ_{2} we get an automorphism of (H_{1}, ψ_{1})^{⊗∞} that cannot be extended to an automorphism of

(H_{1}, ψ_{1})^{⊗∞} ⊕ (H_{1}, ψ_{2})^{⊗∞}.

Similarly to Section 2 we may turn from extensions of automorphisms to isomorphisms of extensions. The system (H_{1}, ψ_{1})^{⊗∞} ⊕ (H_{1}, ψ_{2})^{⊗∞} is an extension of (H_{1}, ψ_{1})^{⊗∞} (in the evident sense). Another vector ψ′_{2} leads to another extension of (H_{1}, ψ_{1})^{⊗∞}. We define an isomorphism between the two extensions as an isomorphism (Θ_{1}, Θ_{∞}) between the toy product systems (H_{1}, ψ_{1})^{⊗∞} ⊕ (H_{1}, ψ_{2})^{⊗∞} and (H_{1}, ψ_{1})^{⊗∞} ⊕ (H_{1}, ψ′_{2})^{⊗∞} whose restriction to (H_{1}, ψ_{1})^{⊗∞} is trivial (the identity):

[figure: (H_{1}, ψ_{1})^{⊗∞} ⊕ (H_{1}, ψ_{2})^{⊗∞}; (Θ_{1},Θ_{∞}); (H_{1}, ψ_{1})^{⊗∞} ⊕ (H_{1}, ψ′_{2})^{⊗∞}; (H_{1}, ψ_{1})^{⊗∞}.]

Clearly, Θ_{1} must be trivial; therefore ψ′_{2} must be equal to ψ_{2}. Otherwise the two extensions are nonisomorphic.

4. Toy models: probability spaces

Definition 4.1. A toy product system of probability spaces is a triple

(Ω_{1}, Ω_{∞}, α),

where Ω_{1}, Ω_{∞} are probability spaces (standard), and α : Ω_{1} × Ω_{∞} → Ω_{∞} is an isomorphism mod 0 (that is, an invertible measure preserving map).

Every toy product system of probability spaces (Ω_{1}, Ω_{∞}, α) leads to a toy product system of Hilbert spaces (H_{1}, H_{∞}, U) as follows:

H_{1} = L_{2}(Ω_{1}); H_{∞} = L_{2}(Ω_{∞});
(Uψ)(·) = ψ(α^{−1}(·)).

Here we use the canonical identification

L_{2}(Ω_{1}) ⊗ L_{2}(Ω_{∞}) = L_{2}(Ω_{1} × Ω_{∞})

and treat a vector ψ ∈ H_{1} ⊗ H_{∞} as an element of L_{2}(Ω_{1} × Ω_{∞}).

An evident example: Ω_{∞} = Ω_{1}^{∞} is the product of an infinite sequence of copies of Ω_{1}. It leads to H_{∞} = (H_{1}, 1)^{⊗∞} where H_{1} = L_{2}(Ω_{1}) and 1 ∈ L_{2}(Ω_{1}) is the constant function, 1(·) = 1.

An uninteresting modification: Ω_{∞} = Ω_{1}^{∞} ×Ω_{0} for some probability space Ω_{0}. It leads to H_{∞} = (H_{1}, 1)^{⊗∞} ⊗ H_{0}, H_{0} = L_{2}(Ω_{0}).

Here is a more interesting example. Let X_{1} : Ω_{1} → {−1, +1} be a measurable function (not a constant). We define Ω_{∞} as the set of all double se-

quences ( \begin{matrix}ω_{1}, & ω_{2}, & ... \\ s_{1}, & s_{2}, & ... \end{matrix} ) such that ω_{k} ∈ Ω_{1}, s_{k} ∈ \{−1, +1\} and s_{k} = s_{k+1}X_{1}(ω_{k})

for all k. Sequences (ω_{1}, ω_{2}, …) ∈ Ω_{1}^{∞} are endowed with the product measure. The conditional distribution of the sequence (s_{1}, s_{2}, …), given (ω_{1}, ω_{2}, …), must be concentrated on the two sequences obeying the relation s_{k} = s_{k+1}X_{1}(ω_{k}). We give to these two sequences equal conditional probabilities, 0.5 to each. Thus, Ω_{∞} is endowed with a probability measure. The map α : Ω_{1} × Ω_{∞} → Ω_{∞} is defined by

α(ω_{1}, (\begin{matrix}ω_{2}, & ω_{3}, & … \\ s_{2}, & s_{3}, & … \end{matrix}) ) = (\begin{matrix}ω_{1}, & ω_{2}, & ω_{3}, & … \\ s_{2}X_{1}(ω_{1}), & s_{2}, & s_{3}, & … \end{matrix}) .

Clearly, α is measure preserving.

This system (Ω_{1}, Ω_{∞}, α) leads to a system (H_{1}, H_{∞}, U) of the form

(H_{1}, ψ_{1})^{⊗∞} ⊕ (H_{1}, ψ_{2})^{⊗∞}

(up to isomorphism), as explained below. We have

H_{1} = L_{2}(Ω_{1}), H_{∞} = L_{2}(Ω_{∞}),

(Uψ) (\begin{matrix}ω_{1}, & ω_{2}, & ω_{3}, & … \\ s_{1}, & s_{2}, & s_{3}, & … \end{matrix}) = ψ(ω_{1}, (\begin{matrix}ω_{2}, & ω_{3}, & … \\ s_{2}, & s_{3}, & … \end{matrix}) ).

The equation U(ψ ⊗ ξ) = ξ becomes

ψ(ω_{1})ξ (\begin{matrix}ω_{2}, & ω_{3}, & … \\ s_{2}, & s_{3}, & … \end{matrix}) = ξ (\begin{matrix}ω_{1}, & ω_{2}, & ω_{3}, & … \\ s_{1}, & s_{2}, & s_{3}, & … \end{matrix}) .

One solution is evident: ψ = 1_{Ω_{1}}, ξ = 1_{Ω_{∞}}. A less evident solution is, ψ = X_{1}, ξ = S_{1}, where S_{1} is defined by S_{1}( \begin{matrix}ω_{1}, & ω_{2}, & ... \\ s_{1}, & s_{2}, & ... \end{matrix} ) = s_{1}. (The equation is satisfied due to the relation X_{1}(ω_{1})s_{2} = s_{1}.) We consider the system (H′_{1}, H′_{∞}, U′) where H′_{1} = H_{1} = L_{2}(Ω_{1}), H′_{∞} = (H′_{1}, 1_{Ω_{1}})^{⊗∞} ⊕ (H′_{1}, X_{1})^{⊗∞} (U′ being defined naturally) and construct an isomorphism (Θ_{1}, Θ_{∞}) between (H_{1}, H_{∞}, U) and (H′_{1}, H′_{∞}, U′) such that

Θ_{∞}1_{Ω_{∞}} = 1_{Ω_{1}}^{⊗∞},
Θ_{∞}S_{1} = X_{1}^{⊗∞}.

To this end we consider an arbitrary n and ξ ∈ L_{2}(Ω_{1}^{n}) = H_{1}^{⊗n}, define φ, ψ ∈ L_{2}(Ω_{∞}) by

φ (\begin{matrix}ω_{1}, & ω_{2}, & … \\ s_{1}, & s_{2}, & … \end{matrix}) = ξ(ω_{1}, …, ω_{n}),
ψ (\begin{matrix}ω_{1}, & ω_{2}, & … \\ s_{1}, & s_{2}, & … \end{matrix}) = s_{n+1}ξ(ω_{1}, …, ω_{n})

and, using the relation (or rather, the natural isomorphism)

H′_{∞} = (H′_{1})^{⊗n} ⊗ H′_{∞},

we let

Θ_{∞}φ = ξ ⊗ 1_{Ω_{1}}^{⊗∞}, Θ_{∞}ψ = ξ ⊗ X_{1}^{⊗∞},

thus defining a unitary Θ_{∞} : H_{∞} → H′_{∞}. (Further details are left to the reader.)

A more general construction is introduced in Section 5.

5. Binary extensions: probability spaces

Definition 5.1. (a) An extension of a probability space Ω consists of an-

other probability space Ω̃ and a measure preserving map γ : Ω̃ → Ω.

(b) Two extensions (Ω̃, γ) and (Ω̃′, γ′) of a probability space Ω are isomorphic, if there exists an invertible (mod 0) measure preserving map θ : Ω̃ → Ω̃′ such that the diagram

\xymatrix{Ω̃ \ar[dr]_{γ} \ar[rr]^{θ} &  & Ω̃′ \ar[dl]^{γ′} \\  & Ω}

is commutative. (Such θ will be called an isomorphism of extensions.)

(c) An extension of a probability space Ω is binary, if it is isomorphic to (Ω×Ω_{±}, γ), where Ω_{±} = {−1, +1} consists of two equiprobable atoms, and γ : Ω × Ω_{±} → Ω is the projection, (ω, s) ↦ ω.

By a well-known theorem of V. Rokhlin, an extension is binary if and only if conditional measures consist of two atoms of probability 0.5. However, this fact will not be used.

Interchanging the two atoms we get an involution on Ω̃. Denoting it by ω̃ ↦ −ω̃ we have

−ω̃ ≠ ω̃, −(−ω̃) = ω̃, γ(−ω̃) = γ(ω̃) for ω̃ ∈ Ω̃;

these properties characterize the involution. In the case Ω̃ = Ω × Ω_{±} we have −(ω, s) = (ω, −s) for ω ∈ Ω, s = ±1.

An isomorphism between two binary extensions boils down to an automorphism of (Ω × Ω_{±}, γ). The general form of such automorphism is (ω, s) ↦ (ω, sU(ω)) for ω ∈ Ω, s = ±1; here U runs over measurable functions Ω → {−1, +1}. The automorphism commutes with the involution, thus, every isomorphism of extensions intertwines the involutions,

θ(−ω̃) = −θ(ω̃) for ω̃ ∈ Ω̃.

Definition 5.2. (a) An inductive system of probability spaces consists of probability spaces Ω_{n} and measure preserving maps β_{n} : Ω_{n} → Ω_{n+1} for n = 1, 2, …

(b) Let (Ω_{n}, β_{n})_{n} and (Ω′_{n}, β′_{n})_{n} be two inductive systems of probability spaces. A morphism from (Ω_{n}, β_{n})_{n} to (Ω′_{n}, β′_{n})_{n} is a sequence of measure preserving maps γ_{n} : Ω_{n} → Ω′_{n} such that the infinite diagram

\xymatrix{Ω_{1} \ar[d]^{γ_{1}} \ar[r]^{β_{1}} & Ω_{2} \ar[d]^{γ_{2}} \ar[r]^{β_{2}} & … \\ Ω′_{1} \ar[r]^{β′_{1}} & Ω′_{2} \ar[r]^{β′_{2}} & …}

is commutative. If each γ_{n} is invertible, the morphism is an isomorphism.

(c) A morphism (γ_{n})_{n} is binary, if for every n the extension (Ω_{n}, γ_{n}) of Ω′_{n} is binary, and each β_{n} intertwines the corresponding involutions,

β_{n}(−ω_{n}) = −β_{n}(ω_{n}) for ω_{n} ∈ Ω_{n}.

Given a binary morphism (γ_{n})_{n} from (Ω_{n}, β_{n})_{n} to (Ω′_{n}, β′_{n})_{n}, we say that (Ω_{n}, β_{n})_{n} is a binary extension of (Ω′_{n}, β′_{n})_{n} (according to (γ_{n})_{n}).

Definition 5.3. Let (Ω_{n}, β_{n})_{n} be an inductive system of probability spaces, (Ω̃_{n}, β̃_{n})_{n} its binary extension (according to (γ_{n})_{n}), and (Ω̃′_{n}, β̃′_{n})_{n} another binary extension of (Ω_{n}, β_{n})_{n} (according to (γ′_{n})_{n}). An isomorphism between the two binary extensions is an isomorphism (θ_{n})_{n} between (Ω̃_{n}, β̃_{n})_{n} and (Ω̃′_{n}, β̃′_{n})_{n} treated as inductive systems of probability spaces, satisfying the following condition: for each n the diagram

\xymatrix{Ω̃_{n} \ar[dr]_{γ_{n}} \ar[rr]^{θ_{n}} &  & Ω̃′_{n} \ar[dl]^{γ′_{n}} \\  & Ω_{n}}

is commutative.

In other words, an isomorphism between the two binary extensions of the inductive system is a sequence (θ_{n})_{n} where each θ_{n} is an isomorphism between the two binary extensions (Ω̃_{n}, γ̃_{n}) and (Ω̃′_{n}, γ′_{n}) of the probability space Ω_{n}, such that the diagram

\xymatrix{Ω̃_{n} \ar[d]^{θ_{n}} \ar[r]^{β̃_{n}} & Ω̃_{n+1} \ar[d]^{θ_{n+1}} \\ Ω̃′_{n} \ar[r]^{β̃′_{n}} & Ω̃′_{n+1}}

is commutative for every n.

Lemma 5.4. Let (Ω_{n}, β_{n})_{n} be an inductive system of probability spaces.

(a) Let X_{n} : Ω_{n} → {−1, +1} be measurable functions, and

(5.5)  \begin{matrix}Ω̃_{n} = Ω_{n} × Ω_{±}, \\ \begin{matrix}β̃_{n}(ω_{n}, s_{n}) & = (β_{n}(ω_{n}), s_{n}X_{n}(ω_{n})) \\ γ_{n}(ω_{n}, s_{n}) & = ω_{n} \end{matrix}\} for ω_{n} ∈ Ω_{n}, s_{n} = ±1. \end{matrix}

Then (Ω̃_{n}, β̃_{n}) is a binary extension of (Ω_{n}, β_{n})_{n} (according to (γ_{n})_{n}).

(b) Every binary extension of (Ω_{n}, β_{n})_{n} is isomorphic to the extension of the form (5.5), for some (X_{n})_{n}.

Proof. (a) Clearly, β̃_{n} and γ_{n} are measure preserving, γ_{n} is binary, and γ_{n+1}(β̃_{n}(ω_{n}, s_{n})) = β_{n}(ω_{n}) = β_{n}(γ_{n}(ω_{n}, s_{n})).

(b) Let (Ω̃_{n}, β̃_{n})_{n} be a binary extension of (Ω_{n}, β_{n})_{n} according to (γ_{n})_{n}. Without loss of generality we assume Ω̃_{n} = Ω_{n} × Ω_{±} and γ_{n}(ω_{n}, s_{n}) = ω_{n}. The relations

γ_{n+1}(β̃_{n}(ω_{n}, s_{n})) = β_{n}(γ_{n}(ω_{n}, s_{n})) = β_{n}(ω_{n}),
β̃_{n}(−ω_{n}) = −β̃_{n}(ω_{n}),

show that β̃_{n} is of the form β̃_{n}(ω_{n}, s_{n}) = (β_{n}(ω_{n}), s_{n}X_{n}(ω_{n})) for some measurable X_{n} : Ω_{n} → {−1, +1}. □

Given an inductive system (Ω_{n}, β_{n})_{n} of probability spaces and two sequences (X_{n})_{n}, (Y_{n})_{n} of measurable functions X_{n}, Y_{n} : Ω_{n} → {−1, +1}, the construction (5.5) gives us two binary extensions of (Ω_{n}, β_{n})_{n}. One extension, (Ω̃_{n}, β̃_{n})_{n}, (γ_{n})_{n}, corresponds to (X_{n})_{n}, the other extension, (Ω̃′_{n}, β̃′_{n})_{n}, (γ′_{n})_{n}, corresponds to (Y_{n})_{n}. We want to know, whether they are isomorphic or not.

For each n separately, the two binary extensions of the probability space Ω_{n} coincide: Ω̃_{n} = Ω_{n} × Ω_{±} = Ω̃′_{n}, γ_{n}(ω_{n}, s_{n}) = ω_{n} = γ′_{n}(ω_{n}, s_{n}). Every isomorphism θ_{n} between them is of the form

θ_{n}(ω_{n}, s_{n}) = (ω_{n}, s_{n}U_{n}(ω_{n})) for ω_{n} ∈ Ω_{n}, s_{n} = ±1,

where U_{n} : Ω_{n} → {−1, +1} is a measurable function. In order to form an isomorphism between the binary extensions of the inductive system, these θ_{n} must satisfy the condition θ_{n+1}(β̃_{n}(ω̃_{n})) = β̃′_{n}(θ_{n}(ω̃_{n})), that is (recall (5.5)),

X_{n}(ω_{n})U_{n+1}(β_{n}(ω_{n})) = U_{n}(ω_{n})Y_{n}(ω_{n}) for ω_{n} ∈ Ω_{n}.

Given an inductive system (Ω_{n}, β_{n})_{n} of probability spaces, we consider the commutative group G((Ω_{n}, β_{n})_{n}) of all sequences f = (f_{n})_{n} of measurable functions f_{n} : Ω_{n} → {−1, +1} treated mod 0; the group operation is the pointwise multiplication. We define the shift homomorphism T : G((Ω_{n}, β_{n})_{n}) → G((Ω_{n}, β_{n})_{n}) by

(T f)_{n}(ω_{n}) = f_{n+1}(β_{n}(ω_{n})) for ω_{n} ∈ Ω_{n}.

According to (5.5), every X ∈ G((Ω_{n}, β_{n})_{n}) leads to a binary extension of (Ω_{n}, β_{n})_{n}. We summarize the previous paragraph as follows.

Lemma 5.6. The binary extensions corresponding to X, Y ∈ G((Ω_{n}, β_{n})_{n}) are isomorphic if and only if XT (U) = Y U for some U ∈ G((Ω_{n}, β_{n})_{n}).

6. Binary extensions: Hilbert spaces

Given an extension of a probability space, γ : Ω̃ → Ω, we have a natural embedding of Hilbert spaces, L_{2}(Ω) ⊂ L_{2}(Ω̃), and a natural action of the commutative algebra L_{∞}(Ω) on L_{2}(Ω̃). (L_{2} and L_{∞} over ℂ are meant.) Assume that the extension is binary. Then the embedded subspace and its orthogonal complement are the ‘even’ and ‘odd’ subspaces w.r.t. the involution ω̃ ↦ −ω̃ on Ω̃; that is,

ψ ∈ L_{2}(Ω) if and only if ψ(−ω̃) = ψ(ω̃) for almost all ω̃ ∈ Ω̃;

ψ ∈ L_{2}(Ω̃) ⊖ L_{2}(Ω) if and only if ψ(−ω̃) = −ψ(ω̃) for almost all ω̃ ∈ Ω̃.

Lemma 6.1. Let γ : Ω̃ → Ω and γ′ : Ω̃′ → Ω be two binary extensions of a probability space Ω. Then the following two conditions on a unitary operator Θ : L_{2}(Ω̃′) → L_{2}(Ω̃) are equivalent:

(a) Θ is trivial on L_{2}(Ω), and intertwines the two actions of L_{∞}(Ω). In other words,

Θψ = ψ for all ψ ∈ L_{2}(Ω),
Θ(h · ψ) = h · (Θψ) for all ψ ∈ L_{2}(Ω̃′), h ∈ L_{∞}(Ω).

(b) There exists an isomorphism of extensions θ : Ω̃ → Ω̃′ and h ∈ L_{∞}(Ω), |h(·)| = 1, such that

Θψ = ψ ∘ θ for all ψ ∈ L_{2}(Ω),
Θψ = h · (ψ ∘ θ) for all ψ ∈ L_{2}(Ω̃′) ⊖ L_{2}(Ω).

Proof. (b) ⟹ (a): evident. (a) ⟹ (b): Without loss of generality we assume that Ω̃ = Ω̃′ = Ω×Ω_{±} and γ(ω, s) = γ′(ω, s) = ω. The Hilbert space L_{2}(Ω̃)⊖L_{2}(Ω) consists of functions of the form (ω, s) ↦ sf(ω) where f runs over L_{2}(Ω). Thus, L_{2}(Ω̃) ⊖ L_{2}(Ω) is naturally isomorphic to L_{2}(Ω), and the isomorphism intertwines the actions of L_{∞}(Ω). The operator Θ maps L_{2}(Ω̃′)⊖L_{2}(Ω) onto L_{2}(Ω̃)⊖L_{2}(Ω) and leads to an operator L_{2}(Ω) → L_{2}(Ω) that commutes with L_{∞}(Ω) and is therefore the multiplication by a function h ∈ L_{∞}(Ω). □

An inductive system of probability spaces (Ω_{n}, β_{n})_{n} leads evidently to a decreasing sequence of Hilbert spaces,

\xymatrix{L_{2}(Ω_{1}) & L_{2}(Ω_{2}) \ar@{^{(}->}[l] & … \ar@{^{(}->}[l]}

Similarly, a morphism from (Ω_{n}, β_{n})_{n} to (Ω′_{n}, β′_{n})_{n} leads to a commutative diagram of Hilbert space embeddings

\xymatrix{L_{2}(Ω_{1}) & L_{2}(Ω_{2}) \ar@{^{(}->}[l] & … \ar@{^{(}->}[l] \\ L_{2}(Ω′_{1}) \ar@{^{(}->}[u] & L_{2}(Ω′_{2}) \ar@{^{(}->}[u] \ar@{^{(}->}[l] & … \ar@{^{(}->}[l]}

The commutative algebra L_{∞}(Ω′_{n}) acts on L_{2}(Ω′_{n}) and L_{2}(Ω_{n}), and the embedding L_{2}(Ω′_{n}) → L_{2}(Ω_{n}) intertwines these two actions.

Lemma 6.2. Let (Ω_{n}, β_{n})_{n} be an inductive system of probability spaces, (Ω̃_{n}, β̃_{n})_{n} its binary extension (according to (γ_{n})_{n}), and (Ω̃′_{n}, β̃′_{n})_{n} another binary extension of (Ω_{n}, β_{n})_{n} (according to (γ′_{n})_{n}). Then the following two conditions are equivalent:

(a) The two binary extensions are isomorphic.

(b) There exist unitary operators

Θ_{n} : L_{2}(Ω̃′_{n}) → L_{2}(Ω̃_{n})

such that for every n, Θ_{n} intertwines the actions of L_{∞}(Ω_{n}) on L_{2}(Ω̃_{n}) and L_{2}(Ω̃′_{n}), and the following two diagrams are commutative:

\xymatrix{L_{2}(Ω̃_{n}) &  & L_{2}(Ω̃′_{n}) \ar[ll]_{Θ_{n}} & L_{2}(Ω̃_{n}) & L_{2}(Ω̃_{n+1}) \ar@{^{(}->}[l] \\  & L_{2}(Ω_{n}) \ar@{^{(}->}[ul] \ar@{^{(}->}[ur] &  & L_{2}(Ω̃′_{n}) \ar@{^{(}->}[u]_{Θ_{n}} & L_{2}(Ω̃′_{n+1}). \ar@{^{(}->}[u]_{Θ_{n+1}} \ar@{^{(}->}[l]}

Proof. (a) ⟹ (b): evident. (b) ⟹ (a): For each n separately we have two binary extensions (Ω̃_{n}, γ_{n}), (Ω̃′_{n}, γ′_{n}) of the probability space Ω_{n}, and a unitary operator Θ_{n} : L_{2}(Ω̃′_{n}) → L_{2}(Ω̃_{n}) that satisfies Condition (a) of Lemma 6.1. On the other hand, due to Lemma 5.4 we may assume that

Ω̃_{n} = Ω_{n} × Ω_{±} = Ω̃′_{n},
γ_{n}(ω_{n}, s_{n}) = ω_{n} = γ′_{n}(ω_{n}, s_{n}),
β̃_{n}(ω_{n}, s_{n}) = (β_{n}(ω_{n}), s_{n}X_{n}(ω_{n})),
β̃′_{n}(ω_{n}, s_{n}) = (β_{n}(ω_{n}), s_{n}Y_{n}(ω_{n})).

Now Lemma 6.1 gives us h_{n} ∈ L_{∞}(Ω_{n}), |h_{n}(·)| = 1, such that

Θ_{n}ψ = h_{n} · ψ

for all ψ ∈ L_{2}(Ω_{n} × Ω_{±}) ⊖ L_{2}(Ω_{n}). In other words, if ψ(ω_{n}, s_{n}) = s_{n}f(ω_{n}) then (Θ_{n}ψ)(ω_{n}, s_{n}) = s_{n}f(ω_{n})h_{n}(ω_{n}); here f runs over L_{2}(Ω_{n}). By commutativity of the second diagram, (Θ_{n+1}ψ)∘β̃_{n} = Θ_{n}(ψ∘β̃′_{n}) for ψ ∈ L_{2}(Ω̃′_{n+1}). For the case ψ(ω_{n+1}, s_{n+1}) = s_{n+1}f(ω_{n+1}) we have, first,

((Θ_{n+1}ψ) ∘ β̃_{n})(ω_{n}, s_{n}) = (Θ_{n+1}ψ)(β_{n}(ω_{n}), s_{n}X_{n}(ω_{n}))
= s_{n}X_{n}(ω_{n})f(β_{n}(ω_{n}))h_{n+1}(β_{n}(ω_{n})),

and second,

(ψ ∘ β̃′_{n})(ω_{n}, s_{n}) = ψ(β_{n}(ω_{n}), s_{n}Y_{n}(ω_{n})) = s_{n}Y_{n}(ω_{n})f(β_{n}(ω_{n})),
Θ_{n}(ψ ∘ β̃′_{n})(ω_{n}, s_{n}) = s_{n}Y_{n}(ω_{n})f(β_{n}(ω_{n}))h_{n}(ω_{n}).

They are equal, which means that X_{n}(ω_{n})h_{n+1}(β_{n}(ω_{n})) = Y_{n}(ω_{n})h_{n}(ω_{n}), that is,

(h_{n+1} ∘ β_{n}) · X_{n} = h_{n} · Y_{n}.

By Lemma 5.6 it is sufficient to find measurable functions

U_{n} : Ω_{n} → {−1, +1}

such that

(U_{n+1} ∘ β_{n}) · X_{n} = U_{n} · Y_{n} for all n.

We choose a Borel function φ : 𝕋 → {−1, +1}, where 𝕋 = {z ∈ ℂ : |z| = 1}, such that φ(−z) = −φ(z) for all z ∈ 𝕋. For example, φ(e^{iα}) = +1 for α ∈ [0, π) but −1 for α ∈ [π, 2π). The functions U_{n}(·) = φ(h_{n}(·)) satisfy the needed equation, since X_{n}(·) = ±1, Y_{n}(·) = ±1. □

7. Products of binary extensions

Definitions and statements of this section are used only in Section 11 (in the proof of Lemma 11.10).

Special measures are taken in the next definition in order to keep the product binary (rather than quaternary).

Definition 7.1. Let (Ω̃_{k}, γ_{k}) be a binary extension of a probability space Ω_{k} for k = 1, 2; Ω = Ω_{1} × Ω_{2}; and A ⊂ Ω a measurable set. The product of these two binary extensions (according to A) is the extension (Ω̃, γ) of Ω defined as follows:

Ω̃ = \underbrace{\{(ω̃_{1}, ω_{2}) : (γ_{1}(ω̃_{1}), ω_{2}) ∈ A\}}_{Ã} ⊎ \underbrace{\{(ω_{1}, ω̃_{2}) : (ω_{1}, γ_{2}(ω̃_{2})) ∈ Ω \ A\}}_{Ω̃\Ã} ,

the measure on Ã is induced from (the product measure on) Ω̃_{1} × Ω_{2}, on Ω̃ \ Ã — from Ω_{1} × Ω̃_{2};

γ(ω̃_{1}, ω_{2}) = (γ_{1}(ω̃_{1}), ω_{2}), γ(ω_{1}, ω̃_{2}) = (ω_{1}, γ_{2}(ω̃_{2})).

Here and henceforth ω_{k} runs over Ω_{k}, and ω̃_{k} runs over Ω̃_{k}.

Clearly, the extension (Ω̃, γ) is binary.

Let a binary extension (Ω̃, γ) of Ω = Ω_{1} ×Ω_{2} be the product of two binary extensions (Ω̃_{k}, γ_{k}), k = 1, 2 (according to a given A ⊂ Ω). Then we have a natural embedding of Hilbert spaces,

(7.2)  L_{2}(Ω̃) ⊂ L_{2}(Ω̃_{1}) ⊗ L_{2}(Ω̃_{2});

it arises from the natural measure preserving map Ω̃_{1} × Ω̃_{2} → Ω̃,

(ω̃_{1}, ω̃_{2}) ↦ \begin{cases}(ω̃_{1}, γ_{2}(ω̃_{2})) & if (γ_{1}(ω̃_{1}), γ_{2}(ω̃_{2})) ∈ A, \\ (γ_{1}(ω̃_{1}), ω̃_{2}) & otherwise. \end{cases}

The restriction of the embedding (7.2) to L_{2}(Ω) is just the tensor product of the two embeddings L_{2}(Ω_{k}) ⊂ L_{2}(Ω̃_{k}), k = 1, 2, since the corresponding composition map Ω̃_{1} × Ω̃_{2} → Ω̃ → Ω is just γ_{1} × γ_{2}.

The projection map Ã → Ω̃_{1}, (ω̃_{1}, ω_{2}) ↦ ω̃_{1}, need not be measure preserving, but anyway, generates a sub-σ-field F_{1} on Ã. Lemma 7.3. Let (Ω̃_{k}, γ_{k}) and (Ω̃′_{k}, γ′_{k}) be two binary extensions of a probability space Ω_{k} (for k = 1, 2), Ω = Ω_{1} × Ω_{2}, A ⊂ Ω a measurable set, Θ_{k} : L_{2}(Ω̃_{k}) → L_{2}(Ω̃′_{k}) unitary operators, each satisfying Condition (a) of Lemma 6.1. Then Θ_{1} × Θ_{2} maps L_{2}(Ω̃) onto L_{2}(Ω̃′), L_{2}(Ã) onto L_{2}(Ã′), and L_{2}(Ã, F_{1}) onto L_{2}(Ã′, F′_{1}).

It is meant that

L_{2}(Ã, F_{1}) ⊂ L_{2}(Ã) ⊂ L_{2}(Ã) ⊕ L_{2}(Ω̃ \ Ã) = L_{2}(Ω̃) ⊂ L_{2}(Ω̃_{1}) ⊗ L_{2}(Ω̃_{2}),

L_{2}(Ã′, F′_{1}) ⊂ L_{2}(Ã′) ⊕ L_{2}(Ω̃′ \ Ã′) = L_{2}(Ω̃′) ⊂ L_{2}(Ω̃′_{1}) ⊗ L_{2}(Ω̃′_{2}). The reader may prove Lemma 7.3 via Lemma 6.1, but the proof below

does not use Lemma 6.1.

Proof. The operator Θ = Θ_{1} ⊗ Θ_{2} intertwines the actions of L_{∞}(Ω_{1}) and L_{∞}(Ω_{2}), therefore, also the actions of L_{∞}(Ω_{1} × Ω_{2}). In particular,

Θ1Ãψ = 1Ã′Θψ for ψ ∈ L_{2}(Ω̃_{1} × Ω̃_{2}).

The space L_{2}(Ã) is the closure of linear combinations of vectors of the form ψ = 1Ã(φ_{1} ⊗ φ_{2}), where φ_{1} ∈ L_{2}(Ω̃_{1}) and φ_{2} ∈ L_{2}(Ω_{2}). For such ψ we have

Θψ = Θ1Ã(φ_{1} ⊗ φ_{2}) = 1Ã′Θ(φ_{1} ⊗ φ_{2}) = 1Ã′(Θ_{1}φ_{1} ⊗ Θ_{2}φ_{2}) ∈ L_{2}(Ã′),

since Θ_{1}φ_{1} ∈ L_{2}(Ω̃′_{1}) and Θ_{2}φ_{2} = φ_{2} ∈ L_{2}(Ω_{2}). Therefore

Θ(L_{2}(Ã)) ⊂ L_{2}(Ã′).

The special case φ_{2} = 1 gives Θ(L_{2}(Ã, F_{1})) ⊂ L_{2}(Ã′, F′_{1}). The same holds for Θ^{−1}, thus, the inclusions are in fact equalities. Similarly,

Θ(L_{2}(Ω̃ \ Ã)) = L_{2}(Ω̃′ \ Ã′).

It follows that Θ(L_{2}(Ω̃)) = L_{2}(Ω̃′). □

8. Some necessary conditions of isomorphism

Let μ_{1} be a probability measure on the space ℝ^{∞} (of all infinite sequences of reals), β : ℝ^{∞} → ℝ^{∞} the shift, β(x_{1}, x_{2}, …) = (x_{2}, x_{3}, …), and μ_{n} the image of μ_{1} under β^{n−1}. Probability spaces Ω_{n} = (ℝ^{∞}, μ_{n}) with maps β_{n} = β are an inductive system of probability spaces.

Let Borel functions f_{n} : ℝ → {−1, +1} be given. We define

X_{n} : Ω_{n} → {−1, +1},
X_{n}(x_{n}, x_{n+1}, …) = f_{n}(x_{n}),

and consider the corresponding binary extension of (Ω_{n}, β_{n})_{n}. Another sequence of functions g_{n} : ℝ → {−1, +1} leads to another binary extension. According to Lemma 5.6 the two binary extensions are isomorphic if and only if there exist U_{n} : Ω_{n} → {−1, +1} such that

(8.1)  U_{n+1}(x_{n+1}, x_{n+2}, …) = U_{n}(x_{n}, x_{n+1}, …)f_{n}(x_{n})g_{n}(x_{n}).

Functions that do not depend on x_{n}, that is, functions of the form

(x_{1}, x_{2}, …) ↦ φ(x_{1}, …, x_{n−1}, x_{n+1}, x_{n+2}, …)

are a subspace H_{n} ⊂ L_{2}(μ_{1}). We consider vectors ψ_{n} ∈ L_{2}(μ_{1}),

(8.2)  ψ_{n}(x_{1}, x_{2}, …) = f_{n}(x_{n})g_{n}(x_{n}),

and the distance between ψ_{n} and H_{n}.

Lemma 8.3. The condition

dist(ψ_{n}, H_{n}) → 0 as n → ∞

is necessary for the two binary extensions to be isomorphic.

Proof. Let U_{n} satisfy (8.1), then

U_{n}(x_{n}, x_{n+1}, …) = U_{1}(x_{1}, x_{2}, …)h_{1}(x_{1}) … h_{n−1}(x_{n−1}),

where h_{n}(x) = f_{n}(x)g_{n}(x). We have

ψ_{n}(x_{1}, x_{2}, …) = h_{n}(x_{n}) = U_{n}(x_{n}, x_{n+1}, …)U_{n+1}(x_{n+1}, x_{n+2}, …)

= U_{1}(x_{1}, x_{2}, …)h_{1}(x_{1}) … h_{n−1}(x_{n−1})U_{n+1}(x_{n+1}, x_{n+2}, …).

Taking into account that H_{n} is invariant under multiplication by any (bounded measurable) function of x_{1}, …, x_{n−1} and x_{n+1}, x_{n+2}, … we see that

dist(ψ_{n}, H_{n}) = dist(U_{1}, H_{n}).

The latter converges to 0, since H_{n} contains all functions of x_{1}, …, x_{n−1}. □

The conditional distribution of x_{n} given x_{1}, …, x_{n−1} and x_{n+1}, x_{n+2}, … (assuming that (x_{1}, x_{2}, …) is distributed μ_{1}) is a probability measure ν_{n} on ℝ; this ν_{n} is random in the sense that it depends on x_{1}, …, x_{n−1} and x_{n+1}, x_{n+2}, … (whose distribution is the marginal of μ_{1}).

Here is a useful condition on μ_{1}:

(8.4)  ∃ε > 0 ∀n ℙ(ν_{n} is ε-good) ≥ ε,

where a probability measure ν on ℝ is called ε-good if

(8.5)  ∃x ∈ ℝ ∀A ν(A) ≥ ε mes(A ∩ (x, x + ε))

(A runs over Borel subsets of ℝ; ‘mes’ stands for Lebesgue measure).

Usually ν has a density; then (8.5) requires the density to exceed ε on some interval of length ε.

Lemma 8.6. Let μ_{1} satisfy (8.4), and numbers ε_{n} ∈ (0, ∞) satisfy ε_{n} → 0. Then there exist Borel functions f_{n} : ℝ → {−1, +1} such that for every c ∈ ℝ \ {0}, defining g_{n} : ℝ → {−1, +1} by g_{n}(x) = f_{n}(x + cε_{n}) we get ψ_{n} (see (8.2)) violating the necessary condition of Lemma 8.3 (and therefore, (f_{n})_{n} and (g_{n})_{n} lead to two nonisomorphic binary extensions).

Proof. We take λ_{1}, λ_{2}, … such that

λ_{n}ε_{n} = \begin{cases}1 & for n odd, \\ \sqrt{2} & for n even, \end{cases}

and define

f_{n}(x) = σ(λ_{n}x),

where

σ(x) = \begin{cases}−1 & for x ∈ ∪_{k∈ℤ}[k − 0.5, k), \\ +1 & for x ∈ ∪_{k∈ℤ}[k, k + 0.5). \end{cases}

Let c ∈ ℝ be given, c ≠ 0. It is sufficient to prove that at least one of two claims

lim sup_{n} dist(ψ_{2n}, H_{2n}) > 0, lim sup_{n} dist(ψ_{2n−1}, H_{2n−1}) > 0

holds. Here

ψ_{n}(x_{1}, x_{2}, …) = h_{n}(x_{n}) = f_{n}(x_{n})g_{n}(x_{n}) = f_{n}(x_{n})f_{n}(x_{n} + cε_{n})
= σ(λ_{n}x_{n})σ(λ_{n}x_{n} + cλ_{n}ε_{n}).

The function h_{n} is periodic, with period 1/λ_{n}. The mean value M_{n} of h_{n} over the period is

M_{n} = λ_{n} ∫_{0}^{1/λ_{n}} h_{n}(x) dx = ∫_{0}^{1} σ(u)σ(u + cλ_{n}ε_{n}) du.

It reaches ±1 when 2cλ_{n}ε_{n} ∈ ℤ; otherwise −1 < M_{n} < 1. The relations 2c ∈ ℤ and 2c\sqrt{2} ∈ ℤ are incompatible, therefore at least one of two claims

sup_{n} |M_{2n}| < 1, sup_{n} |M_{2n−1}| < 1

holds. (Of course, M_{2n} and M_{2n−1} do not depend on n, but this fact does not matter.) It is sufficient to prove that

sup_{n} |M_{2n}| < 1 implies lim sup_{n} dist(ψ_{2n}, H_{2n}) > 0,
sup_{n} |M_{2n−1}| < 1 implies lim sup_{n} dist(ψ_{2n−1}, H_{2n−1}) > 0.

The former implication will be proved (the latter is similar). Assume the contrary: sup_{n} |M_{2n}| < 1 and dist(ψ_{2n}, H_{2n}) → 0.

For any probability measure ν on ℝ, the squared distance in the space L_{2}(ν) between the function h_{n} and the one-dimensional space of constant functions is

∫ (h_{n} − ∫ h_{n} dν)^{2} dν = ∫ h_{n}^{2} dν − ( ∫ h_{n} dν)^{2} = 1 − ( ∫ h_{n} dν)^{2}.

We use the random measure ν_{n}, take the average and recall the definition of H_{n}:

𝔼 (1 − ( ∫ h_{n} dν_{n})^{2}) = dist^{2}(ψ_{n}, H_{n}).

Taking into account that dist(ψ_{2n}, H_{2n}) → 0 we see that | ∫ h_{2n} dν_{2n}| → 1 in probability. In order to get a contradiction to (8.4) it is sufficient to prove that lim sup_{n} sup_{ν} | ∫ h_{2n} dν| < 1, where ν runs over all ε-good measures (recall (8.4) and (8.5)). Or, equivalently,

lim inf_{n} inf_{ν} ν(h_{2n}^{−1}(−1)) > 0, lim inf_{n} inf_{ν} ν(h_{2n}^{−1}(+1)) > 0.

The former will be proved (the latter is similar). By (8.5),

ν(h_{2n}^{−1}(−1)) ≥ ε mes(h_{2n}^{−1}(−1) ∩ (x, x + ε)).

For large n the period 1/λ_{2n} of the function h_{2n} is ≪ ε, therefore

\begin{aligned}\frac{1}{ε} mes(h_{2n}^{−1}(−1) ∩ (x, x + ε)) & ≥ \frac{1}{2} · λ_{2n} ∫_{0}^{1/λ_{2n}} \frac{1 − h_{2n}(x)}{2} dx \\ & = \frac{1}{2} · \frac{1 − M_{2n}}{2} \\ & ≥ \frac{1}{4}(1 − sup_{n} |M_{2n}|) > 0. \end{aligned}  □

Remark 8.7. The functions f_{n} constructed in the proof of Lemma 8.6 depend only on the numbers ε_{n}, not on the measure μ_{1}.

Let a probability measure μ on C[0, 1] be given. Random variables A(t) on the probability space Ω = (C[0, 1], μ) defined by A(t)(a) = a(t) for a ∈ C[0, 1], t ∈ [0, 1], are a random process. For every n the restriction map C[0, 1] → C[0, 3^{−n}] sends μ to some μ_{n}. Probability spaces

Ω_{n} = (C[0, 3^{−n}], μ_{n})

with restriction maps are an inductive system.

Given Borel functions f_{n} : ℝ → {−1, +1}, we define random variables X_{n} : Ω_{n} → {−1, +1} by X_{n} = f_{n}(A(2 · 3^{−n−1})). The corresponding binary extension may be visualized as follows. We consider pairs (a, s) of a function a ∈ C[0, 1] and another function s : (0, 1] → {−1, +1} constant on each [2 · 3^{−n−1}, 2 · 3^{−n}) and such that s(2 · 3^{−n}−)s(2 · 3^{−n}) = f_{n−1}(a(2 · 3^{−n})) for all n. We get a pair of random processes A(·), S(·) satisfying

\frac{S(2 · 3^{−n})}{S(2 · 3^{−n}−)} = f_{n−1}(A(2 · 3^{−n})).

Their restrictions to [0, 3^{−n}] give Ω̃_{n}. For each t (separately), the random variable S(t) is independent of the process A(·) and takes on the two equiprobable values ±1.

As before, given also g_{n} : ℝ → {−1, +1} (thus, another binary extension), we define ψ_{n} ∈ L_{2}(Ω) by

ψ_{n} = f_{n}(A(2 · 3^{−n−1}))g_{n}(A(2 · 3^{−n−1})).

We consider the subspaces H_{n} ⊂ L_{2}(Ω) consisting of functions of A(t) for t ∈ [0, 3^{−n−1}] ∪ [3^{−n}, 1] only (in other words, functions of the restrictions of sample paths to [0, 3^{−n−1}] ∪ [3^{−n}, 1]).

Lemma 8.8. The condition dist(ψ_{n}, H_{n}) → 0 is necessary for the two binary extensions to be isomorphic.

The proof, similar to the proof of Lemma 8.3, is left to the reader.

Similarly, C[−1, 1] may be used (instead of C[0, 1]), with

Ω_{n} = (C[−1, 3^{−n}], μ_{n});

the process S(·) jumps at 2 · 3^{−n}, as before. Now H_{n} consists of functions of the restriction A|_{[−1,3^{−n−1}]∪[3^{−n},1]} of A to [−1, 3^{−n−1}] ∪ [3^{−n}, 1] (rather than [0, 3^{−n−1}] ∪ [3^{−n}, 1]). Lemma 8.8 remains true.

The conditional distribution of A(2 · 3^{−n−1}) given A|_{[−1,3^{−n−1}]∪[3^{−n},1]} is too concentrated (when n is large) for being ε-good (recall (8.5)). A useful condition on μ stipulates rescaling by 3^{n/2}:

(8.9) there exists ε > 0 such that for every n,

the conditional distribution of 3^{n/2}A(2 · 3^{−n−1}) given A|_{[−1,3^{−n−1}]∪[3^{−n},1]} is ε-good with probability ≥ ε.

Here is a counterpart of Lemma 8.6 for ε_{n} = 3^{−n/2}.

Lemma 8.10. Let μ satisfy (8.9). Then there exist Borel functions f_{n} : ℝ → {−1, +1} such that for every c ∈ ℝ \ {0}, defining g_{n} : ℝ → {−1, +1} by g_{n}(x) = f_{n}(x+3^{−n}c) we get ψ_{n} = f_{n}(A(2·3^{−n−1}))g_{n}(A(2·3^{−n−1})) violating the necessary condition of Lemma 8.8 (and therefore, two nonisomorphic binary extensions).

The proof, similar to the proof of Lemma 8.6, is left to the reader.

9. A binary extension of Brownian motion

The space C[0, 1] of all continuous functions b : [0, 1] → ℝ, endowed with the Wiener measure W, is a probability space. Random variables B(t) on (C[0, 1], W), defined for t ∈ [0, 1] by B(t)(b) = b(t), are the Brownian motion on [0, 1]. Almost surely, a Brownian sample path on [0, 1] has a unique (global) minimum,

min_{t∈[0,1]} B(t) = B(τ),

τ being a measurable function on (C[0, 1], W), 0 < τ(·) < 1 a.s.

We define another random process A, on the time interval [−1, 1], by

A(t) = B(min(1, τ + t)) − B(τ) for t ∈ [0, 1],
A(t) = B(max(0, τ + t)) − B(τ) for t ∈ [−1, 0].

A W-measurable map C[0, 1] → C[−1, 1] is thus introduced. The map is one-to-one (mod 0), since B(·) is nonconstant on every time interval, almost surely.

Proposition 9.1. The process A satisfies (8.9).

The proof is given after three lemmas.

The conditional distribution of the process B given the restriction A|_{[−1,ε]} (for a given ε ∈ (0, 1)) is the same as the conditional distribution of the process B given τ and B|_{[0,τ+ε]}, since the two corresponding measurable partitions of (C[0, 1], W) are equal (mod 0). This conditional distribution is a probability measure on the set of Brownian sample paths b such that

b(t) = x(t) for t ∈ [0, s + ε],
b(t) > x(s) for t ∈ [s + ε, 1];

here s ∈ (0, 1 − ε) is a given value of τ, and x ∈ C[0, s + ε] is a given sample path of B|_{[0,τ+ε]}; of course, s is the unique minimizer of x. We assume that s < 1 − ε, since the other case is trivial (the conditional distribution is a single atom).

The corresponding conditional distribution of B|_{[s+ε,1]} is a probability measure on the set of functions b ∈ C[s + ε, 1] such that

b(s + ε) = x(s + ε),
b(t) > x(s) for t ∈ [s + ε, 1].

This set depends only on the three numbers s+ε, x(s+ε), and x(s). One may guess that the considered measure on this set also depends on these three numbers only (rather than the whole function x). The following well-known lemma confirms the guess and gives a simple description of the measure.

Lemma 9.2. The conditional distribution of B|_{[s+ε,1]} given that τ = s and B_{[0,τ+ε]} = x is equal to the conditional distribution of B|_{[s+ε,1]} given that B(s + ε) = x(s + ε) and B(t) > x(s) for t ∈ [s + ε, 1].

Proof. We take n such that \frac{1}{n} < ε. Let k ∈ \{1, …, n − 1\}. The conditional distribution of B|_{[ \frac{k}{n} ,1]} given B|_{[0, \frac{k}{n}]} depends only on B(\frac{k}{n}) (by the Markov property of B) and is just the distribution of the Brownian motion starting from (\frac{k}{n}, B(\frac{k}{n})). Therefore the conditional distribution of B|_{[ \frac{k}{n} ,1]} given both

B|_{[0, \frac{k}{n}]} and \frac{k−1}{n} < τ < \frac{k}{n} is the distribution of the Brownian motion starting

from (\frac{k}{n}, B(\frac{k}{n})) and conditioned to stay above the minimum of the given path on [0, \frac{k}{n}]. (Indeed, a measurable partition of the whole probability space induces a measurable partition of a given subset of positive probability, and conditional measures for the former partition induce conditional measures for the latter partition.) Now it is easy to condition further on B|_{[ \frac{k}{n} ,τ+ε]} and combine all k together. □

Lemma 9.2 gives the conditional distribution of B|_{[τ+ε,1]} given τ and B|_{[0,τ+ε]}. Now we turn to the conditional distribution of B|_{[τ+ε,τ+3ε]} given τ, B|_{[0,τ+ε]} and B|_{[τ+3ε,1]} (in the case τ + 3ε < 1). We are especially interested in B(τ + 2ε).

Lemma 9.3. The conditional distribution of B(τ +2ε)−B(τ), given τ (such that τ + 3ε < 1), B|_{[0,τ+ε]} and B|_{[τ+3ε,1]}, has the density

x ↦ \frac{(1 − e^{−2ax/ε})(1 − e^{−2bx/ε})}{1 − e^{−ab/ε}} · \frac{1}{\sqrt{πε}} exp (− \frac{1}{ε}(x − \frac{a + b}{2})^{2}) for x > 0,

where a = B(τ + ε) − B(τ) and b = B(τ + 3ε) − B(τ).

Proof. Using Lemma 9.2 we turn to an equivalent question: a Brownian motion starting from (s + ε, B(s + ε)) is conditioned to stay above B(s) on [s+ε, s+3ε], and is known on [s+3ε, 1] (which means another conditioning, of course); we need the (conditional) distribution of B(s + 2ε). Omitting for a while the condition B_{[s+ε,s+3ε]}(·) > B(s) we get the so-called Brownian bridge, — the Brownian motion on [s+ε, s+3ε] with given boundary values B(s + ε), B(s + 3ε). (Later we’ll condition the bridge to stay above B(s).)

For the bridge, B(s + 2ε) has normal distribution N(\frac{B(s+ε)+B(s+3ε)}{2}, \frac{ε}{2}). Given B(s + 2ε) we get two independent bridges, one on [s + ε, s + 2ε], the other on [s + 2ε, s + 3ε]. The bridge on [s + ε, s + 2ε] stays above B(s) with the probability (calculated via the reflection principle)

\frac{p_{ε}(a − x) − p_{ε}(a + x)}{p_{ε}(a − x)} = 1 − exp (− \frac{2}{ε}ax),

where a = B(s + ε) − B(s), x = B(s + 2ε) − B(s), b = B(s + 3ε) − B(s), and

p_{ε}(u) = \frac{1}{\sqrt{2πε}} exp (− \frac{u^{2}}{2ε}).

It remains to write similar formulas on [s+2ε, s+3ε] and all of [s+ε, s+3ε], and apply the Bayes formula

p_{X|A}(x) = \frac{ℙ(A|X = x)p_{X}(x)}{ℙ(A)}

for the conditional density p_{X|A}(·) of a random variable X given an event A. Namely, X = B(s + 2ε) − B(s) ∼ N(\frac{a+b}{2}, \frac{ε}{2}), p_{X}(x) = p_{ε/2}(x − \frac{a+b}{2}), A is the event B_{[s+ε,s+3ε]}(·) > B(s), ℙ(A|X = x) = (1 − e^{−2ax/ε})(1 − e^{−2bx/ε}), ℙ(A) = 1 − e^{−ab/ε}. □

Lemma 9.4. There exists ε > 0 such that for all a, b ∈ (0, ∞) the probability measure on (0, ∞) that has the density

x ↦ \frac{(1 − e^{−2ax})(1 − e^{−2bx})}{1 − e^{−ab}} · \frac{1}{\sqrt{π}} exp ( − (x − \frac{a + b}{2})^{2})

is ε-good (as defined by (8.5)).

Proof. It is sufficient to prove that

inf_{a,b∈(0,∞)} inf_{x∈[ \frac{a+b}{2} +1, \frac{a+b}{2} +2]} p_{a,b}(x) > 0,

where p_{a,b}(·) is the given density. Assume the contrary: there exist a_{n}, b_{n}, x_{n} such that b_{n} ≥ a_{n} > 0, \frac{a_{n}+b_{n}}{2} + 1 ≤ x_{n} ≤ \frac{a_{n}+b_{n}}{2} + 2, and p_{a_{n},b_{n}}(x_{n}) → 0. Then

\frac{(1 − e^{−2a_{n}x_{n}})(1 − e^{−2b_{n}x_{n}})}{1 − e^{−a_{n}b_{n}}} → 0.

It follows that 1 − e^{−2a_{n}x_{n}} → 0, a_{n}x_{n} → 0, a_{n}(\frac{a_{n}+b_{n}}{2} + 1) → 0, a_{n} → 0, a_{n}b_{n} → 0, and

\frac{(1 − e^{−2a_{n}x_{n}})(1 − e^{−2b_{n}x_{n}})}{a_{n}b_{n}} → 0 .

Therefore

\frac{1 − e^{−2a_{n}}}{a_{n}} · \frac{1 − e^{−2b_{n}}}{b_{n}} → 0; \frac{1 − e^{−2b_{n}}}{b_{n}} → 0;

b_{n} → ∞, b_{n}x_{n} → ∞; 1 − e^{−2b_{n}x_{n}} → 1;

\frac{1 − e^{−2a_{n}x_{n}}}{a_{n}b_{n}} → 0; \frac{1 − e^{−a_{n}b_{n}}}{a_{n}b_{n}} → 0,

in contradiction to a_{n}b_{n} → 0. □

Proof of Proposition 9.1. Lemma 9.3 (for ε = 3^{−n−1}) gives us the conditional distribution of A(2 · 3^{−n−1}) given τ and A|_{[−1,3^{−n−1}]∪[3^{−n},1]}, but only for the case τ + 3^{−n} < 1. Lemma 9.4 states that the corresponding distribution of 3^{n/2}A(2 · 3^{−n−1}) is ε-good. It remains to note that ℙ(τ + 3^{−n} < 1) ≥ ℙ(τ < 2/3) ≥ ε. □

Combining Proposition 9.1 and Lemma 8.10 we get a binary extension of the inductive system (of probability spaces) formed by the restrictions A|_{[−1,3^{−n}]} of the process A. In terms of the Brownian motion B the inductive system is formed by B|_{[0,τ+3^{−n}]}, and the binary extension may be visualized by a random function S : (τ, 1) → {−1, +1} constant on

[τ + 2 · 3^{−n−1}, τ + 2 · 3^{−n}) ∩ (0, 1)

for each n and such that

(9.5)  \frac{S(τ + 2 · 3^{−n})}{S(τ + 2 · 3^{−n}−)} = f_{n−1}(B(τ + 2 · 3^{−n}) − B(τ))

for all n such that τ + 2 · 3^{−n} < 1. Here f_{n} : ℝ → {−1, +1} are the functions given by Lemma 8.10. They are constructed as to make the binary extension sensitive to drift in the following sense. For every c ∈ ℝ \ {0} the binary extension constructed via

f_{n}(B(τ + 2 · 3^{−n}) − B(τ) + c · 2 · 3^{−n})

is not isomorphic to that for c = 0.

10. A new noise extending the white noise

This is a noise richer than white noise: in addition to the increments of a Brownian motion B it carries a countable collection of independent Bernoulli random variables which are attached to the local minima of B.

J. Warren [11, the end]

… magically, this independent random variable has appeared from somewhere! Indeed, it really has appeared from thin air, because… it is not present at time 0!

L.C.G. Rogers, D. Williams [7, p. 156]

The two ideas mentioned above will be combined; at every local minimum of the Brownian motion B, a new random variable will appear from thin air. That is, the binary extension, performed in Section 9 at the global minimum, will be performed at every local minimum, thus achieving locality and stationarity required from a noise, while retaining the drift sensitivity achieved in Section 9 (as will be shown in Section 11).

A new random sign attached to a local minimum at τ may be thought of as a random choice of one of the two functions S : (τ, τ + ε_{1}) → {−1, +1} constant on [τ + ε_{n+1}, τ + ε_{n}) (for each n) and such that

(10.1)  S(τ + ε_{n}) = S(τ + ε_{n}−)f_{n}(B(τ + ε_{n}) − B(τ))

(the numbers ε_{n} ↓ 0 and the functions f_{n} : ℝ → {−1, +1} being chosen appropriately). Given a time interval (0, t), for each local minimizer τ ∈ (0, t) we describe the new random sign by the value S(t) (of the corresponding function S), denoted however by η_{t}(τ). Relation (10.1) turns into the relation (10.8) between η_{s}(τ) and η_{s+t}(τ).

Before attaching something to the local minima we enumerate them. For every time interval (a, b) ⊂ ℝ there exists a measurable enumeration of local minima on (a, b), — a sequence of F_{a,b}^{white}-measurable random variables τ_{1}, τ_{2}, ⋯: Ω → (a, b) such that for almost all ω the Brownian path t ↦ B_{t}(ω) has a local minimum at each τ_{k}(ω), no other local minima exist on (a, b), and the numbers τ_{1}(ω), τ_{2}(ω), … are pairwise different a.s. Here is a simple construction for (a, b) = (0, 1) taken from [9, 2e]. First, τ_{1}(ω) is the minimizer on the whole (0, 1) (unique a.s.). Second, if τ_{1}(ω) ∈ (0, 1/2) then τ_{2}(ω) is the minimizer on (1/2, 1), otherwise — on (0, 1/2). Third, τ_{3}(ω) is the minimizer on the first of the four intervals (0, 1/4), (1/4, 1/2), (1/2, 3/4) and (3/4, 1) that contains neither τ_{1}(ω) nor τ_{2}(ω). And so on.

All measurable enumerations (τ′_{k})_{k} result from one of them (τ_{k})_{k} in the sense that

τ′_{k}(ω) = τ_{σ_{ω}(k)}(ω) a.s.

for some (unique, in fact) random permutation σ : Ω → S_{∞}, that is, an F_{a,b}^{white}-measurable random variable σ valued in the group S_{∞} of all bijective maps {1, 2, … } → {1, 2, … } (equipped with its natural Borel σ-field). See also [9, 2e].

Each τ_{k} is a measurable selector of the set of all local minimizers; for short, let us say just a selected minimum. Here is the general form of a selected minimum τ in terms of a given enumeration (τ_{k})_{k}:

(10.2) τ(ω) = τ_{k}(ω) for ω ∈ A_{k},

where (A_{1}, A_{2}, …) is a countable measurable partition of Ω.

Every selected minimum may serve as (say) the first element of some enumeration.

Given two adjacent time intervals (a, b) and (b, c), we may choose measurable enumerations (τ′_{k})_{k} and (τ′′_{k})_{k} of local minima on (a, b) and (b, c) respectively, and combine them into a measurable enumeration (τ_{k})_{k} on (a, c), say,

(10.3)  τ_{2k−1} = τ′_{k}, τ_{2k} = τ′′_{k} for k = 1, 2, …

taking into account that the point b is a.s. not a local minimizer. Now we can attach independent random signs to the local minima. Let

Ω_{1}^{white} ⊂ C[0, 1]

be a set of full Wiener measure such that for every ω_{1}^{white} ∈ Ω_{1}^{white} the set LocMin(ω_{1}^{white}) of all local minimizers of the path ω_{1}^{white} is a dense countable subset of (0, 1). We introduce the set {−1, +1}^{LocMin(ω_{1}^{white})} of all functions η_{1} : LocMin(ω_{1}^{white}) → {−1, +1} and consider the disjoint union Ω_{1} of these sets over all ω_{1}^{white},

Ω_{1} = {(ω_{1}^{white}, η_{1}) : ω_{1}^{white} ∈ Ω_{1}^{white}, η_{1} ∈ {−1, +1}^{LocMin(ω_{1}^{white})}} .

Every measurable enumeration (τ_{k})_{k} of the local minima on (0, 1) gives us a one-to-one correspondence

Ω_{1} ↔ Ω_{1}^{white} × {−1, +1}^{∞},
(ω_{1}^{white}, η_{1}) ↔ (ω_{1}^{white}, (η_{1}(τ_{k}(ω_{1}^{white})))_{k});

here {−1, +1}^{∞} = {−1, +1}^{{1,2,... }} is the set of all infinite sequences of numbers ±1. (As usual, a set of Wiener measure 0 in Ω_{1}^{white} may be neglected.) We take the uniform probability distribution m on {−1, +1} (giving equal probabilities 0.5 to −1 and +1), equip {−1, +1}^{∞} with the product measure m^{∞}, and Ω_{1}^{white}×{−1, +1}^{∞} — with the Wiener measure multiplied by m^{∞}. Then, using the one-to-one correspondence, we transfer the probability measure (and the underlying σ-field) to Ω_{1}. The choice of an enumeration (τ_{k})_{k} does not matter, since m^{∞} is invariant under permutations.

Now Ω_{1} is a probability space. Similarly, Ω_{t} becomes a probability space for every t ∈ (0, ∞). Given s, t ∈ (0, ∞), we get a natural isomorphism

Ω_{s} × Ω_{t} ⟷ Ω_{s+t}, (10.4)

((ω_{s}^{white}, η_{s}), (ω_{t}^{white}, η_{t})) ⟷ (ω_{s+t}^{white}, η_{s+t})

where (ω_{s}^{white}, ω_{t}^{white}) ⟷ ω_{s+t}^{white} is the usual composition of Brownian paths, and

(10.5)  η_{s+t}(τ) = \begin{cases}η_{s}(τ) & if τ < s, \\ η_{t}(τ − s) & if τ > s. \end{cases}

(The notation is not good for the case s = t, since ω_{s} and ω_{t} are still treated as different variables; hopefully it is not too confusing.) The composition (η_{s}, η_{t}) ⟷ η_{s+t} is described conveniently in terms of an enumeration of the form (10.3) for a = 0, b = s, c = s + t:

(10.6) η_{s+t}(τ_{2k−1}) = η_{s}(τ′_{k}), η_{s+t}(τ_{2k}) = η_{t}(τ′′_{k} − s) for k = 1, 2, …

(of course, all these η and τ depend implicitly on the underlying ω^{white}).

We have a noise (an extension of the white noise). It is described above via probability spaces Ω_{t} satisfying Ω_{s} × Ω_{t} = Ω_{s+t} rather than sub-σ-fields F_{s,t} (on a single Ω) satisfying F_{r,s} ⊗F_{s,t} = F_{r,t}, but these are two equivalent languages (see [9, 3c1 and 3c6]), and the corresponding Arveson system is just H_{t} = L_{2}(Ω_{t}).

However, it is not yet the new, drift sensitive noise that we need. Rather, it is Warren’s noise of splitting. The binary extension performed at each τ should follow the construction of Section 9. To this end we retain the probability spaces Ω_{t} constructed before, but replace the straightforward isomorphisms (10.4)–(10.5) with less evident, ‘twisted’ isomorphisms. Namely, (10.5) is replaced with

η_{s+t}(τ) = η_{t}(τ − s) if τ > s, (10.7)

η_{s+t}(τ) = η_{s}(τ) ∏_{n:τ+ε_{n}∈(s,s+t]} f_{n}(B(τ + ε_{n}) − B(τ)) if τ < s. (10.8)

As before, all these η and τ depend implicitly on the underlying ω^{white}, and B(s)(ω_{t}^{white}) = ω_{t}^{white}(s) for s ∈ [0, t].

The new noise is thus constructed. Its parameters (ε_{n})_{n} and (f_{n})_{n} will be chosen later. (In fact, ε_{n} = 2 · 3^{−n−1}, and f_{n} are given by Lemma 8.10.)

The classical part of the new noise is exhausted by the white noise, which can be proved via the predictable representation property, see [9, 4d].

In order to examine the impact of drift on the new noise we need the relation

(10.9)  LocMin(ω_{t}^{white}) = LocMin(θ_{t}^{λ}(ω_{t}^{white}))

(for all t, λ and almost all ω_{t}^{white} ∈ Ω_{t}^{white}); as before, θ_{t}^{λ} : C[0, t] → C[0, t] is the drift transformation, (θ_{t}^{λ}b)(s) = b(s) − 2λs. The relation (10.9) follows from the well-known fact that all local minima of the Brownian motion are sharp (a.s.) in the sense that

\frac{B(t) − B(τ)}{|t − τ|} → ∞ as t → τ, t ≠ τ

whenever τ is a local minimizer. See [4, Section 2.10, Items 7,8]. (In fact, |t − τ| may be replaced with \sqrt{|t − τ|}/ ln^{2} |t − τ|.)

It is easy to guess that a drift corresponds to a shift of the functions f_{n}. The proof (rather boring) is given below.

Lemma 10.10. Let numbers λ ∈ ℝ, ε_{n} ↓ 0 and Borel functions

f_{n}, g_{n} : ℝ → {−1, +1}

satisfy

g_{n}(x) = f_{n}(x + 2λε_{n}) for all x ∈ ℝ and n.

Let two extensions of the white noise be constructed as before, one corresponding to (f_{n})_{n} and (ε_{n})_{n}, the other corresponding to (g_{n})_{n} and (ε_{n})_{n}. Then the second extension results from the first one by the drift 2λ (as defined in Section 2), up to isomorphism of extensions.

Proof. The probability spaces Ω_{t} and measure preserving maps Ω_{t} → Ω_{t}^{white} are the same for both extensions, however, the corresponding isomorphisms α_{f}, α_{g} : Ω_{s} × Ω_{t} → Ω_{s+t} differ; α_{f}, used in the first extension, involves f_{n} (recall (10.8)), while α_{g}, used in the second extension, involves g_{n} instead of f_{n}.

We introduce the third extension, resulting from the first one by the drift 2λ, and seek an isomorphism between the second and third extensions.

The third extension uses the same Ω_{t} but with probability measures P′_{t} different from the probability measures P_{t} used by the first and second extensions; namely,

\frac{dP′_{t}}{dP_{t}} = D_{t} = exp(2λB_{t} − 2λ^{2}t).

The white noise extended by the third extension is generated by the Brownian motion B′_{t} = B_{t} − 2λt. Note also that the third extension uses α_{f}.

The probability space Ω_{t} consists of pairs (ω_{t}^{white}, η_{t}) where

ω_{t}^{white} ∈ Ω_{t}^{white} ⊂ C[0, t]

and η_{t} ∈ {−1, +1}^{LocMin(ω_{t}^{white})}. The drift transformation θ_{t}^{λ} may be treated as a measure preserving map

θ_{t}^{λ} : (Ω_{t}^{white}, D_{t} · W_{t}) → (Ω_{t}^{white}, W_{t}).

Using (10.9) we define θ̃_{t}^{λ} : Ω_{t} → Ω_{t} by θ̃_{t}^{λ}(ω_{t}^{white}, η_{t}) = (θ_{t}^{λ}ω_{t}^{white}, η_{t}) and get a measure preserving map

θ̃_{t}^{λ} : (Ω_{t}, P′_{t}) → (Ω_{t}, P_{t}).

Clearly, B′_{s} = B_{s} ∘ θ̃_{t}^{λ} for s ∈ [0, t]. It remains to check that θ̃_{s}^{λ} × θ̃_{t}^{λ} = θ̃_{s+t}^{λ} in the sense that the diagram

\xymatrix{Ω_{s} × Ω_{t} \ar[d]^{α_{f}} \ar[r]^{θ̃_{s}^{λ}×θ̃_{t}^{λ}} & Ω_{s} × Ω_{t} \ar[d]^{α_{g}} \\ Ω_{s+t} \ar[r]^{θ̃_{s+t}^{λ}} & Ω_{s+t}}

is commutative. Let ω_{s} = (ω_{s}^{white}, η_{s}) ∈ Ω_{s} and ω_{t} = (ω_{t}^{white}, η_{t}) ∈ Ω_{t}. We have α_{f}(ω_{s}, ω_{t}) = (ω_{s+t}^{white}, η_{s+t}), where ω_{s+t}^{white} is the usual composition of ω_{s}^{white} and ω_{t}^{white}, while η_{s+t} is obtained from η_{s} and η_{t} according to (10.7), (10.8). Thus,

θ̃_{s+t}^{λ}(α_{f}(ω_{s}, ω_{t})) = θ̃_{s+t}^{λ}(ω_{s+t}^{white}, η_{s+t}) = (θ_{s+t}^{λ}(ω_{s+t}^{white}), η_{s+t}).

On the other hand,

(θ̃_{s}^{λ} × θ̃_{t}^{λ})(ω_{s}, ω_{t}) = (θ̃_{s}^{λ}(ω_{s}), θ̃_{t}^{λ}(ω_{t})) = ((θ_{s}^{λ}(ω_{s}^{white}), η_{s}), (θ_{t}^{λ}(ω_{t}^{white}), η_{t})).

Clearly, α_{g}((θ_{s}^{λ}(ω_{s}^{white}), η_{s}), (θ_{t}^{λ}(ω_{t}^{white}), η_{t})) = (θ_{s+t}^{λ}(ω_{s+t}^{white}), η′_{s+t}) for some η′_{s+t} (since θ_{s}^{λ} × θ_{t}^{λ} = θ_{s+t}^{λ}). Finally, η′_{s+t} = η_{s+t} by (10.7), (10.8) and the equality

g_{n}(θ_{s+t}^{λ}(ω_{s+t}^{white})(τ + ε_{n}) − θ_{s+t}^{λ}(ω_{s+t}^{white})(τ))

= g_{n}(ω_{s+t}^{white}(τ + ε_{n}) − ω_{s+t}^{white}(τ) − 2λε_{n})

= f_{n}(ω_{s+t}^{white}(τ + ε_{n}) − ω_{s+t}^{white}(τ)). □

11. The binary extension inside the new noise

According to Section 9, the Brownian motion B leads to an inductive system of probability spaces formed by the restrictions of B to the time intervals [0, τ + 3^{−n}] ∩ [0, 1], where τ is the (global) minimizer of B on [0, 1]. Further, every sequence (f_{n})_{n} of Borel functions f_{n} : ℝ → {−1, +1} leads to a binary extension of this inductive system. The extension is formed by the restrictions of B and S_{f} to [0, τ + 3^{−n}] ∩ [0, 1]; here S_{f} : (τ, 1) → {−1, +1} is a random function satisfying (9.5).

On the other hand, according to Section 10, (f_{n})_{n} (in combination with ε_{n} = 2 · 3^{−n−1}) leads to a noise that extends the white noise. The noise is formed by the Brownian motion B and the random variables η_{t}(τ); here τ runs over all local minimizers of B on (0, t). In turn, the noise leads to an Arveson system that extends the type I_{1} Arveson system of the white noise.

These constructions of Sections 9 and 10 are related as follows.

Proposition 11.1. If two sequences (f_{n})_{n}, (g_{n})_{n} of Borel functions

ℝ → {−1, +1}

lead to isomorphic extensions of the type I_{1} Arveson system (of the white noise), then they lead to isomorphic binary extensions of the inductive system of probability spaces.

The proof is given after the proof of Proposition 11.4.

Proof of Theorem 1.10. The binary extension, constructed in Section 9 using the functions f_{n} given by Lemma 8.10 (combined with Proposition 9.1), is not isomorphic to the extension that corresponds to the shifted functions g_{n}(x) = f_{n}(x + 3^{−n}c), unless c = 0. By Proposition 11.1, (f_{n})_{n} and (g_{n})_{n} lead to nonisomorphic extensions (constructed in Section 10) of the type I_{1} Arveson system (of the white noise). By Lemma 10.10, these nonisomorphic extensions result from one another by a drift. This drift sensitivity implies Theorem 1.10 by Corollary 2.2. □

Comparing (9.5) and (10.8) we see that the function t ↦ η_{t}(τ) behaves like the function S_{f}. (Here τ is the global minimizer of B on [0, 1].) In other words, we may let for some (therefore, all) n such that τ + 3^{−n} < 1,

S_{f}(τ + 3^{−n}) = η_{τ+3^{−n}}(τ),

thus defining a measure preserving map from the probability space Ω^{noise(f)} of the noise on [0, 1] to the probability space Ω^{bin(f)} of the binary extension on [0, 1];

Ω^{noise(f)} → Ω^{bin(f)};

here f = (f_{n})_{n} is the given sequence of functions. Accordingly, we have a natural embedding of Hilbert spaces,

L_{2}(Ω^{bin(f)}) → L_{2}(Ω^{noise(f)}).

Striving to prove Proposition 11.1 we assume existence of an isomorphism Θ = (Θ_{t})_{t} between the two Arveson systems,

(11.2) Θ_{t} : L_{2}(Ω_{t}^{noise(f)}) → L_{2}(Ω_{t}^{noise(g)}),

(11.3) Θ_{t} is trivial on L_{2}(Ω_{t}^{white}).

Note that Ω^{noise(f)} = Ω_{1}^{noise(f)}.

Proposition 11.4. Θ_{1} maps the subspace L_{2}(Ω^{bin(f)}) of L_{2}(Ω^{noise(f)}) onto the subspace L_{2}(Ω^{bin(g)}) of L_{2}(Ω^{noise(g)}).

The proof is given after Lemma 11.5.

The structure of L_{2}(Ω^{noise(f)}) is easy to describe:

L_{2}(Ω^{noise(f)}) = H_{0}^{f} ⊕ H_{1}^{f} ⊕ H_{2}^{f} ⊕ ⋯,

where H_{n}^{f} (called the n-th superchaos space) consists of the random variables of the form

∑_{k_{1}<···<k_{n}} η_{1}(τ_{k_{1}}) … η_{1}(τ_{k_{n}})φ_{k_{1},...,k_{n}},

φ_{k_{1},...,k_{n}} ∈ L_{2}(Ω_{1}^{white}), ∑_{k_{1}<···<k_{n}} ∥φ_{k_{1},...,k_{n}}∥^{2} < ∞,

where (τ_{k})_{k} is a measurable enumeration of the local minimizers of B on (0, 1) (the choice of the enumeration does not matter). See [10, (3.1)] for the case f_{n}(·) = 1 (Warren’s noise of splitting); the same argument works in general. Note that H_{0}^{f} = L_{2}(Ω_{1}^{white}).

It is well-known that the superchaos spaces may be described in terms of the Arveson system, and therefore Θ_{1} maps H_{n}^{f} onto H_{n}^{g}. We need the first superchaos space only; here is a simple argument for this case:

H_{1} = {ψ ∈ L_{2}(Ω^{noise}) : ∀t ∈ (0, 1) ψ = Q_{0,t}ψ + Q_{t,1}ψ};

here Q_{0,t} is the orthogonal projection of L_{2}(Ω^{noise}) = L_{2}(Ω_{t}^{noise})⊗L_{2}(Ω_{1−t}^{noise}) onto the subspace L_{2}(Ω_{t}^{noise}) ⊗ L_{2}(Ω_{1−t}^{white}), and Q_{t,1} onto

L_{2}(Ω_{t}^{white}) ⊗ L_{2}(Ω_{1−t}^{noise}).

We have

Θ_{1}(L_{2}(Ω_{t}^{noise(f)}) ⊗ L_{2}(Ω_{1−t}^{white})) = Θ_{t}(L_{2}(Ω_{t}^{noise(f)})) ⊗ Θ_{1−t}(L_{2}(Ω_{1−t}^{white}))
= L_{2}(Ω_{t}^{noise(g)}) ⊗ L_{2}(Ω_{1−t}^{white})

by (11.3); therefore Θ_{1}Q_{0,t}^{f} = Q_{0,t}^{g}Θ_{1}. Similarly, Θ_{1}Q_{t,1}^{f} = Q_{t,1}^{g}Θ_{1}. It follows that

Θ_{1}H_{1}^{f} = H_{1}^{g}.

Similarly, L_{2}(Ω_{t}^{noise}) = H_{0}(t) ⊕ H_{1}(t) ⊕ H_{2}(t) ⊕ ⋯ (the upper index, be it f or g, is omitted). Identifying L_{2}(Ω_{1}) with L_{2}(Ω_{t}) ⊗ L_{2}(Ω_{1−t}) we have

H_{1} = \underbrace{H_{1}(t) ⊗ H_{0}(1 − t)}_{Q_{0,t}H_{1}} ⊕ \underbrace{H_{0}(t) ⊗ H_{1}(1 − t)}_{Q_{t,1}H_{1}} .

The commutative algebra L_{∞}(Ω_{1}^{white}) acts naturally on H_{1}:

h · ∑_{k} η_{1}(τ_{k})φ_{k} = ∑_{k} η_{1}(τ_{k})h · φ_{k} for h ∈ L_{∞}(Ω_{1}^{white}).

Also the commutative algebra L_{∞}(0, 1) acts naturally on H_{1}. In particular, 1_{(0,t)} acts as Q_{0,t}, and 1_{(t,1)} acts as Q_{t,1}. In general,

h · ∑_{k} η_{1}(τ_{k})φ_{k} = ∑_{k} η_{1}(τ_{k})h(τ_{k})φ_{k} for h ∈ L_{∞}(0, 1).

(The choice of enumeration (τ_{k})_{k} does not matter.) The two actions commute, and may be combined into the action of L_{∞}(μ) (on H_{1}) for some measure μ on Ω_{1}^{white} × (0, 1):

h · ∑_{k} η_{1}(τ_{k})φ_{k}(·) = ∑_{k} η_{1}(τ_{k})h(·, τ_{k})φ_{k}(·) for h ∈ L_{∞}(μ).

The measure μ may be chosen as

∫ h dμ = 𝔼 ∑_{k} \frac{1}{k^{2}} h(B, τ_{k})

(or anything equivalent).

Lemma 11.5. The diagram

\xymatrix{H_{1}^{f} \ar[d]^{h} \ar[r]^{Θ_{1}} & H_{1}^{g} \ar[d]^{h} \\ H_{1}^{f} \ar[r]^{Θ_{1}} & H_{1}^{g}}

is commutative for every h ∈ L_{∞}(μ).

Proof. Given [a, b] ⊂ [0, 1], we define a subalgebra Γ(a, b) ⊂ L_{∞}(μ) as consisting of the functions of the form

h(ω_{0,1}^{white}, t) = \begin{cases}h′(ω_{0,a}^{white}, ω_{b,1}^{white}) & for t ∈ (a, b), \\ 0 & for t ∈ (0, a) ∪ (b, 1), \end{cases}

where h′ ∈ L_{∞}(Ω_{a}^{white} × Ω_{1−b}^{white}), and ω_{0,1}^{white} ∈ Ω_{1}^{white} is treated as the triple (ω_{0,a}^{white}, ω_{a,b}^{white}, ω_{b,1}^{white}) according to the natural isomorphism between Ω_{1}^{white} and Ω_{a}^{white} × Ω_{b−a}^{white} × Ω_{1−b}^{white}. For each n = 1, 2, … we define a subalgebra Γ_{n} ⊂ L_{∞}(μ) by

Γ_{n} = ∑_{k=1}^{2^{n}} Γ(\frac{k − 1}{2^{n}}, \frac{k}{2^{n}} ).

It is easy to see that Γ_{n} corresponds to a measurable partition; in other words, Γ_{n} = L_{∞}(Ω_{1}^{white} × (0, 1), E_{n}, μ) for some sub-σ-field E_{n} of the σ-field E of all μ-measurable sets. We have E_{n} ↑ E, that is, E_{1} ⊂ E_{2} ⊂ ⋯ and E is the least sub-σ-field containing all E_{n}, which follows from the fact that Γ_{1} ∪Γ_{2} ∪… contains a countable set that separates points of Ω_{1}^{white} ×(0, 1).

If Θ_{1}h_{n} = h_{n}Θ_{1} (as operators H_{1}^{f} → H_{1}^{g}) for all n, and h_{n} → h almost everywhere, and sup_{n} ∥h_{n}∥_{∞} < ∞, then Θ_{1}h = hΘ_{1}. Thus, it is sufficient to prove the equality Θ_{1}h = hΘ_{1} for all h ∈ Γ_{1} ∪ Γ_{2} ∪ . . .. Without loss of generality we may assume that h ∈ Γ(a, b) for some a, b. Moreover, I assume that b = 1, leaving the general case to the reader. Thus,

h(ω_{0,1}^{white}, t) = h′(ω_{0,a}^{white})1_{(a,1)}(t).

We recall that Θ_{1} = Θ_{a}⊗Θ_{1−a}, H_{1} = H_{1}(a)⊗H_{0}(1−a)⊕H_{0}(a)⊗H_{1}(1−a), and note that

Θ_{1}(H_{1}^{f}(a) ⊗ H_{0}^{f}(1 − a)) = H_{1}^{g}(a) ⊗ H_{0}^{g}(1 − a),
Θ_{1}(H_{0}^{f}(a) ⊗ H_{1}^{f}(1 − a)) = H_{0}^{g}(a) ⊗ H_{1}^{g}(1 − a).

The subspaces H_{1}^{f}(a) ⊗ H_{0}^{f}(1 − a) and H_{1}^{g}(a) ⊗ H_{0}^{g}(1 − a) are annihilated by h, thus, Θ_{1}h and hΘ_{1} both vanish on H_{1}^{f}(a) ⊗ H_{0}^{f}(1 − a). On the other subspace, H_{0}^{f}(a) ⊗ H_{1}^{f}(1 − a), h acts as h′ ⊗ 1, while Θ_{1} acts as 1 ⊗ Θ_{1−a}. Therefore Θ_{1}h = hΘ_{1}. □

Proof of Proposition 11.4. We apply Lemma 11.5 to h ∈ L_{∞}(μ) defined by

h(ω_{1}^{white}, t) = \begin{cases}1 & if τ(ω_{1}^{white}) = t, \\ 0 & otherwise, \end{cases}

where τ is the (global) minimizer on (0, 1). This function acts on H_{1} as the projection onto the subspace

{η_{1}(τ)φ : φ ∈ L_{2}(Ω_{1}^{white})} = L_{2}(Ω^{bin}) ⊖ L_{2}(Ω^{white}).  □

Proof of Proposition 11.1. We have two binary extensions, (Ω_{n}^{bin(f)}, β̃_{n}^{f})_{n} and (Ω_{n}^{bin(g)}, β̃_{n}^{g})_{n}, of an inductive system (Ω_{n}^{white}, β_{n})_{n} of probability spaces (according to (γ_{n}^{f})_{n} and (γ_{n}^{g})_{n} respectively). Their isomorphism is ensured by Lemma 6.2, provided that Condition 6.2(b) is satisfied by some unitary operators Θ_{n}^{bin} : L_{2}(Ω_{n}^{bin(f)}) → L_{2}(Ω_{n}^{bin(g)}). Using the natural embeddings L_{2}(Ω_{n}^{bin(f)}) ⊂ L_{2}(Ω^{bin(f)}) and L_{2}(Ω_{n}^{bin(g)}) ⊂ L_{2}(Ω^{bin(g)}) we define all Θ_{n}^{bin} as restrictions of a single operator Θ^{bin} : L_{2}(Ω^{bin(f)}) → L_{2}(Ω^{bin(g)}). Using Proposition 11.4 we define Θ^{bin} as the restriction of Θ_{1} to L_{2}(Ω^{bin(f)}). It remains to prove that

(11.6) Θ^{bin}(L_{2}(Ω_{n}^{bin(f)})) = L_{2}(Ω_{n}^{bin(g)}),

Θ_{n}^{bin} intertwines the actions of L_{∞}(Ω_{n}^{white})
on L_{2}(Ω_{n}^{bin(f)}) and L_{2}(Ω_{n}^{bin(g)}),

(11.7)

(11.8) Θ_{n}^{bin} is trivial on L_{2}(Ω_{n}^{white})

for all n.

By (11.3), Θ^{bin} is trivial on L_{2}(Ω^{white}); (11.8) follows.

By Lemma 11.5, Θ^{bin} intertwines the actions of L_{∞}(Ω^{white}) on L_{2}(Ω^{bin(f)}) and L_{2}(Ω^{bin(g)}); (11.7) follows.

The proof of (11.6) is the point of Proposition 11.9 below. □

Proposition 11.9. The operator Θ^{bin} maps the subspace

L_{2}(Ω_{n}^{bin(f)}) ⊂ L_{2}(Ω^{bin(f)})

onto the subspace

L_{2}(Ω_{n}^{bin(g)}) ⊂ L_{2}(Ω^{bin(g)}).

The proof is given after Lemma 11.10.

Recall that the elements of L_{2}(Ω_{n}^{bin(f)}) are functions of the restrictions of B and S_{f} to [0, τ + 3^{−n}] ∩ [0, 1].

For a given t ∈ (0, 1) we consider the sub-σ-field F_{t}^{f} on Ω^{bin(f)}, generated by the restrictions of B and S_{f} to [0, t]. The elements of the subspace L_{2}(Ω^{bin(f)}, F_{t}^{f}) are functions of B|_{[0,t]} and S_{f}|_{[0,t]} (that is, S_{f}|_{(τ,t]}).

We know L_{∞}(Ω^{white}) acts on L_{2}(Ω^{bin(f)}). In particular, for 0 < r < s < 1, the function 1_{(r,s)}(τ) (that is, the indicator of {ω^{white} : r < τ(ω^{white}) < s}) acts as the projection onto a subspace H_{r,s}^{f} of L_{2}(Ω^{bin(f)}). The same holds for g. We have Θ^{bin}(H_{r,s}^{f}) ⊂ H_{r,s}^{g}, since Θ^{bin} intertwines the two actions of L_{∞}(Ω^{white}). We define

H_{r,s,t}^{f} = H_{r,s}^{f} ∩ L_{2}(Ω^{bin(f)}, F_{t}^{f}) for 0 < r < s < t < 1.

Lemma 11.10. Θ^{bin}(H_{r,s,t}^{f}) ⊂ H_{r,s,t}^{g}.

Proof. The binary extension Ω^{bin(f)} is constructed on the time interval (0, 1), but the same can be made on the time interval (0, t), giving a binary extension Ω^{bin(f,t)} of Ω_{t}^{white}, using the (global) minimizer τ_{t} on (0, t); sometimes τ_{t} = τ_{1}, sometimes τ_{t} ≠ τ_{1}.

The binary extension Ω^{bin(f)} is the product (recall Definition 7.1) of two binary extensions, Ω^{bin(f,t)} and Ω^{bin(f,1−t)}, according to the set

A ⊂ Ω_{1}^{white} = Ω_{t}^{white} × Ω_{1−t}^{white},
A = {ω_{1}^{white} : τ_{1}(ω_{1}^{white}) = τ_{t}(ω_{1}^{white}|_{[0,t]})} .

We know that Θ_{1} = Θ_{t} ⊗ Θ_{1−t}. Similarly to Proposition 11.4,

Θ_{t}(L_{2}(Ω^{bin(f,t)})) = L_{2}(Ω^{bin(g,t)});

we define Θ^{bin,t} : L_{2}(Ω^{bin(f,t)}) → L_{2}(Ω^{bin(g,t)}) as the restriction of Θ_{t} and observe that Θ^{bin} is the restriction of Θ^{bin,t} ⊗ Θ^{bin,1−t} to

L_{2}(Ω^{bin(f)}) ⊂ L_{2}(Ω^{bin(f,t)}) ⊗ L_{2}(Ω^{bin(f,1−t)})

(recall (7.2)).

By Lemma 7.3, Θ^{bin}(L_{2}(Ã, F_{1})) = L_{2}(Ã′, F′_{1}), where the sets Ã ⊂ Ω^{bin(f)}, Ã′ ⊂ Ω^{bin(g)} correspond to the inequality τ < t, the sub-σ-field F_{1} on Ã is induced by the sub-σ-field F_{t}^{f} on Ω^{bin(f)}, and F′_{1} on Ã′ — by F_{t}^{g}.

Taking into account that (r, s) ⊂ (0, t) we get H_{r,s,t} ⊂ L_{2}(Ã, F_{1}). Therefore Θ^{bin}(H_{r,s,t}^{f}) ⊂ L_{2}(Ã′, F′_{1}). On the other hand,

Θ^{bin}(H_{r,s,t}^{f}) ⊂ Θ^{bin}(H_{r,s}^{f}) ⊂ H_{r,s}^{g}.

It remains to note that L_{2}(Ã′, F′_{1}) ∩ H_{r,s}^{g} ⊂ H_{r,s,t}^{g}. □

Proof of Proposition 11.9. If r, s, t and n satisfy t ≤ r + 3^{−n} then

H_{r,s,t}^{g} ⊂ L_{2}(Ω_{n}^{bin(g)})

(since t ≤ τ(·) + 3^{−n} for all relevant points), and therefore

Θ^{bin}(H_{r,s,t}^{f}) ⊂ L_{2}(Ω_{n}^{bin(g)}).

The elements of L_{2}(Ω_{n}^{bin(f)}) are functions of the restrictions of B and S_{f} to [0, τ + 3^{−n}] ∩ [0, 1]. For every N such that \frac{1}{N} < 3^{−n} consider the

On automorphisms of Arveson systems 575

functions of the restrictions of B and S_{f} to [0, τ + 3^{−n} − \frac{1}{N}] ∩ [0, 1]; these are L_{2}(Ω_{n}^{bin(f)}, E_{N}) for some sub-σ-field E_{N}, and

⋃_{N} L_{2}(Ω_{n}^{bin(f)}, E_{N}) is dense in L_{2}(Ω_{n}^{bin(f)}),

since E_{N} ↑ E (a similar argument is used in the proof of Lemma 11.5; note that S_{f} jumps at τ +2·3^{−n}, not τ +3^{−n}). In order to prove Proposition 11.9 it remains to prove that

Θ^{bin}(L_{2}(Ω_{n}^{bin(f)}, E_{N})) ⊂ L_{2}(Ω^{bin(g)})

for all N (satisfying \frac{1}{N} < 3^{−n}).

Clearly,

L_{2}(Ω^{bin(f)}) = H_{0, \frac{1}{N}} ⊕ ⋯ ⊕ H_{\frac{N−1}{N} ,1}

(for every N). Every ψ ∈ L_{2}(Ω^{bin(f)}) is of the form

ψ = ψ_{1} + ⋯ + ψ_{N}, ψ_{k} ∈ H_{\frac{k−1}{N} , \frac{k}{N}} .

If ψ ∈ L_{2}(Ω_{n}^{bin(f)}, E_{N}) then ψ_{k} ∈ L_{2}(Ω^{bin(f)}, F_{\frac{k−1}{N} +3^{−n}}) (since

τ(·) + 3^{−n} − \frac{1}{N} < \frac{k − 1}{N} + 3^{−n}

for all relevant points), thus, ψ_{k} ∈ H_{\frac{k−1}{N} , \frac{k}{N} , \frac{k−1}{N} +3^{−n}}^{f}. Taking into account

that Θ^{bin}(H_{\frac{k−1}{N} , \frac{k}{N} , \frac{k−1}{N} +3^{−n}}^{f}) ⊂ L_{2}(Ω_{n}^{bin(g)}) we see that

Θ^{bin}(ψ) ∈ L_{2}(Ω_{n}^{bin(g)}).  □

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School of Mathematics, Tel Aviv University, Tel Aviv 69978, Israel tsirel@post.tau.ac.il http://www.tau.ac.il/˜ tsirel/

This paper is available via http://nyjm.albany.edu/j/2008/14-25.html.