Markovianity and the Thompson Group 𝐹

We show that representations of the Thompson group 𝐹 in the automorphisms of a noncommutative probability space yield a large class of bilateral stationary noncommutative Markov processes. As a partial converse, bilateral stationary Markov processes in tensor dilation form yield representations of 𝐹...

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Published in:	Symmetry, Integrability and Geometry: Methods and Applications
Date:	2022
Main Authors:	Köstler, Claus, Krishnan, Arundhathi
Format:	Article
Language:	English
Published:	Інститут математики НАН України 2022
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/211821
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Cite this:	Markovianity and the Thompson Group 𝐹. Claus Köstler and Arundhathi Krishnan. SIGMA 18 (2022), 083, 27 pages

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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author	Köstler, Claus Krishnan, Arundhathi
author_facet	Köstler, Claus Krishnan, Arundhathi
citation_txt	Markovianity and the Thompson Group 𝐹. Claus Köstler and Arundhathi Krishnan. SIGMA 18 (2022), 083, 27 pages
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container_title	Symmetry, Integrability and Geometry: Methods and Applications
description	We show that representations of the Thompson group 𝐹 in the automorphisms of a noncommutative probability space yield a large class of bilateral stationary noncommutative Markov processes. As a partial converse, bilateral stationary Markov processes in tensor dilation form yield representations of 𝐹. As an application, and building on a result of Kümmerer, we canonically associate a representation of 𝐹 to a bilateral stationary Markov process in classical probability.
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 083, 27 pages Markovianity and the Thompson Group F Claus KÖSTLER a and Arundhathi KRISHNAN b a) School of Mathematical Sciences, University College Cork, Cork, Ireland E-mail: claus@ucc.ie b) Department of Pure Mathematics, University of Waterloo, Ontario, Canada E-mail: arundhathi.krishnan@uwaterloo.ca Received April 08, 2022, in final form October 07, 2022; Published online October 27, 2022 https://doi.org/10.3842/SIGMA.2022.083 Abstract. We show that representations of the Thompson group F in the automorphisms of a noncommutative probability space yield a large class of bilateral stationary noncom- mutative Markov processes. As a partial converse, bilateral stationary Markov processes in tensor dilation form yield representations of F . As an application, and building on a result of Kümmerer, we canonically associate a representation of F to a bilateral stationary Markov process in classical probability. Key words: noncommutative stationary Markov processes; representations of Thompson group F 2020 Mathematics Subject Classification: 46L53; 60J05; 60G09; 20M30 1 Introduction The Thompson group F was introduced by Richard Thompson in the 1960s and many of its un- usual, interesting properties [6, 7] have been deeply studied over the past decades, in particular due to the still open conjecture of its nonamenability. Recently Vaughan Jones provided a new approach to the construction of (unitary) representations of the Thompson group F which is mo- tivated by the link between subfactor theory and conformal field theory (see [1, 4, 5, 12, 13, 14]). Independently, another approach to the representation theory of the Thompson group F is motivated by recent progress in the study of distributional invariance principles and symme- tries in noncommutative probability (see [8, 16] and [17, Introduction]). More precisely, a close relation between certain representations of the Thompson monoid F+ and unilateral noncom- mutative stationary Markov processes is established in [17]. The goal of the present paper is to demonstrate that this connection appropriately extends to one between representations of the Thompson group F and bilateral stationary noncommutative Markov processes (in the sense of Kümmerer [18]). Throughout we will mainly focus on a conceptual framework that is relevant in the operator algebraic reformulation of stationary Markov processes in classical probability theory. One of our main results is Theorem 3.9 which is about the construction of a local Markov filtration and a bilateral stationary Markov process from a given representation of the Thompson group F . Going beyond the framework of Markovianity, this construction is further deepened in Theorem 3.13 and Corollary 3.14, to obtain rich triangular arrays of commuting squares. A main result in the converse direction is Theorem 4.5, where we provide a canonical construction of This paper is a contribution to the Special Issue on Non-Commutative Algebra, Probability and Analysis in Ac- tion. The full collection is available at https://www.emis.de/journals/SIGMA/non-commutative-probability.html mailto:claus@ucc.ie mailto:arundhathi.krishnan@uwaterloo.ca https://doi.org/10.3842/SIGMA.2022.083 https://www.emis.de/journals/SIGMA/non-commutative-probability.html 2 C. Köstler and A. Krishnan a representation of the Thompson group F from a given bilateral stationary noncommutative Markov process in tensor dilation form. Finally, we apply this canonical construction to bilateral stationary Markov processes in classical probability. We establish in Theorem 4.8 that, for a given Markov transition operator, there exists a representation of the Thompson group F such that this Markov transition operator is the compression of a represented generator of the Thompson group F . We keep the presentation of our results on the connection between representations of the Thompson group F and Markovianity as close as possible to our treatment for the Thompson monoid F+ in [17]. Here we focus on the dynamical systems approach for noncommutative stationary processes and deliberately omit reformulations in terms of noncommutative random variables. In parts this is attributed to the fact that usually the noncommutative probability space generated by a bilateral stationary Markov sequence of noncommutative random variables turns out to be “too small” to accommodate a representation of the Thompson group F . This is in contrast to the situation in [17], where unilateral stationary Markov sequences generate a noncommutative probability space which is large enough to support a representation of the Thompson monoid F+. Some of these conceptual differences are further discussed and illustrated in the closing Section 4.4. Therein we constrain ourselves to the basics of the construction of representations of the Thompson group F from a given Markov transition operator and postpone a more-in-depth structural discussion to the future. Let us outline the content of this paper. Section 2 starts with providing definitions, notation and some background results on the Thompson group F (see Section 2.1). The basics of noncom- mutative probability spaces and Markov maps are given in Section 2.2. We review in Section 2.3 the notion of commuting squares from subfactor theory, as it underlies the present concept of Markovianity in noncommutative probability. Furthermore, we provide the notion of a local Markov filtration which allows us to define Markovianity on the level of von Neumann subal- gebras without any reference to noncommutative random variables. Finally, we review some results on noncommutative stationary processes in Section 2.4. Here we will meet bilateral non- commutative stationary Markov processes and Markov dilations in the sense of Kümmerer [18] as well as bilateral noncommutative stationary Bernoulli shifts. We investigate in Section 3 how representations of the Thompson group F in the auto- morphisms of noncommutative probability spaces yield bilateral noncommutative stationary Markov processes. Section 3.1 introduces the generating property of representations of F in Definition 3.1. This property ensures that the fixed point algebras of the represented generators of F form a tower which generates the noncommutative probability space, see Proposition 3.5. This tower of fixed point algebras equips the noncommutative probability space with a filtration which, using actions of the represented generators, can be further upgraded to become a local Markov filtration. Section 3.2 considers certain noncommutative stationary processes which are adapted to this local Markov filtration. The closing Section 4 shows that representations of F can be obtained from an important class of bilateral stationary noncommutative Markov processes. To be more precise, in Sec- tion 4.1 we provide elementary constructions of the Thompson group F in the automorphisms of a tensor product von Neumann algebra. This extends the representation of the Thompson monoid F+ obtained in [17] and also provides examples of bilateral noncommutative Markov and Bernoulli shifts. We show in Section 4.2 that Markov processes in tensor dilation form give rise to representations of F . Finally, in Section 4.3 we use a result of Kümmerer to show that, given a bilateral stationary Markov process in the classical case, we can obtain representations of F such that the associated transition operator is the compression of a represented generator of F . We provide more details to further motivate the construction of these representations in Section 4.4, also pointing out differences between the unilateral and bilateral cases in the process. Markovianity and the Thompson Group F 3 2 Preliminaries 2.1 The Thompson group F The Thompson group F , originally introduced by Richard Thompson in 1965 as a certain group of piece-wise linear homeomorphisms on the interval [0, 1], is known to have the infinite presen- tation F := ⟨g0, g1, g2, . . . \| gkgℓ = gℓ+1gk for 0 ≤ k < ℓ <∞⟩. We note that we work throughout with generators gk which correspond to the inverses of the generators usually used in the literature (e.g., [3]). Let e ∈ F denote the neutral element. As it is well-known, F is finitely generated with F = ⟨g0, g1⟩. Furthermore, as shown for example in [3, Theorem 1.3.7], an element e ̸= g ∈ F has the unique normal form g = g−b00 · · · g−bkk gakk · · · ga00 , (2.1) where a0, . . . , ak, b0, . . . , bk ∈ N0, k ≥ 0 and (i) exactly one of ak and bk is non-zero, (ii) if ai ̸= 0 and bi ̸= 0, then ai+1 ̸= 0 or bi+1 ̸= 0. As the defining relations of this presentation of F involve no inverse generators, one can associate to it the monoid F+ = ⟨g0, g1, g2, . . . \| gkgℓ = gℓ+1gk for 0 ≤ k < ℓ <∞⟩+, (2.2) referred to as the Thompson monoid F+. We remark that, alternatively, the generators of this monoid can be obtained as morphisms (in the inductive limit) of the category of finite binary forests, see for example [3, 13]. Definition 2.1. Let m,n ∈ N0 with m ≤ n be fixed. The (m,n)-partial shift shm,n is the group homomorphism on F defined by shm,n(gk) = { gm if k = 0, gn+k if k ≥ 1. We remark that the map shm,n preserves all defining relations of F and is thus well-defined as a group homomorphism. Lemma 2.2. The group homomorphisms shm,n on F are injective for all m,n ∈ N0. Proof. It suffices to show that shm,n(g) = e implies g = e. Let g ∈ F have the (unique) normal form as stated in (2.1). Thus, by the definition of the partial shifts, shm,n(g) = g−b0m · · · g−bkn+k g ak n+k · · · g a0 m . Thus shm,n(g) = e if and only if gakn+k · · · g a0 m = gbkn+k · · · g b0 m . Since the elements on both sides of the last equation are in normal form, its uniqueness implies ai = bi for all i. But this entails g = e. ■ 4 C. Köstler and A. Krishnan 2.2 Noncommutative probability spaces and Markov maps Throughout, a noncommutative probability space (M, ψ) consists of a von Neumann algebra M and a faithful normal state ψ on M. The identity of M will be denoted by 1M, or simply by 1 when the context is clear. Throughout, ∨ i∈I Mi denotes the von Neumann algebra generated by the family of von Neumann algebras {Mi}i∈I ⊂ M for I ⊂ Z. If M is abelian and acts on a separable Hilbert space, then (M, ψ) is isomorphic to ( L∞(Ω,Σ, µ), ∫ Ω · dµ ) for some standard probability space (Ω,Σ, µ). Definition 2.3. An endomorphism α of a noncommutative probability space (M, ψ) is a ∗- homomorphism on M satisfying the following additional properties: (i) ψ ◦ α = ψ (stationarity), (ii) α and the modular automorphism group σψt commute for all t ∈ R (modularity). The set of endomorphisms of (M, ψ) is denoted by End(M, ψ). We note that an endomorphism of (M, ψ) is automatically injective. In this paper, we will chiefly work with the automorphisms of (M, ψ) denoted by Aut(M, ψ). Note that α ∈ End(M, ψ) automatically satisfies α(1M) = 1M (unitality). Indeed, the -homomorphism property and stationarity of α entails ψ ( (α(1M)− 1M)∗(α(1M)− 1M) ) = 0. Now the faithfulness of ψ ensures α(1M)− 1M = 0. Definition 2.4. Let (M, ψ) and (N , φ) be two noncommutative probability spaces. A linear map T : M → N is called a (ψ,φ)-Markov map if the following conditions are satisfied: (i) T is completely positive, (ii) T is unital, (iii) φ ◦ T = ψ, (iv) T ◦ σψt = σφt ◦ T , for all t ∈ R. Here σψ and σφ denote the modular automorphism groups of (M, ψ) and (N , φ), respectively. If (M, ψ) = (N , φ), we say that T is a ψ-Markov map on M. Conditions (i) to (iii) imply that a Markov map is automatically normal. The condition (iv) is equivalent to the condition that a unique Markov map T ∗ : (N , φ) → (M, ψ) exists such that ψ ( T ∗(y)x ) = φ ( y T (x) ) , x ∈ M, y ∈ N . The Markov map T ∗ is called the adjoint of T and T is called self-adjoint if T = T ∗. We note that condition (iv) is automatically satisfied whenever ψ and φ are tracial, in particular for abelian von Neumann algebras M and N . Furthermore, we note that any T ∈ End(M, ψ) is automatically a ψ-Markov map and, in particular, any T ∈ Aut(M, ψ) is a ψ-Markov map with adjoint T ∗ = T−1. We recall for the convenience of the reader the definition of conditional expectations in the present framework of noncommutative probability spaces. Definition 2.5. Let (M, ψ) be a noncommutative probability space, and N be a von Neumann subalgebra of M. A linear map E : M → N is called a conditional expectation if it satisfies the following conditions: Markovianity and the Thompson Group F 5 (i) E(x) = x for all x ∈ N , (ii) ∥E(x)∥ ≤ ∥x∥ for all x ∈ M, (iii) ψ ◦ E = ψ. Such a conditional expectation exists if and only if N is globally invariant under the modular automorphism group of (M, ψ) (see [23, 24, 25]). The von Neumann subalgebra N is called ψ-conditioned if this condition is satisfied. Note that such a conditional expectation is auto- matically normal and uniquely determined by ψ. In particular, a conditional expectation is a Markov map and satisfies the module property E(axb) = aE(x)b for a, b ∈ N and x ∈ M. 2.3 Noncommutative independence and Markovianity We recall some equivalent properties as they serve to define commuting squares in subfactor theory (see for example [10, 15, 22]) and as they are familiar from conditional independence in classical probability. Proposition 2.6. Let M0, M1, M2 be ψ-conditioned von Neumann subalgebras of the proba- bility space (M, ψ) such that M0 ⊂ (M1 ∩M2). Then the following are equivalent: (i) EM0(xy) = EM0(x)EM0(y) for all x ∈ M1 and y ∈ M2, (ii) EM1EM2 = EM0, (iii) EM1(M2) = M0, (iv) EM1EM2 = EM2EM1 and M1 ∩M2 = M0. In particular, it holds that M0 = M1 ∩ M2 if one and thus all of these four assertions are satisfied. Proof. The case of tracial ψ is proved in [10, Proposition 4.2.1]. The non-tracial case follows from this, after some minor modifications of the arguments therein. ■ Definition 2.7. The inclusions M2 ⊂ M ∪ ∪ M0 ⊂ M1 as given in Proposition 2.6 are said to form a commuting square (of von Neumann algebras) if one (and thus all) of the equivalent conditions (i) to (iv) are satisfied in Proposition 2.6. Notation 2.8. We write I < J for two subsets I, J ⊂ Z if i < j for all i ∈ I and j ∈ J . The cardinality of I is denoted by \|I\|. For N ∈ Z, we denote by I+N the shifted set {i+N \| i ∈ I}. Finally, I(Z) denotes the set of all “intervals” of Z, i.e., sets of the form [m,n] := {m,m + 1, . . . , n}, [m,∞) := {m,m+ 1, . . .} or (−∞,m] := {. . . ,m− 1,m} for −∞ < m ≤ n <∞. We next address the basic notions of Markovianity in noncommutative probability. Com- monly, Markovianity is understood as a property of random variables relative to a filtration of the underlying probability space. Our investigations from the viewpoint of distributional invariance principles reveal that the phenomenon of “Markovianity” emerges without reference to any stochastic process already on the level of a family of von Neumann subalgebras, indexed by the partially ordered set of all “intervals” I(Z). As commonly the index set of a filtration is understood to be totally ordered [27], we refer to such families with partially ordered index sets as “local filtrations”. 6 C. Köstler and A. Krishnan Definition 2.9. A family of ψ-conditioned von Neumann subalgebras M• ≡ {MI}I∈I(Z) of the probability space (M, ψ) is called a local filtration (of (M, ψ)) if I ⊂ J =⇒ MI ⊂ MJ (isotony). The isotony property ensures that one has the inclusions MI ⊂ M ∪ ∪ MK ⊂ MJ for I, J,K ∈ I(Z) with K ⊂ (I ∩ J). Finally, let N• ≡ {NI}I∈I(Z) be another local filtration of (M, ψ). Then N• is said to be coarser than M• if NI ⊂ MI for all I ∈ I(Z) and we denote this by N• ≺ M•. Occasionally we will address N• also as a local subfiltration of M•. Definition 2.10. Let M• ≡ {MI}I∈I(Z) be a local filtration of (M, ψ). M• is said to be Markovian if the inclusions M(−∞,n] ⊂ M ∪ ∪ M[n,n] ⊂ M[n,∞) form a commuting square for each n ∈ Z. Cast as commuting squares, Markovianity of the local filtration M• has many equivalent formulations, see Proposition 2.6. In particular, it holds that EM(−∞,n] EM[n,∞) = EM[n,n] for all n ∈ Z. Here EMI denotes the ψ-preserving normal conditional expectation from M onto MI . 2.4 Noncommutative stationary processes and dilations We introduce bilateral noncommutative stationary processes, as they underlie the approach to distributional invariance principles in [9, 16]. Furthermore, we present dilations of Markov maps using Kümmerer’s approach to noncommutative stationary Markov processes [18]. The existence of such dilations is actually equivalent to the factoralizability of Markov maps (see [2, 11]). Definition 2.11. A bilateral stationary process (M, ψ, α,A0) consists of a probability space (M, ψ), a ψ-conditioned subalgebra A0 ⊂ M, and an automorphism α ∈ Aut(M, ψ). The sequence (ιn)n∈Z : (A0, ψ0) → (M, ψ), ιn := αn\|A0 = αnι0, is called the sequence of random variables associated to (M, ψ, α,A0). Here ψ0 denotes the restriction of ψ from M to A0 and ι0 denotes the inclusion map of A0 in M. The stationary process (M, ψ, α,A0) is called minimal if∨ i∈Z αi(A0) = M. Definition 2.12. The (not necessarily minimal) stationary process (M, ψ, α,A0) is called a (bi- lateral noncommutative) stationary Markov process if its canonical local filtration{ AI := ∨ i∈I αi(A0) } I∈I(Z) Markovianity and the Thompson Group F 7 is Markovian. If this process is minimal, then the endomorphism α is also called a Markov shift with generator A0. Furthermore, the associated ψ0-Markov map T = ι∗0αι0 on A0 is called the transition operator of the stationary Markov process. Here ι0 denotes the inclusion map of A0 in M, and ψ0 is the restriction of ψ to A0. The next lemma gives a simplified condition to check that a bilateral stationary process is a Markov process. Lemma 2.13. Let (M, ψ, α,A0) be a bilateral stationary process with canonical local filtration {AI := ∨ i∈I α i(A0)}I∈I(Z). Suppose P(−∞,0]P[0,∞) = P[0,0], where PI denotes the ψ-preserving normal conditional expectation from M onto AI . Then {AI}I∈I(Z) is a local Markov filtration and (M, ψ, α,A0) is a bilateral stationary Markov process. Proof. For all k ∈ Z and I ∈ I(Z), we have αkPI = PI+kα k (see [18, Remark 2.1.4]). Hence, for each n ∈ Z, P(−∞,0]P[0,∞) = P[0,0] ⇐⇒ αnP(−∞,0]P[0,∞)α −n = αnP[0,0]α −n ⇐⇒ P(−∞,n]P[n,∞) = P[n,n], which is the required Markovianity for the local filtration {AI}I∈I(Z). ■ Definition 2.14 ([18, Definition 2.1.1]). Let (A, φ) be a probability space. A φ-Markov map T on A is said to admit a (bilateral state-preserving) dilation if there exists a probability space (M, ψ), an automorphism α ∈ Aut(M, ψ) and a (φ,ψ)-Markov map ι0 : A → M such that Tn = ι∗0α nι0 for all n ∈ N0. Such a dilation of T is denoted by the quadruple (M, ψ, α, ι0) and is said to be minimal if M = ∨ n∈Z α nι0(A). (M, ψ, α, ι0) is called a dilation of first order if the equality T = ι∗0αι0 alone holds. Actually it follows from the case n = 0 that the (φ,ψ)-Markov map ι0 is a random vari- able from (A, φ) to (M, ψ) such that ι0ι ∗ 0 is the ψ-preserving conditional expectation from M onto ι0(A). Definition 2.15 ([18, Definition 2.2.4]). The dilation (M, ψ, α, ι0) of the φ-Markov map T on A (as introduced in Definition 2.14) is said to be a (bilateral state-preserving) Markov dilation if the local filtration { AI := ∨ n∈I α nι0(A) } I∈I(Z) is Markovian. Remark 2.16. A dilation of a φ-Markov map T on A may not be a Markov dilation. This is discussed in [21, Section 3], where it is shown that Varilly has constructed a dilation in [26] which is not a Markov dilation. We are grateful to B. Kümmerer for bringing this to our attention [20]. Note that this does not contradict the result that the existence of a dilation and the existence of a Markov dilation are equivalent (see [11, Theorem 4.4] or [17, Theorem 2.6.8]). Definition 2.17 ([18, Definition 4.1.3]). Let (A, φ) be a probability space and T be a φ-Markov map on A. A dilation of first order (M, ψ, α, ι0) of T is called a tensor dilation if the conditional expectation ι0ι ∗ 0 : M → ι0(A) is of tensor type, that is, there exists a von Neumann subalgebra C of M with faithful normal state χ such that M = A ⊗ C and (ι0ι ∗ 0)(a ⊗ x) = χ(x)(a ⊗ 1C) for all a ∈ A, x ∈ C. 8 C. Köstler and A. Krishnan Let us next relate the above bilateral notions of dilations and stationary processes. It is immediate that a dilation (M, ψ, α, ι0) of the φ-Markov map T on A gives rise to the stationary process (M, ψ, α, ι0(A)). Furthermore, this stationary process is Markovian if and only if the dilation is a Markov dilation, as evident from the definitions. Conversely, a stationary Markov process yields a dilation (and thus a Markov dilation) as it was shown by Kümmerer, stated below for the convenience of the reader. Proposition 2.18 ([18, Proposition 2.2.7]). Let (M, ψ, α,A0) be a bilateral noncommutative stationary Markov process and T = ι∗0αι0 be the corresponding transition operator where ι0 is the inclusion map of A0 into M. Then (M, ψ, α, ι0) is a dilation of T . In other words, the following diagram commutes for all n ∈ N0: (A0, ψ0) (A0, ψ0) (M, ψ) (M, ψ). Tn ι0 ι∗0 αn Here ψ0 denotes the restriction of ψ to A0. We close this section by providing a noncommutative notion of operator-valued Bernoulli shifts. The definition of such shifts stems from investigations of Kümmerer on the structure of noncommutative Markov processes in [18], and such shifts can also be seen to emerge from the noncommutative extended de Finetti theorem in [16]. In the following, Mβ := {x ∈ M \| β(x) = x} denotes the fixed point algebra of β ∈ Aut(M, ψ). Note that Mβ is automatically a ψ-conditioned von Neumann subalgebra. Definition 2.19. The minimal stationary process (M, ψ, β,B0) with canonical local filtration{ BI = ∨ i∈I β i 0(B0) } I∈I(Z) is called a bilateral noncommutative Bernoulli shift with generator B0 if Mβ ⊂ B0 and BI ⊂ M ∪ ∪ Mβ ⊂ BJ forms a commuting square for any I, J ∈ I(Z) with I ∩ J = ∅. It is easy to see that a noncommutative Bernoulli shift (M, ψ, β,B0) is a minimal stationary Markov process where the corresponding transition operator ι∗0βι0 is a conditional expectation (onto Mβ, the fixed point algebra of β). Here ι0 denotes the inclusion map of B0 into M. 3 Markovianity from representations of F We show that bilateral stationary Markov processes can be obtained from representations of the Thompson group F in the automorphisms of a noncommutative probability space. Most of the results in this section follow closely those of [17, Section 4], suitably adapted to the bilateral case. Let us fix some notation, as it will be used throughout this section. We assume that the probability space (M, ψ) is equipped with the representation ρ : F → Aut(M, ψ). For brevity of notion, especially in proofs, the represented generators of F are also denoted by αn := ρ(gn) ∈ Aut(M, ψ), Markovianity and the Thompson Group F 9 with fixed point algebras given by Mαn := {x ∈ M \| αn(x) = x}, for 0 ≤ n < ∞. Of course, Mαn = Mα−1 n . Furthermore, the intersections of fixed point algebras Mn := ⋂ k≥n+1 Mαk give the tower of von Neumann subalgebras Mρ(F ) ⊂ M0 ⊂ M1 ⊂ M2 ⊂ · · · ⊂ M∞ := ∨ n≥0 Mn ⊂ M. From the viewpoint of noncommutative probability theory, this tower provides a filtration of the noncommutative probability space (M, ψ). The canonical local filtration of a stationary process (M, ψ, α0,A0) will be seen to be a local subfiltration of a local Markov filtration whenever the ψ-conditioned von Neumann subalgebra A0 is well-localized, to be more precise: contained in the intersection of fixed point algebras M0. It is worthwhile to emphasize that, depending on the choice of the generator A0, the canonical local filtration of this stationary process may not be Markovian. Section 3.2 investigates in detail conditions under which the canonical local filtration of a stationary process (M, ψ, α0,A0) is Markovian. 3.1 Representations with a generating property An immediate consequence of the relations between generators of the Thompson group F is the adaptedness of the endomorphism α0 to the tower of (intersected) fixed point algebras: α0(Mn) ⊂ Mn+1 for all n ∈ N0. To see this, note that if x ∈ Mn and k ≥ n + 2, then αkα0(x) = α0αk−1(x) = α0x. On the other hand, if x ∈ Mn and k ≥ n, then αkα −1 0 (x) = α−1 0 αk+1(x) = α−1 0 (x). This gives that α−1 0 (Mn) ⊂ Mn−1 for n ≥ 1. Hence, actually α0(Mn) = Mn+1 for all n ∈ N0. We also note that α−1 0 (M0) ⊂ M0. Thus, generalizing terminology from classical probability, the random variables ι0 := Id \|M0 : M0 → M0 ⊂ M, ι1 := α0\|M0 : M0 → M1 ⊂ M, ι2 := α2 0\|M0 : M0 → M2 ⊂ M, ... ιn := αn0 \|M0 : M0 → Mn ⊂ M are adapted to the filtration M0 ⊂ M1 ⊂ M2 ⊂ · · · , and α0 is the time evolution of the stationary process (M, ψ, α0,M0). An immediate question is whether a representation of the Thompson group F restricts to the von Neumann subalgebra M∞. Definition 3.1. The representation ρ : F → Aut(M, ψ) is said to have the generating property if M∞ = M. As shown in Proposition 3.5 below, this generating property entails that each intersected fixed point algebra Mn = ⋂ k≥n+1Mαk equals the single fixed point algebra Mαn+1 . Thus the generating property tremendously simplifies the form of the tower M0 ⊂ M1 ⊂ · · · , and our next result shows that this can always be achieved by restriction. 10 C. Köstler and A. Krishnan Proposition 3.2. The representation ρ : F → Aut(M, ψ) restricts to the generating represen- tation ρgen : F → Aut(M∞, ψ∞) such that αn(M∞) ⊂ M∞ and EM∞EMαn = EMαnEM∞ for all n ∈ N0. Here ψ∞ denotes the restriction of the state ψ to M∞. EMαn and EM∞ denote the unique ψ-preserving normal conditional expectations onto Mαn and M∞ respectively. Proof. We show that αi(Mn) ⊂ Mn+1 for all i, n ≥ 0. Let x ∈ Mn. If i ≥ n + 1 then αi(x) = x is immediate from the definition of Mn. If i < n + 1 then, using the relations for the generators of the Thompson group, αi(x) = αiαk+1(x) = αk+2αi(x) for any k ≥ n, thus αi(x) ∈ Mn+1. Consequently, αi maps ⋃ n≥0Mn into itself for any i ∈ N0. It is also easily verified that α−1 i (Mn) ⊂ Mn for all i and n ≥ 0. Now a standard approximation argument shows that M∞ is invariant under αi and α −1 i for any i ∈ N0. Consequently, the representation ρ restricts to M∞ and, of course, this restriction ρgen has the generating property. Since M∞ is globally invariant under the modular automorphism group of (M, ψ), there ex- ists the (unique) ψ-preserving normal conditional expectation EM∞ from M onto M∞. In par- ticular, ρgen(gn) = αn\|M∞ commutes with the modular automorphism group of (M∞, ψ∞) which ensures ρgen(gn) ∈ Aut(M∞, ψ∞). Finally, that EM∞ and EMαn commute is concluded from EM∞αnEM∞ = αnEM∞ , which implies EMαnEM∞ = EM∞EMαn by routine arguments, and an application of the mean ergodic theorem (see for example [16, Theorem 8.3]), EMαn = lim N→∞ 1 N N∑ i=1 αin, where the limit is taken in the pointwise strong operator topology. ■ Lemma 3.3. With the notations as above, Mk = Mαk+1 ∩M∞ for all k ∈ N0. Proof. For the sake of brevity of notation, let Qn = EMαn denote the ψ-preserving normal conditional expectation from M onto Mαn . Let us first make the following observation: if x ∈ M∞, then Qn(x) ∈ M∞ for every n ∈ N0. Indeed, by Proposition 3.2, x ∈ M∞ implies αn(x) ∈ M∞ and thus 1 M ∑M i=1 α i n(x) ∈ M∞ for allM ≥ 1. As Qn(x) = limM→∞ 1 M ∑M i=1 α i k(x) in the strong operator topology, this ensures Qn(x) ∈ M∞. By the definition of Mk and M∞, it is clear that Mk ⊂ Mαk+1 ∩M∞. In order to show the reverse inclusion, it suffices to show that QnQk\|M∞ = Qk\|M∞ for 0 ≤ k < n < ∞. We claim that, for 0 ≤ k < n, QnQk\|M∞ = Qk\|M∞ ⇐⇒ QkQnQk\|M∞ = Qk\|M∞ . Indeed this equivalence is immediate from ψ ( (QnQk −Qk)(y ∗)(QnQk −Qk)(x) ) = ψ ( y∗(QkQn −Qk)(QnQk −Qk)(x) ) = ψ ( y∗(Qk −QkQnQk)(x) ) for all x, y ∈ M∞. We are left to prove QkQnQk\|M∞ = Qk\|M∞ for k < n. For this purpose we express the conditional expectations Qk and Qn as mean ergodic limits in the pointwise strong operator topology and calculate QkQnQk\|M∞ = lim M→∞ lim N→∞ 1 MN M∑ i=1 N∑ j=1 αikα j nQk\|M∞ Markovianity and the Thompson Group F 11 = lim M→∞ lim N→∞ 1 MN M∑ i=1 N∑ j=1 αjn+iα i kQk\|M∞ = lim M→∞ lim N→∞ 1 MN M∑ i=1 N∑ j=1 αjn+iQk\|M∞ = lim M→∞ 1 M M∑ i=1 Qn+iQk\|M∞ = Qk\|M∞ . The last equality is ensured as x ∈ M∞ implies that Qk(x) ∈ M∞, hence as Mρ(F ) ⊂ M0 ⊂ · · · ⊂ M∞ = ∨n≥0Mn, there exists sufficiently large i0 such that Qn+iQk(x) = Qk(x) for all i ≥ i0. Thus lim M→∞ 1 M M∑ i=1 Qn+iQk\|M∞ = IdQk\|M∞ in the pointwise strong operator topology. ■ Corollary 3.4. With notations as introduced at the beginning of the present Section 3, the following set of inclusions forms a commuting square for every n ∈ N0: Mαn+1 ⊂ M ∪ ∪ Mn ⊂ M∞. Proof. Let Qn and EM∞ be the ψ-preserving normal conditional expectations from M onto Mαn and M∞ respectively for n ∈ N0. For n ∈ N0, by Proposition 3.2, Qn+1EM∞ = EM∞Qn+1 and by Lemma 3.3, Mn = Mαn+1 ∩M∞. By (iv) of Proposition 2.6, we get a com- muting square. ■ Proposition 3.5. If the representation ρ : F → Aut(M, ψ) has the generating property then the following equality holds for all n ∈ N0: Mn = Mρ(gn+1). In other words, one has the tower of fixed point algebras Mρ(F+) ⊂ Mρ(g0) ⊂ Mρ(g1) ⊂ Mρ(g2) ⊂ · · · ⊂ M = ∨ n≥0 Mρ(gn). Proof. If the representation ρ is generating, then M∞ = M. Hence Mn = Mαn+1 for all n ∈ N0 as a consequence of Lemma 3.3. ■ The following intertwining property will be crucial for obtaining stationary Markov processes from representations of the Thompson group F . Proposition 3.6. Suppose ρ : F → Aut(M, ψ) is a (not necessarily generating) representation of F . Then with αn = ρ(gn), the following equality holds: αkQn = Qn+1αk for all 0 ≤ k < n <∞. Here Qn denotes the ψ-preserving normal conditional expectation from M onto the fixed point algebra Mαn of the represented generator αn ∈ Aut(M, ψ). 12 C. Köstler and A. Krishnan Proof. An application of the mean ergodic theorem and the relations between the generators of the Thompson group F yield that, for k < n, αkQn = lim N→∞ 1 N N∑ i=1 αkα i n = lim N→∞ 1 N N∑ i=1 αin+1αk = Qn+1αk. Here the limits are taken in the pointwise strong operator topology. ■ 3.2 Commuting squares and Markovianity for stationary processes Given the representation ρ : F → Aut(M, ψ), with represented generators αn := ρ(gn), for n ∈ N0, we recall that Mn = ⋂ k≥n+1 Mαk , denotes the intersected fixed point algebras. Throughout this section, let A0 be a ψ-conditioned von Neumann subalgebra of M0. Then (M, ψ, α0,A0) is a (bilateral noncommutative) station- ary process with generating algebra A0 (as introduced in Definition 2.11). Its canonical local filtration is denoted by A• ≡ {AI}I∈I(Z), where AI := ∨ i∈I αi0(A0), and an “interval” I ∈ I(Z) is written as [m,n] := {i ∈ Z \| m ≤ i ≤ n} or [m,∞) := {i ∈ Z \| m ≤ i} or (−∞, n] := {i ∈ Z \| i ≤ n}. Furthermore, PI will denote the ψ-preserving normal conditional expectation from M onto AI . Note that the endomorphism α0 acts compatibly on the local filtration, i.e., α0(AI) = AI+1 for all I ∈ I(Z), where I + 1 := {i+ 1 \| i ∈ I}. We record a simple, but important, observation obtained from the relations of F on stationary processes to which we will frequently appeal. Proposition 3.7. Let (M, ψ, α0,A0) be the (bilateral noncommutative) stationary process with A0 a ψ-conditioned subalgebra of M0. Then it holds that A(−∞,n] ⊂ Mn for all n ∈ N0. Proof. As A0 ⊂ M0, it holds that αn(x) = x for any x ∈ A0 and n ∈ N. Thus using the defining relations of F we get for 0 ≤ k ≤ n < ℓ, αℓα k 0(x) = αk0αℓ−k(x) = αk0(x). On the other hand, for k < 0 and ℓ ≥ 1, αℓα k 0(x) = αk0αℓ−k(x) = αk0(x). Hence A(−∞,n] = ∨ i∈(−∞,n] αi0(A0) ⊂ M0 ⊂ Mn for all n ∈ N0. ■ We next observe that the generating property of the representation ρ can be concluded from the minimality of a stationary process. Proposition 3.8. Suppose the representation ρ : F → Aut(M, ψ) and A0 ⊂ M0 are given. If the stationary process (M, ψ, α0,A0) is minimal, then ρ is generating. Markovianity and the Thompson Group F 13 Proof. For the stationary process (M, ψ, α0,A0), recall that A(−∞,∞) = ∨ i∈Z α i 0(A0) and minimality implies A(−∞,∞) = M. By Proposition 3.7, A(−∞,n] ⊂ Mn for all n ∈ N0. Thus M = ∨ n≥0A(−∞,n] ⊂ ∨ n≥0Mn = M∞. We conclude from this that the representation ρ has the generating property, i.e., M∞ = M. ■ In the following results, it is not assumed that the stationary process is minimal or that the representation ρ is generating unless explicitly mentioned. Theorem 3.9. Suppose ρ : F → Aut(M, ψ) is a representation with αn := ρ(gn) as before. Let A0 ⊂ M0 and A[0,∞) := ∨ n∈N0 αn0 (A0) be von Neumann subalgebras of (M, ψ) such that the inclusions Mα1 ⊂ M ∪ ∪ A0 ⊂ A[0,∞) form a commuting square. Then the family of von Neumann subalgebras A• ≡ {AI}I∈I(Z), with AI := ∨ i∈I αi0(A0), is a local Markov filtration and (M, ψ, α0,A0) is a (bilateral) stationary Markov process. Proof. Let Qn and PI denote the ψ-preserving normal conditional expectations from M onto Mαn and AI respectively. Note that the commuting square condition implies Q1P[0,∞) = P[0,0]. From Proposition 3.7, A(−∞,0] ⊂ M0 ⊂ Mα1 . Hence we get P(−∞,0]P[0,∞) = P(−∞,0]Q1P[0,∞) (since A(−∞,0] ⊂ Mα1) = P(−∞,0]P[0,0]P[0,∞) (by commuting square condition) = P[0,0] (as A[0,0] ⊂ A(−∞,0] and A[0,0] ⊂ A[0,∞)). Thus, by Lemma 2.13, {AI}I∈I(Z) is a local Markov filtration and (M, ψ, α0,A0) is a bilateral stationary Markov process. ■ Corollary 3.10. Suppose ρ : F → Aut(M, ψ) is a representation with α0 = ρ(g0). Then the quadruple (M, ψ, α0,M0) is a bilateral stationary Markov process. Proof. We know from Corollary 3.4 that the following is a commuting square: Mα1 ⊂ M ∪ ∪ M0 ⊂ M∞. Let {MI}I∈I(Z) denote the local filtration given by MI = ∨ i∈I α i 0(M0) and PI be the cor- responding conditional expectations. As M(−∞,n] ⊂ Mn for all n ∈ N0, it is easily ver- ified that M(−∞,∞) ⊂ M∞. Let P0 := P[0,0] be the ψ-preserving conditional expectation from M onto M0. Then from the commuting square above, we have EM∞Q1 = P0, where EM∞ is of course the conditional expectation onto M∞. This in turn gives P(−∞,∞)Q1 = P(−∞,∞)EM∞Q1 = P(−∞,∞)P0 = P0. Hence we get that M0 is a von Neumann subalgebra of M such that Mα1 ⊂ M ∪ ∪ M0 ⊂ M[0,∞) forms a commuting square. By Theorem 3.9, (M, ψ, α0,M0) is a stationary Markov process. ■ 14 C. Köstler and A. Krishnan Corollary 3.11. Suppose ρ : F → Aut(M, ψ) is a representation with αm = ρ(gm), for m ∈ N0. Then the quadruple (M, ψ, αm,Mn) is a bilateral stationary Markov process for any 0 ≤ m ≤ n <∞. Proof. Consider the representation ρm,n := ρ ◦ shm,n : F → Aut(M, ψ), where shm,n denotes the (m,n)-partial shift as introduced in Definition 2.1. We observe that ρm,n(g0) = ρ(gm) and ρm,n(gk) = ρ(gn+k) for all k ≥ 1. In particular, we get⋂ k≥1 Mρm,n(gk) = ⋂ k≥1 Mρ(gk+n) = ⋂ k≥n+1 Mρ(gk) = Mn. Thus Corollary 3.10 applies for the (m,n)-shifted representation ρm,n, and its application com- pletes the proof. ■ Corollary 3.12. Suppose ρ : F → Aut(M, ψ) is a generating representation. Then the quadru- ple ( M, ψ, αm,Mαn+1 ) is a bilateral stationary Markov process for any 0 ≤ m ≤ n <∞. Proof. If the representation ρ is generating, then Mαn+1 = Mn. Hence the result follows by Corollary 3.11. ■ Theorem 3.13. Let the probability space (M, ψ) be equipped with the representation ρ : F → Aut(M, ψ) and the local filtration A• ≡ {AI}I∈I(Z), where AI := ∨ i∈I ρ(g i 0)(A0) for some ψ- conditioned von Neumann subalgebra A0 of M0 = ⋂ k≥1Mρ(gk). Further suppose the inclusions Mρ(gk+1) ⊂ M ∪ ∪ A[0,k] ⊂ A[0,∞) form a commuting square for every k ≥ 0. Then each cell in the following infinite triangular array of inclusions is a commuting square: · · · ⊂ A(−∞,−2] ⊂ A(−∞,−1] ⊂ A(−∞,0] ⊂ A(−∞,1] ⊂ A(−∞,2] ⊂ · · · ⊂ A(−∞,∞) ∪ ∪ ∪ ∪ ∪ · · · ∪ ... ... ... ... ... · · · ... ∪ ∪ ∪ ∪ ∪ · · · ∪ A[−2,−2] ⊂ A[−2,−1] ⊂ A[−2,0] ⊂ A[−2,1] ⊂ A[−2,2] ⊂ · · · ⊂ A[−2,∞) ∪ ∪ ∪ ∪ ∪ A[−1,−1] ⊂ A[−1,0] ⊂ A[−1,1] ⊂ A[−1,2] ⊂ · · · ⊂ A[−1,∞) ∪ ∪ ∪ ∪ A[0,0] ⊂ A[0,1] ⊂ A[0,2] ⊂ · · · ⊂ A[0,∞) ∪ ∪ ∪ A[1,1] ⊂ A[1,2] ⊂ · · · ⊂ A[1,∞) ∪ ∪ A[2,2] ⊂ · · · ⊂ A[2,∞) ∪ ... In particular, A• is a local Markov filtration. Proof. All claimed inclusions in the triangular array are clear from the definition of A[m,n]. We recall from Proposition 3.7 that αk0(A0) ⊂ Mαn+1 for k ≤ n. Hence A[m,n] ⊂ Mαn+1 for all m ≤ n. Next we show that, for −∞ < m < n <∞, the cell of inclusions A[m,n] ⊂ A[m,n+1] ∪ ∪ A[m+1,n] ⊂ A[m+1,n+1] Markovianity and the Thompson Group F 15 forms a commuting square. So, as PI denotes the normal ψ-preserving conditional expecta- tion from M onto AI , we need to show P[m,n]P[m+1,n+1] = P[m+1,n]. As αm0 PIα −m 0 = PI+m for all m ∈ Z, it suffices to show that, for all n ∈ N, P[0,n]P[1,n+1] = P[1,n] or, equivalently, P[0,n]α0P[0,n] = α0P[0,n−1]. We calculate P[0,n]α0P[0,n] = P[0,n]Qn+1α0P[0,n] = P[0,n]α0QnP[0,n] = P[0,n]α0QnP[0,∞)P[0,n] = P[0,n]α0P[0,n−1]P[0,n] = P[0,n]α0P[0,n−1] = α0P[0,n−1]. Here we have used P[0,n] = P[0,n]Qn+1, the intertwining properties of α0 and the commuting square assumption QnP[0,∞) = P[0,n−1]. Thus each cell of inclusions in this triangular array forms a commuting square. ■ More generally, we may consider a probability space which is equipped both with a local filtration and a representation of the Thompson group F , and formulate compatiblity conditions between the local filtration and the representation such that one obtains rich commuting square structures. Corollary 3.14. Suppose the probability space (M, ψ) is equipped with a local filtration N• ≡ {NI}I∈I(Z) and a representation ρ : F → Aut(M, ψ) such that (i) ρ(g0)(NI) = NI+1 for all I ∈ I(Z) (compatibility), (ii) N[0,n] ⊂ Mρ(gn+1) for all n ∈ N0 (adaptedness), (iii) the inclusions Mρ(gk+1) ⊂ M ∪ ∪ N[0,k] ⊂ N[0,∞) form a commuting square for all k ∈ N0. Then each cell in the following infinite triangular array of inclusions is a commuting square: · · · ⊂ N(−∞,−2] ⊂ N(−∞,−1] ⊂ N(−∞,0] ⊂ N(−∞,1] ⊂ N(−∞,2] ⊂ · · · ⊂ N(−∞,∞) ∪ ∪ ∪ ∪ ∪ · · · ∪ ... ... ... ... ... · · · ... ∪ ∪ ∪ ∪ ∪ · · · ∪ N[−2,−2] ⊂ N[−2,−1] ⊂ N[−2,0] ⊂ N[−2,1] ⊂ N[−2,2] ⊂ · · · ⊂ N[−2,∞) ∪ ∪ ∪ ∪ ∪ N[−1,−1] ⊂ N[−1,0] ⊂ N[−1,1] ⊂ N[−1,2] ⊂ · · · ⊂ N[−1,∞) ∪ ∪ ∪ ∪ N[0,0] ⊂ N[0,1] ⊂ N[0,2] ⊂ · · · ⊂ N[0,∞) ∪ ∪ ∪ N[1,1] ⊂ N[1,2] ⊂ · · · ⊂ N[1,∞) ∪ ∪ N[2,2] ⊂ · · · ⊂ N[2,∞) ∪ ... In particular, N• is a local Markov filtration. 16 C. Köstler and A. Krishnan Proof. Let PI be the normal ψ-preserving conditional expectation onto NI . Let αn = ρ(gn) and Qn be the normal ψ-preserving conditional expectation onto Mαn as before. We observe that N = N[0,0] ⊂ Mα1 by the adaptedness condition (ii). This adaptedness property also gives us N[0,n] ⊂ Mαn+1 , and thus P[0,n] = P[0,n]Qn+1, for any n ∈ N0. The rest of the proof follows the arguments used in the proof of Theorem 3.13. ■ 4 Constructions of representations of F from stationary Markov processes This section is about how to construct representations of the Thompson group F as they arise in noncommutative probability theory. It will be seen that a large class of bilateral stationary Markov processes in tensor dilation form (see Definition 2.17) will give rise to representations of F . In particular, this will establish that a Markov map on a probability space (A, φ) with A a commutative von Neumann algebra can be written as a compressed represented generator of F . 4.1 An illustrative example Let (A, φ) and (C, χ) be noncommutative probability spaces. We have already shown in [17] how to obtain a representation of the Thompson monoid F+ and a unilateral stationary Markov process on ( A ⊗ C⊗N0 , φ ⊗ χ⊗N0 ) . In general, especially for C finite-dimensional, this tensor product model for a noncommutative probability space is “too small” to accommodate a rep- resentation of the Thompson group F . Also, even though the extension ( A ⊗ C⊗Z , φ ⊗ χ⊗Z ) suffices to set up a bilateral extension of a unilateral stationary Markov process (see for example [18, Section 4.2.2]), it would still be “too small” for canonically extending a represention of the monoid F+ to one of the group F . This motivates the following model build on two given noncommutative probability spaces (A, φ) and (C, χ). Throughout this final section, consider the infinite von Neumann algebraic tensor product with respect to an infinite tensor product state given by (M, ψ) := ( A⊗ C⊗N20 , φ⊗ χ ⊗N20 ) . This probability space can be equipped with a representation of the Thompson group F . Also it can be used to set up a bilateral noncommutative Bernoulli shift and, more generally, a bilateral stationary noncommutative Markov process. We start with providing a representation of the Thompson group F . For k ∈ N0, let βk be the automorphisms of M defined on the weak-total set of finite elementary tensors in M as β0 ( a⊗ ( ⊗ (i,j)∈N2 0 xi,j )) := a⊗ ( ⊗ (i,j)∈N2 0 yi,j ) with yi,j =  x2i+1,j if j = 0, x2i,j−1 if j = 1, xi,j−1 if j ≥ 2 and βk ( a⊗ ( ⊗ (i,j)∈N2 0 xi,j )) := a⊗ ( ⊗ (i,j)∈N2 0 yi,j ) with yi,j =  xi,j if j ≤ k − 1, x2i+1,j if j = k, x2i,j−1 if j = k + 1, xi,j−1 if j ≥ k + 1 for k ∈ N. It is evident from these two definitions that the actions of β0 and β1 are induced from corresponding shifts on the index set N2 0, as visualized graphically in Figure 1. Markovianity and the Thompson Group F 17 β0 =̂ ... ... ... ... • • • • · · · • • • • · · · • • • • · · · • • • • · · · • • • • · · · • • • • · · · ↑ i • • • • · · · ■ • • • • · · · j−→ β1 =̂ ... ... ... ... ... • • • • • · · · • • • • • · · · • • • • • · · · • • • • • · · · • • • • • · · · • • • • • · · · ↑ i • • • • • · · · ■ • • • • • · · · j−→ Figure 1. Visualization of the action of the automorphisms β0 (left) and β1 (right). Here ■ denotes an element of A and • denotes an element of C, and the blue arrows indicate how the automorphisms act as shifts when considered on the index set N2 0. We note that the fixed point algebras Mβ0 and Mβ1 of β0 and β1 are given by, respectively, Mβ0 = A⊗ 1 ⊗N0 C ⊗ 1 ⊗N0 C ⊗ 1 ⊗N0 C ⊗ · · · , (4.1) Mβ1 = A⊗ C⊗N0 ⊗ 1 ⊗N0 C ⊗ 1 ⊗N0 C ⊗ · · · . (4.2) Let B0 := β−1 0 ( A⊗ 1 ⊗N0 C ⊗ C⊗N0 ⊗ 1 ⊗N0 C ⊗ · · · ) which can be thought of as the “present” von Neumann subalgebra at time n = 0 of the explicit form ... ... ... ⊗ ⊗ ⊗ 1C 1C 1C ⊗ ⊗ ⊗ C 1C 1C ⊗ ⊗ ⊗ 1C 1C 1C ⊗ ⊗ ⊗ A ⊗ C ⊗ 1C ⊗ 1C ⊗ · · · . Proposition 4.1. The maps gn 7→ ρB(gn) := βn, with n ∈ N0, extend multiplicatively to a representation ρB : F → Aut(M, ψ) which has the generating property. Further, (M, ψ, β0,B0) is a bilateral noncommutative Bernoulli shift with generator B0. Proof. For 0 ≤ k < ℓ < ∞, the relations βkβℓ = βℓ+1βk are verified in a straightforward computation on finite elementary tensors. Since ψ ◦ βn = ψ, the maps gn 7→ ρB(gn) := βn extend to a representation of F in Aut(M, ψ). The generating property of this representation will follow from the minimality of the stationary process by Proposition 3.8. Indeed, let BI :=∨ i∈I β i 0(B0) for I ∈ I(Z) and note that B[0,0] = B0. Clearly BZ = M, hence the stationary process (M, ψ, β0,B0) is minimal. We are left to show that this minimal stationary process 18 C. Köstler and A. Krishnan is a bilateral noncommutative Bernoulli shift. Clearly, Mβ0 ⊂ B0. We are left to verify the factorization Q0(xy) = Q0(x)Q0(y) for any x ∈ BI , y ∈ BJ whenever I ∩ J = ∅. Here Q0 is the ψ-preserving normal conditional expectation from M onto Mβ0 which is of the tensor type Q0 ( a⊗ ( ⊗ (i,j)∈N2 0 xi,j )) = a⊗ ( ⊗ (i,j)∈N2 0 χ(xi,j)1C ) for finite elementary tensors in M. Now the required factorization easily follows by observing that distinct powers of the “time evolution” β0 send elements of B0 to elements which are supported by disjoint index sets in N2 0. ■ To obtain more general representations of the Thompson group F , we can further “perturb” the automorphisms βn. Here we focus on a very particular case of such perturbations, as it will turn out to be useful when constructing representations of F from bilateral stationary noncommutative Markov processes. Given an automorphism γ ∈ Aut(A ⊗ C, φ ⊗ χ), let γ0 ∈ Aut(M, ψ) denote its natural extension such that γ0 ( a⊗ ( ⊗ (i,j)∈N2 0 xi,j )) = γ(a⊗ x00)⊗ ( ⊗ (i,j)∈N2 0\{(0,0)} xi,j ) . Furthermore, let α0 := γ0 ◦ β0, αn := βn for all n ≥ 1. Proposition 4.2. The maps gn 7→ ρM (gn) := αn, with n ∈ N0, extend multiplicatively to a representation ρM : F → Aut(M, ψ) which has the generating property. Further, the quadruple (M, ψ, α0,Mα1) is a bilateral noncommutative stationary Markov process. Proof. For 1 ≤ k < ℓ, the relations αkαℓ = αℓ+1αk are those of the βn-s from Proposition 4.1. The relations α0αℓ = αℓ+1α0 for l > 0 are verified on finite elementary tensors by a straight- forward computation. Similar arguments as used in the proof of Proposition 4.1 ensure that the maps gn 7→ ρM (gn) := αn extend multiplicatively to a representation ρM : F → Aut(M, ψ). Its generating property is again immediate from the minimality of the stationary process by Proposition 3.8. Finally, the Markovianity of the bilateral stationary process (M, ψ, α0,Mα1) follows from Corollary 3.12. ■ Given the stationary Markov process (M, ψ, α0,Mα1) (from Proposition 4.2), a restriction of the generating algebra Mα1 to a von Neumann subalgebra A0 provides a candidate for another stationary Markov process. Viewing the Markov shift α0 as a “perturbation” of the Bernoulli shift β0, the subalgebra A0 = Mβ0 is an interesting choice. Proposition 4.3. The quadruple ( M, ψ, α0,Mβ0 ) is a bilateral noncommutative stationary Markov process. Proof. We recall from (4.1) that Mβ0 = A⊗ 1 ⊗N0 C ⊗ 1 ⊗N0 C ⊗ 1 ⊗N0 C ⊗ · · · . Markovianity and the Thompson Group F 19 Let PI denote the ψ-preserving normal conditional expectation from M onto AI :=∨ i∈I α i 0 ( Mβ0 ) for an interval I ⊂ Z. By Lemma 2.13, it suffices to verify the Markov property P(−∞,0]P[0,∞) = P[0,0]. For this purpose we use the von Neumann subalgebra D0 := ... ... ... ⊗ ⊗ ⊗ 1C 1C 1C · · · ⊗ ⊗ ⊗ A ⊗ C ⊗ 1C ⊗ 1C ⊗ · · · and the tensor shift β0 to generate the “past algebra” D< := ∨ i<0 β i 0(D0) and the “future algebra” D≥ := ∨ i≥0 β i 0(D0). One has the inclusions A(−∞,0] ⊂ D<, A[0,∞) ⊂ D≥, D< ∩ D≥ = Mβ0 . Here we used for the first inclusion that α0 = γ0 ◦ β0 and thus α−1 0 = β−1 0 ◦ γ−1 0 . The second inclusion is immediate from the definitions of the von Neumann algebras. Finally, the claimed intersection property is readily deduced from the underlying tensor product structure. Let ED< and ED≥ denote the ψ-preserving normal conditional expectations from M onto D< and D≥, respectively. We observe that ED<ED≥ = P[0,0] is immediately deduced from the tensor product structure of the probability space (M, ψ). But this allows us to compute P(−∞,0]P[0,∞) = P(−∞,0]ED<ED≥P[0,∞) = P(−∞,0]P[0,0]P[0,∞) = P[0,0]. ■ Remark 4.4. The above constructed bilateral noncommutative stationary Markov process( M, ψ, α0,Mβ0 ) is not minimal, as the von Neumann algebra AZ generated by αn0 ( Mβ0 ) for all n ∈ Z is clearly contained in the subalgebra ... ... ... ⊗ ⊗ ⊗ C 1C 1C ⊗ ⊗ ⊗ C 1C 1C ⊗ ⊗ ⊗ C 1C 1C ⊗ ⊗ ⊗ A ⊗ C ⊗ C ⊗ C ⊗ · · · . The subalgebra AZ is invariant under the action of α0 = ρM (g0) and its inverse, but it fails to be invariant under the action of the inverse of α1 = ρM (g1). This illustrates that the von Neumann algebra of a bilateral stationary Markov process may be “too small” to carry a representation of the Thompson group F such that its Markov shift represents the generator g0 ∈ F . 4.2 Constructions of representations of F from stationary Markov processes The following theorem uses the tensor product construction of the present section to show that automorphisms on tensor products give representations of F such that the compressed automorphism is equal to a compressed represented generator. Throughout this section we will use the following notion of an embedding for two noncom- mutative probability spaces (A, φ) and (M, ψ). An embedding ι : (A, φ) → (M, ψ) is a (φ,ψ)- Markov map ι : A → M which is also a ∗-homomorphism. Furthermore, recall the notion of a dilation of first order from Definition 2.14. 20 C. Köstler and A. Krishnan Theorem 4.5. Suppose γ ∈ Aut(A⊗ C, φ⊗ χ) and let ι0 be the canonical embedding of (A, φ) into (A ⊗ C, φ ⊗ χ). Then there exists a noncommutative probability space (M, ψ), generating representations ρB, ρM : F → Aut(M, ψ) and an embedding κ : (A⊗C, φ⊗χ) → (M, ψ) such that (i) κι0(A) = MρB(g0), (ii) ι∗0γ nι0 = ι∗0κ ∗ρM (gn0 )κι0 for all n ∈ N0. In particular, ( M, ψ, ρM (g0),MρB(g0) ) is a bilateral noncommutative stationary Markov process. Proof. We take (M, ψ) := ( A⊗ C⊗N20 , φ⊗ χ ⊗N20 ) and let κ be the natural embedding of (A⊗C, φ⊗χ) into (M, ψ). We construct two representa- tions of the Thompson group F as done for the illustrative example in Section 4.1. That is, we define the representation ρB : F → Aut(M, ψ) as ρB(gn) := βn for n ≥ 0 (see Proposition 4.1) and the representation ρM : F → Aut(M, ψ) as ρM (gn) := αn with α0 = γ0 ◦β0 and αn = βn for n ≥ 1 (see Proposition 4.2). The generating property of these two representations ρB and ρM has already been verified in Propositions 4.1 and 4.2. We recall from Section 4.1 that γ0 is the natural extension of γ to an automorphism on (M, ψ) which is easily seen to satisfy κ∗γn0 κι0 = γnι0 for all n ∈ N0. (4.3) Note that for the case n = 1, the left hand side of this equation can be written as κ∗γ0κι0 = κ∗γ0β0κι0 = κ∗α0κι0. (4.4) Now Proposition 4.3 ensures that ( M, ψ, α0,Mβ0 ) is a bilateral noncommutative stationary Markov process with κι0(A) = Mβ0 , as claimed in (i) of the theorem. We note that κι0(κι0) ∗ is the ψ-preserving normal conditional expectation from M onto Mβ0 = κι0(A), and by definition, the stationary Markov process ( M, ψ, α0,Mβ0 ) has the transition operator T := κι0(κι0) ∗α0κι0(κι0) ∗. We observe that (4.3) and (4.4) allow us to rewrite T as follows: T = κι0(κι0) ∗α0κι0(κι0) ∗ = κι0ι ∗ 0(κ ∗α0κι0)(κι0) ∗ = κι0ι ∗ 0(κ ∗γ0κι0)ι ∗ 0κ ∗ = κι0ι ∗ 0γι0ι ∗ 0κ ∗. (4.5) On the other hand, Proposition 2.18 gives that T satisfies Tn = κι0(κι0) ∗αn0κι0(κι0) ∗ for all n ∈ N0. (4.6) Hence by (4.5) and (4.6), (κι0ι ∗ 0)γ n(κι0ι ∗ 0) ∗ = [(κι0ι ∗ 0)γ(κι0ι ∗ 0) ∗]n = Tn = κι0(κι0) ∗αn0κι0(κι0) ∗. Simplifying, we get ι∗0γ nι0 = ι∗0κ ∗αn0κι0 for all n ∈ N0, as claimed in (ii) of the theorem. ■ This result builds on an observation related to the existence of Markov dilations already made by Kümmerer in [18, Theorem 4.2.1]: if a φ-Markov map R on A has a tensor dilation of first order (A ⊗ C, φ ⊗ χ, γ, ι0), then this implies the existence of a (Markov) dilation on the noncommutative probability space ( A⊗ C⊗Z , φ⊗ χ⊗Z ) . Here we have utilized this fact and amplified further the dilation to the noncommutative probability space (M, ψ) = ( A⊗C⊗N20 , φ⊗ χ ⊗N20 ) , such that a representation of the Thompson group F can be accommodated. Markovianity and the Thompson Group F 21 4.3 The classical case We state a result of Kümmerer that provides a tensor dilation of any Markov map on a commuta- tive von Neumann algebra. This will allow us to obtain a representation of F as in Theorem 4.5. Notation 4.6. The (non)commutative probability space (L, trλ) is given by the Lebesgue space of essentially bounded functions L := L∞([0, 1], λ) and trλ := ∫ [0,1] ·dλ as the faithful normal state on L. Here λ denotes the Lebesgue measure on the unit interval [0, 1] ⊂ R. Theorem 4.7 ([19, 4.4.2]). Let R be a φ-Markov map on A, where A is a commutative von Neumann algebra with separable predual. Then there exists γ ∈ Aut(A ⊗ L, φ ⊗ trλ) such that (A⊗L, φ⊗ trλ, γ, ι0) is a Markov (tensor) dilation of R. That is, (A⊗L, φ⊗ trλ, γ,A⊗ 1L) is a stationary Markov process, and for all n ∈ N0, Rn = ι∗0 γ nι0, where ι0 : (A, φ) → (A ⊗ L, φ ⊗ trλ) denotes the canonical embedding ι0(a) = a ⊗ 1L such that E0 := ι0 ◦ ι∗0 is the φ⊗ trλ-preserving normal conditional expectation from A⊗L onto A⊗ 1L. A proof of this result on bilateral commutative stationary Markov processes is contained in [19]. For the convenience of the reader, this proof is made available in [17], with minor modifications to the unilateral setting of such processes. This folkore result ensures that, in particular, every transition operator of a commutative stationary Markov process has a dilation of first order, which was the starting assumption of Theorem 4.5. Consequently, we can associate to each classical bilateral stationary Markov process a representation of the Thompson group F . Theorem 4.8. Let (A, φ) be a noncommutative probability space where A is commutative with separable predual, and let R be a φ-Markov map on A. There exists a probability space (M, ψ), generating representations ρB, ρM : F → Aut(M, ψ), and an embedding ι : (A, φ) → (M, ψ) such that (i) ι(A) = MρB(g0), (ii) Rn = ι∗ρM (gn0 )ι for all n ∈ N0. Proof. By Theorem 4.7, there exists γ ∈ Aut(A⊗L, φ⊗trλ) such that (A⊗L, φ⊗trλ, γ,A⊗1L) is a stationary Markov process, and Rn = ι∗0 γ nι0, for all n ∈ N0, where ι0 : (A, φ) → (A⊗L, φ⊗ trλ) denotes the canonical embedding ι0(a) = a⊗ 1L. By Theorem 4.5, there exists a probability space (M, ψ), generating representations ρB, ρM : F → Aut(M, ψ), and an embedding κ : (A⊗L, φ⊗χ) → (M, ψ) such that κ(A⊗1L) = MρB(g0) and ι∗0γ nι0 = ι∗0κ ∗ρM (gn0 )κι0 for all n ∈ N0. The proof is completed by taking ι := κ ◦ ι0, as we get Rn = ι∗0γ nι0 = ι∗0κ ∗ρM (gn0 )κι0 = ι∗ρM (gn0 )ι for all n ∈ N0. ■ 4.4 Further discussion of the classical case We illustrate Theorem 4.8 for a classical stationary Markov process taking values in the finite set [d] := {1, 2, . . . , d} for some d ≥ 2, adapting the classical construction of such processes to our algebraic approach. Consider the unital *-algebra A := Cd ∼= {f : [d] → C}. Then φ(f) := ∑d i=1 qif(i) defines a faithful (normal tracial) state φ on A if and only if ∑d i=1 qi = 1 and 0 < qi < 1 for all 1 ≤ i ≤ d. Now consider the transition operator R : A → A given by the matrix R =  p1,1 p1,2 · · · p1,d p2,1 p2,2 · · · p2,d ... ... . . . ... pd,1 pd,2 · · · pd,d  22 C. Köstler and A. Krishnan for some pi,j ∈ [0, 1] satisfying ∑d j=1 pi,j = 1 for all i = 1, . . . , d. One easily verifies that φ ◦R = φ ⇐⇒ d∑ i=1 qipi,j = qj for all 1 ≤ j ≤ d (stationarity). The usual Daniell–Kolmogorov construction of a stationary Markov process can now be alge- braically reformulated as follows. Here we closely follow the exposition provided in [19]. A state φ̃ is defined on the infinite algebraic tensor product ⊙ ZA by φ̃(· · · ⊗ 1A ⊗ f−m ⊗ f−m+1 ⊗ · · · ⊗ fn−1 ⊗ fn ⊗ 1A ⊗ · · · ) := φ ( f−mR(f−m+1R(· · · fn−1R(fn) · · · )) ) . This state φ̃ extends to a faithful normal state φ̂ on the von Neumann algebraic tensor product Â := ⊗ ZA such that ( Â, φ̂ ) is a noncommutative probability space (in the sense of Section 2.2). Furthermore, the tensor right shift on ⊙ ZA extends to an automorphism Â of ( Â, φ̂ ) . Finally, let ι̂DK : A → Â denote the injection which canonically embeds f ∈ A into the 0-th position of the infinite tensor product Â = ⊗ ZA. Then it can be verified that ( Â, φ̂, T̂ , ι̂DK(A) ) is a minimal stationary Markov process (in the sense of Definition 2.12). However, the Daniell–Kolmogorov construction does not seem to accommodate a represen- tation ρ̂ : F → Aut ( Â, φ̂ ) with ρ̂(g0) = T̂ which satisfies the additional localization prop- erty ι̂DK(A) ⊂ Âρ̂(gn) for n ≥ 1. This observation is connected to the well-known fact that the Daniell–Kolmogorov construction puts all information about a stochastic process into the state φ̂, while the automorphism T̂ is simply implemented by a bilateral tensor shift. Fortunately, Kümmerer’s approach to the construction of stationary Markov processes is more feasible for finding representations of the Thompson group F with properties as ad- dressed above. This open dynamical system approach is alternative to the Daniell–Kolmogorov construction in classical probability; and it is actually independent of it for finite-set-valued processes. As explained in [19], this alternative approach provides a construction which puts some information of the stationary Markov process into the automorphism while simplifying the state (see Theorem 4.7). More specifically, this strategy divides the construction into two steps. One first tries to construct a dilation of first order, and then one attempts in a sec- ond step to extend this first-order dilation to a full (Markov) dilation (see Section 2.4). In fact, as already observed in Section 4.2, this two-step strategy can be further extended to construct a representation of the Thompson group F which encodes the Markovianity of the given stationary process. Let us further discuss this alternative construction for a tensor di- lation for the present example ( A = Cd, φ ) with transition operator R on A. For this pur- pose, recall Notation 4.6. Similar as done for the case d = 2 in [17, Example 3.4.3] and as detailed in [19], one can construct an automorphism γ ∈ Aut(A ⊗ L, φ ⊗ trλ) such that the φ-Markov map R on A has the dilation of first order (A ⊗ L, φ ⊗ trλ, γ, ι0). As before, ι0 denotes the canonical embedding of (A, φ) into (A ⊗ L, φ ⊗ trλ). In other words, the dia- gram (A, φ) (A, φ) (A⊗ L, φ⊗ trλ) (A⊗ L, φ⊗ trλ) R ι0 ι∗0 γ (4.7) commutes. Remark 4.9. All information about the φ-Markov map R on A is contained in the φ ⊗ trλ- preserving automorphism γ on A⊗ L. Generally, AZ := ∨ n∈Z γ n(A⊗ 1L) is strictly contained Markovianity and the Thompson Group F 23 in A ⊗ L. In other words, Theorem 4.7 provides a non-minimal stationary Markov process, in general. Actually, our first step in the construction of a representation of the Thompson group F consists in finding a suitable dilation of first order (4.7). Kümmerer’s Theorem 4.7 guarantees the existence of such dilations. However, we refrain from further discussing the structure of these dilations of first order, as this would go beyond the scope of the present paper. Having arrived at this dilation of first order, several straightforward constructions of station- ary Markov processes are possible. Here we discuss those which are of relevance for obtaining unilateral and bilateral versions of stationary Markov processes, in particular with the view of obtaining suitable representations of the Thompson group F , and its monoid F+, as introduced in (2.2). A unilateral noncommutative stationary Markov process ( M̃, ψ̃, α̃0, ι̃(A) ) is obtained by putting ( M̃, ψ̃ ) := ( A⊗ L⊗N0 , φ⊗ tr ⊗N0 λ ) with α̃0 := γ̃0β̃0, where β̃0(f ⊗ x0 ⊗ x1 ⊗ · · · ) := f ⊗ 1L ⊗ x0 ⊗ x1 ⊗ · · · , γ̃0(f ⊗ x0 ⊗ x1 ⊗ · · · ) := γ(f ⊗ x0)⊗ x1 ⊗ · · · , ι̃(f) := f ⊗ 1L ⊗ 1L ⊗ · · · for f ∈ A, x0, x1, . . . ∈ L. This construction was the subject of [17], as it allows to introduce the representations ρ̃B and ρ̃M of the Thompson monoid F+ by putting ρ̃B(gk) := β̃k for k ≥ 0, (4.8) ρ̃M (gk) := { α̃0 for k = 0, β̃k for k > 0, (4.9) with β̃k(f ⊗ x0 ⊗ · · · ⊗ xk−1 ⊗ xk ⊗ xk+1 ⊗ · · · ) := f ⊗ x0 ⊗ · · · ⊗ xk−1 ⊗ 1L ⊗ xk ⊗ · · · . It is now elementary to verify the relations β̃kβ̃ℓ = β̃ℓ+1β̃k, 0 ≤ k ≤ ℓ <∞, α̃kα̃ℓ = α̃ℓ+1α̃k, 0 ≤ k < ℓ <∞. (4.10) The choices made in (4.8) are canonical for the partial shifts β̃k (see also [8, 17]). The choice made in (4.9) is also canonical from the dynamical systems viewpoint of constructing a stationary Markov process as a local perturbation of a Bernoulli shift. But of course, other choices are possible for ρ̃M (gk) for k ≥ 1, respecting the localization property ι̃(A) ⊂ M̃ρM (gk), without violating the relations of the Thompson monoid F+ (see also [17, Section 5.3]). This construction is nicely illustrated in Figure 2 with actions of injective maps on the set {■} ⊔ N0. Here the set {■} pictures the algebra A (or an element of it), • pictures a copy of the algebra L (or an element of it), and disjoint unions of sets correspond to tensor products in the algebraic formulation. Now the action of the partial shifts β̃0 and β̃1 become injective maps on the set {■} ⊔ N0 which can be visualized by blue arrows. Furthermore, the action of the local automorphism γ̃0 is visualized by a bijection on {■} ⊔ N0 which moves only those elements inside the red ellipse, as indicated in red colour in Figure 2. A similar visualization is immediate for the actions of β̃k for k > 1. We finally note for Figure 2 that ◦ visualizes the one-dimensional subalgebra C1L ⊂ L (or its element 1L) which is actually given by the empty set ∅ on the level of sets. Here we could have omitted these isomorphic embeddings for our visualization, but these embeddings will guide our consecutive amplifications, in particular as relevant for canonically constructing representations of F . As it can be clearly seen in Figure 2, the set {■} ⊔ N0 is invariant for the injections which visualize the actions of β̃k − s and γ̃0. 24 C. Köstler and A. Krishnan ... ... ... ... ◦ ◦ ◦ ◦ · · · ↑ i ◦ ◦ ◦ ◦ · · · ◦ ◦ ◦ ◦ · · · ■ ⟲ • • • • · · · γ̃0 β̃0 j−→ ... ... ... ... ◦ ◦ ◦ ◦ · · · ↑ i ◦ ◦ ◦ ◦ · · · ◦ ◦ ◦ ◦ · · · ■ • • • • · · · β̃1 j−→ Figure 2. Visualization on the set {■}⊔N0 of the action of the one-sided Bernoulli shift β̃0 (blue, left), and the local automorphism γ̃0 (red, left) and the action of the one-sided Bernoulli shift β̃1 (blue, right). Next, we extend the unilateral stationary Markov process ( M̃, ψ̃, α̃0, ι̃(A) ) to the bilateral stationary Markov process ( M̂, ψ̂, α̂0, ι̂(A) ) by putting (M̂, ψ̂) := ( A ⊗ L⊗Z , φ ⊗ tr ⊗Z λ ) with α̂0 := γ̂0β̂0, where β̂0 · · · ⊗ x−1 ⊗ x0⊗ f ⊗ x1 ⊗ · · ·  := · · · ⊗ x−2 ⊗ x−1 ⊗ f ⊗ x0 ⊗ · · · , γ̂0 · · · ⊗ x−1 ⊗ x0⊗ f ⊗ x1 ⊗ · · ·  := · · · ⊗ x−1 ⊗ γ0 x0⊗ f ⊗ x1 ⊗ · · · , ι̂(f) := · · · ⊗ 1L ⊗ 1L⊗ f ⊗ 1L ⊗ · · · for f ∈ A, . . . , x−1, x0, x1, . . . ∈ L. Considering the automorphism α̂0 as a canonical bilateral extension of the endomorphism α̃0, we are interested in identifying bilateral extensions of the other endomorphisms α̃1, α̃2, . . . to automorphisms of ( M̂, ψ̂ ) , now satisfying the relations of the Thompson group F . But this seems to be impossible, as ( M̂, ψ̂ ) provides “too little space” for accommodating such automorphisms. This is illustrated in Figure 3 again on the level of the set {■} ⊔ Z, when visualized as an appropriate subset of {■} ⊔ N2 0. Note that we have made a particular choice of how to embed {■} ⊔ Z into {■} ⊔ N2 0, and there are many other interesting possibilities for choosing such an embedding. This challenge to provide sufficient space for properly extending all partial shifts { β̃k \| k ≥ 0 } ⊂ ( M̃, φ̃ ) is overcome by choosing (M, ψ) = ( A⊗ L⊗N20 , φ⊗ tr ⊗N20 λ ) with the canonical embedding ι : (A, φ) → (M, ψ) given by ι(a) := a ⊗ (⊗ (i,j)∈N2 0 1L ) . This approach has already been detailed in the illustrative example of Section 4.1. For the convenience of the reader, let us repeat how the partial shifts β̃k and the local automorphism γ̃0 on M̃ are extended to automorphisms on M: β0 ( a⊗ ( ⊗ (i,j)∈N2 0 xi,j )) := a⊗ ( ⊗ (i,j)∈N2 0 yi,j ) with yi,j =  x2i+1,j if j = 0, x2i,j−1 if j = 1, xi,j−1 if j ≥ 2, Markovianity and the Thompson Group F 25 ... ... ... ... • ◦ ◦ ◦ · · · ◦ ◦ ◦ ◦ · · · ◦ ◦ ◦ ◦ · · · ↑ i ◦ ◦ ◦ ◦ · · · • ◦ ◦ ◦ · · · ◦ ◦ ◦ ◦ · · · • ◦ ◦ ◦ · · · ■ ⟲ • • • • · · · γ̂0 β̂0 j−→ ... ... ... ... • ◦ ◦ ◦ · · · ◦ ◦ ◦ ◦ · · · ◦ ◦ ◦ ◦ · · · ↑ i ◦ ◦ ◦ ◦ · · · • ◦ ◦ ◦ · · · ◦ ◦ ◦ ◦ · · · • ◦ ◦ ◦ · · · ■ • • • • · · · β̃1 j−→ Figure 3. Visualization on the set {■}⊔Z of the action of the two-sided Bernoulli shift β̂0 and the local automorphism γ̂0 and of the “inability” to extend β̃1 from {■} ⊔N0 to an automorphism β̂1 on {■} ⊔ Z such that the relations of F are satisfied. and, for k ∈ N, βk ( a⊗ ( ⊗ (i,j)∈N2 0 xi,j )) := a⊗ ( ⊗ (i,j)∈N2 0 yi,j ) with yi,j =  xi,j if j ≤ k − 1, x2i+1,j if j = k, x2i,j−1 if j = k + 1, xi,j−1 if j ≥ k + 1. Furthermore, the local perturbation γ ∈ Aut(A,L) is amplified to γ0 ( a⊗ ( ⊗ (i,j)∈N2 0 xi,j )) = γ(a⊗ x00)⊗ ( ⊗ (i,j)∈N2 0\{(0,0)} xi,j ) . We refer the reader to Figure 4 for a visualization of the action of the two-sided shifts β0, β1 and the action of the local automorphism γ0. We address {βk \| k ≥ 0} as a canonical extension of the family { β̃k \| k ≥ 0 } . Of course, there are many other interesting possibilities to arrive at suitable extensions. Now the multiplicative extension of the automorphisms ρB(gk) := βk for k ≥ 0, ρM (gk) := { α0 := γ0β0 for k = 0, αk := βk for k > 0, provides us with two representations ρB, ρM : F → Aut(M, ψ), as it is elementary to verify the relations βkβℓ = βℓ+1βk, 0 ≤ k < ℓ <∞, αkαℓ = αℓ+1αk, 0 ≤ k < ℓ <∞. (4.11) 26 C. Köstler and A. Krishnan ... ... ... ... • • • • · · · ↑ i • • • • · · · • • • • · · · • • • • · · · • • • • · · · • • • • · · · • • • • · · · ■ ⟲ • • • • · · · γ0 β0 j−→ ... ... ... ... ... • • • • • · · · • • • • • · · · • • • • • · · · • • • • • · · · • • • • • · · · • • • • • · · · ↑ i • • • • • · · · ■ • • • • • · · · β1 j−→ Figure 4. Visualization on the set {■} ⊔ N2 0 of the action of the two-sided Bernoulli shift β0, the local automorphism γ0, and the two-sided Bernoulli shift β1. Note that (4.11) fails to be valid for k = ℓ, in contrast to the relations for the partial shifts β̃k in (4.10). We have already verified in Proposition 4.3 that (M, ψ, α0, ι(A)) is a bilateral non- commutative Markov process. The above discussion has provided additional background information on the ideas underlying Theorem 4.8, and on its proof strategy. Acknowledgements The second author was partially supported by a Government of Ireland Postdoctoral Fellowship (Project ID: GOIPD/2018/498). Both authors acknowledge several helpful discussions with B.V. Rajarama Bhat in an early stage of this project. Also the first author would like to thank Persi Diaconis, Gwion Evans, Rolf Gohm, Burkhard Kümmerer and Hans Maassen for several fruitful discussions on Markovianity. Both authors thank the organizers of the conference Non- commutative algebra, Probability and Analysis in Action held at Greifswald in September 2021 in honour of Michael Schürmann. The authors also thank the anonymous referees for their comments. 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Surveys 72 (2017), 257–333, arXiv:1705.06619. https://doi.org/10.1216/RMJ-2017-47-6-1839 http://arxiv.org/abs/1508.03168 https://doi.org/10.1007/s00220-008-0716-x http://arxiv.org/abs/0806.3691 https://doi.org/10.1007/978-1-4613-9641-3 https://doi.org/10.1007/s00220-011-1216-y http://arxiv.org/abs/1009.0778 https://doi.org/10.4171/JCA/1-1-1 http://arxiv.org/abs/1412.7740 https://doi.org/10.1007/s00220-017-2945-3 https://doi.org/10.1007/s00220-017-2945-3 http://arxiv.org/abs/1607.08769 https://doi.org/10.1088/1751-8121/aaa4dd http://arxiv.org/abs/1706.00515 https://doi.org/10.1017/CBO9780511566219 https://doi.org/10.1016/j.jfa.2009.10.021 http://arxiv.org/abs/0806.3621 http://arxiv.org/abs/2009.14811 https://doi.org/10.1016/0022-1236(85)90084-9 https://doi.org/10.1007/BF01205506 https://doi.org/10.2140/pjm.1989.137.181 https://doi.org/10.1016/0022-1236(72)90004-3 https://doi.org/10.1007/978-1-4612-6188-9 https://doi.org/10.1007/978-3-662-10451-4 https://doi.org/10.1007/BF00403239 https://doi.org/10.1070/RM9763 https://doi.org/10.1070/RM9763 http://arxiv.org/abs/1705.06619 1 Introduction 2 Preliminaries 2.1 The Thompson group F 2.2 Noncommutative probability spaces and Markov maps 2.3 Noncommutative independence and Markovianity 2.4 Noncommutative stationary processes and dilations 3 Markovianity from representations of F 3.1 Representations with a generating property 3.2 Commuting squares and Markovianity for stationary processes 4 Constructions of representations of F from stationary Markov processes 4.1 An illustrative example 4.2 Constructions of representations of F from stationary Markov processes 4.3 The classical case 4.4 Further discussion of the classical case References
id	nasplib_isofts_kiev_ua-123456789-211821
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1815-0659
language	English
last_indexed	2026-03-14T15:16:03Z
publishDate	2022
publisher	Інститут математики НАН України
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spelling	Köstler, Claus Krishnan, Arundhathi 2026-01-12T10:19:44Z 2022 Markovianity and the Thompson Group 𝐹. Claus Köstler and Arundhathi Krishnan. SIGMA 18 (2022), 083, 27 pages 1815-0659 2020 Mathematics Subject Classification: 46L53; 60J05; 60G09; 20M30 arXiv:2204.03595 https://nasplib.isofts.kiev.ua/handle/123456789/211821 https://doi.org/10.3842/SIGMA.2022.083 We show that representations of the Thompson group 𝐹 in the automorphisms of a noncommutative probability space yield a large class of bilateral stationary noncommutative Markov processes. As a partial converse, bilateral stationary Markov processes in tensor dilation form yield representations of 𝐹. As an application, and building on a result of Kümmerer, we canonically associate a representation of 𝐹 to a bilateral stationary Markov process in classical probability. The second author was partially supported by a Government of Ireland Postdoctoral Fellowship (Project ID: GOIPD/2018/498). Both authors acknowledge several helpful discussions with B.V. Rajarama Bhat in an early stage of this project. The first author would also like to thank Persi Diaconis, Gwion Evans, Rolf Gohm, Burkhard Kummerer, and Hans Maassen for several fruitful discussions on Markovianity. Both authors thank the organizers of the conference Noncommutative algebra, Probability and Analysis in Action held at Greifswald in September 2021 in honour of Michael Schurmann. The authors also thank the anonymous referees for their comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Markovianity and the Thompson Group 𝐹 Article published earlier
spellingShingle	Markovianity and the Thompson Group 𝐹 Köstler, Claus Krishnan, Arundhathi
title	Markovianity and the Thompson Group 𝐹
title_full	Markovianity and the Thompson Group 𝐹
title_fullStr	Markovianity and the Thompson Group 𝐹
title_full_unstemmed	Markovianity and the Thompson Group 𝐹
title_short	Markovianity and the Thompson Group 𝐹
title_sort	markovianity and the thompson group 𝐹
url	https://nasplib.isofts.kiev.ua/handle/123456789/211821
work_keys_str_mv	AT kostlerclaus markovianityandthethompsongroupf AT krishnanarundhathi markovianityandthethompsongroupf

Markovianity and the Thompson Group 𝐹

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