A Markovian and Roe-algebraic approach to asymptotic expansion in measure
aa r X i v : . [ m a t h . D S ] A ug A MARKOVIAN AND ROE-ALGEBRAIC APPROACH TO ASYMPTOTICEXPANSION IN MEASURE
KANG LI, FEDERICO VIGOLO, AND JIAWEN ZHANGA
BSTRACT . In this paper, we conduct further studies on geometric and analytic prop-erties of asymptotic expansion in measure. More precisely, we develop a machinery ofMarkov expansion and obtain an associated structure theorem for asymptotically expand-ing actions. Based on this, we establish an analytic characterisation for asymptotic expan-sion in terms of the Dru¸tu-Nowak projection and the Roe algebra of the associated warpedcones. As an application, we provide new counterexamples to the coarse Baum-Connesconjecture.
Mathematics Subject Classification (2020): 37A30, 37A15. Secondary: 46H35, 19K56.Keywords: Asymptotic expansion in measure; Coarse Baum-Connes conjecture; Markovexpansion; Spectral gap; Strong ergodicity; Warped cones.
1. I
NTRODUCTION
This paper is the second part of a broader study of the notion of asymptotic expansion inmeasure for measurable actions of countable groups on probability spaces. We introducedthis notion in [26], as a dynamical analogue of a previously defined notion of asymptoticexpansion for metric spaces [25].Asymptotic expansion in measure is a weakening of expansion in measure as definedin [53] and it turns out that—for measure-class-preserving actions—it is also equivalentto the classical notion of strong ergodicity introduced by Schmidt [48] and Connes-Weiss[10] (see [26] for more details).More precisely, a measurable action ρ : Γ y ( X , ν ) of a countable group on a prob-ability space is asymptotically expanding in measure if for each α ∈ ( , ] there exist c α > S α ⊆ Γ such that for every measurable subset A ⊆ X with α ≤ ν ( A ) ≤ we have(1.1) ν (cid:16) [ s ∈ S α s · A (cid:17) > ( + c ) ν ( A ) . The action ρ is called expanding in measure if we can let S α ≡ S for some fixed S .In [26] we studied the general structure theory of asymptotically expanding actions.Most notably, we showed that an action is asymptotically expanding in measure if andonly if it admits an exhaustion by domains of expansion (see Section 2.5 for a more de-tailed account). This fact allowed us to reprove a few recent–and–old results for stronglyergodic actions and it is also a key technical tool for the present paper. In addition, we alsomade explicit the connection between the notion of asymptotic expansion for measurable Date : August 31, 2020.KL has received funding from the European Research Council (ERC) under the European Union’s Hori-zon 2020 research and innovation programme (grant agreement no. 677120-INDEX).FV was supported by the ISF Moked 713510 grant number 2919/19.JZ was supported by the Sino-British Trust Fellowship by Royal Society and NSFC11871342. actions and that of asymptotic expansion for metric spaces. This allowed us to provide arich source of concrete examples of asymptotic expander graphs (see [26] for details).In this paper, we will further study the notion of asymptotic expansion in measurein the context of measure-class-preserving actions (in particular, all the results here de-scribed hold for strongly ergodic actions). Adapting the techniques developed in [24] tothe dynamical setting, we are able to prove some rather striking analytic and geometricproperties of asymptotic expansion in measure. More precisely, we obtain an analyticcharacterisation of asymptotic expansion in measure in terms of quasi-locality of aver-aging projections and Roe algebras of the associated warped cones. As a consequence,we will provide a new source of counterexamples to the coarse Baum–Connes conjecture,which is a central problem in higher index theory (see, e.g. , [39, 58]).To obtain the above results, we develop a spectral characterisation of (asymptotic) ex-pansion in measure which we find of independent interest. This is obtained by associatingmeasure-class-preserving actions with some reversible Markov kernels and by studyingthe resulting Laplacians and averaging operators. The spectral characterisation is obtainedby extending some classical results for Markov processes on finite state-spaces to generalMarkov kernels. We find that this theory provides a solid framework to study spectralproperties of actions that are not necessarily measure-preserving.1.1.
Spectral gaps and Markov expansion.
A probability measure-preserving action ρ : Γ y ( X , ν ) always induces a unitary representation π : Γ y L ( X , ν ) . If Γ is gener-ated by a finite symmetric subset S , the action ρ has a spectral gap if there exists somepositive constant κ > f ∈ L ( X , ν ) with R X f d ν = k f k ≤ κ ∑ s ∈ S k π ( s ) f − f k . This can be seen as an extremely strong version of ergodicity, and it is not very hard toshow that ρ has a spectral gap if and only if it is expanding in measure (this was shownmore or less independently in [7, 16, 20, 51], and was already implicit in earlier works ofK. Schmidt and Connes–Feldmann–Weiss).With the action ρ is associated a Markov operator P ∈ B ( L ( X , ν )) defined by P ≔ | S | ∑ s ∈ S π ( s ) and a Laplacian ∆ ≔ − P ∈ B ( L ( X , ν )) . These operators are self-adjoint,and ρ has a spectral gap if and only if simple ( i.e. , with multiplicity one) isolatedpoint in the spectrum of ∆ (equivalently, 1 is a simple isolated point in the spectrum of P ). This characterisation in terms of self-adjoint operators is crucial to provide explicitexamples of actions with spectral gap, as it opens a door to algebraic and representationtheoretical tools. In fact, this point of view leads to very deep connections between dy-namical systems, analysis and number theory. These connections make the study of thespectral gap property for measure-preserving actions into a very active and important fieldof research ([5, 6, 7, 15, 28, 30]).As an intermediate step toward an analytic study of asymptotic expansion in measure,we set a framework to extend the above connections to the setting of measure-class-pre-serving actions. It follows from the work of Houdayer–Marrakchi–Verraedt [20, Theorem3.2] that expansion in measure is equivalent to (1.2), whenever ρ ( s ) has bounded Radon–Nikodym derivative for every s ∈ S . In turn, (1.2) holds if and only if T ≔ ∑ s ∈ S | − π ( s ) | . However, the spec-trum of the operator T remains difficult to control. It is therefore desirable to producesome spectral condition which can more adequately describe the notion of expansion.In this paper, we provide a rather satisfactory answer to the above need by using Markovkernels and Markov expansion. Our approach is based on a shift in paradigm, and can MARKOVIAN AND ROE-ALGEBRAIC APPROACH TO ASYMPTOTIC EXPANSION 3 be justified by some analogies between finite graphs and dynamical systems. The ori-gin of these analogies lies in the approximation procedure introduced in [51] and furtherdeveloped in [26]. According to this procedure, expansion in measure corresponds to vertex-expansion for finite graphs [51] and, if the action is measure-preserving, the spec-tral gap condition (1.2) can be seen as an analogue of spectral expansion for graphs (seealso [42]). It is a classical result that spectral-expansion is equivalent to edge-expansion ([2, 3, 11]), and it is easy to verify that the latter is equivalent to vertex-expansion. Thiscan be seen as the graph-theoretic analogue of the equivalence between (1.2) and expan-sion in measure for measure-preserving actions.To be more precise, spectral expansion for a regular finite graph G is defined in termsof the spectral gap of the discrete Laplacian ∆ ∈ B ( L ( G , ν )) . Here ν is the counting mea-sure on the set of vertices of G and ∆ is defined as 1 − P , where P is the averaging operator( a.k.a. Markov operator) defined by P f ( v ) ≔ ∑ v ∼ w f ( w ) / degree(v) for f ∈ L ( G , ν ) . Im-portantly, if the graph G is not regular then the discrete Laplacian is no longer self-adjointin B ( L ( G , ν )) . Instead, it is self-adjoint in B ( L ( G , ˜ ν )) , where ˜ ν is a different measurewhich takes into account the degree of each vertex. Spectral expansion is then defined interms of the spectrum of ∆ seen as an operator on L ( G , ˜ ν ) . A more sophisticated way ofrephrasing this is that the (lazy) simple random walk on a finite connected graph G has aunique stationary probability measure ˜ ν . The probability distribution of the n -th step ofsuch a random walk converges exponentially fast to ˜ ν (in the L -norm), and the spectralexpansion measures the rate of exponential convergence.The above discussion can be used as heuristics in the dynamical setting. We remark thatgraphs corresponding to a measure-preserving action are “regular on a large scale”. It istherefore natural to expect a correspondence between spectral expansion and expansion inmeasure. On the other hand, actions that are not measure-preserving correspond to irreg-ular graphs (the “large scale degrees” are governed by the Radon–Nikodym derivatives).This suggests us to search for a spectral characterisation of expansion in measure in termsof some operator in B ( L ( X , ˜ ν )) —where ˜ ν is some stationary measure depending on theRadon–Nikodym derivatives. This is precisely the approach that we take in this paper.Let Γ be a finitely generated group and S ⊆ Γ a finite symmetric generating set con-taining the identity element, and let ρ : Γ y ( X , ν ) be a measure-class-preserving actionwith the Radon–Nikodym derivatives r ( γ , x ) ≔ d γ − ∗ ν d ν ( x ) . It turns out that the measure ˜ ν defined by d ˜ ν ( x ) ≔ ∑ s ∈ S r ( s , x ) d ν ( x ) is a stationary measure for the reversible Markov kernel Π ( x , - ) ≔ ∑ s ∈ S r ( s , x ) ∑ s ∈ S r ( s , x ) δ s · x . Naturally associated to Π , there are a Markov operator P and a Laplacian ∆ = − P . Both of these are self-adjoint operators in B ( L ( X , ˜ ν )) —we defer to Section 3 forpreliminaries and definitions regarding Markov kernels. Every measurable subset A ⊆ X Assuming that the graphs have uniformly bounded degree. To give a somewhat precise meaning to the notion of “regular on a large scale” it is necessary touse the terminology of [26, 51]: given any measurable subset A ⊆ X and a sufficiently fine approximation [ A ] P , the ratio | ∂ [ A ] P | / | [ A ] P | will be roughly equal to ν ( S · A ) / ν ( A ) . If ρ is measure-preserving and A isdisjoint from s · A for every s ∈ S , then the latter ratio is equal to | S | . That is, the approximating graphs are“ | S | -regular on a large scale”. KANG LI, FEDERICO VIGOLO, AND JIAWEN ZHANG has a natural notion of “measure of the boundary” | ∂ Π ( A ) | ∈ R ≥ (Definition 3.2 or [22]),and we say that ρ is Markov expanding if there is a c > | ∂ Π ( A ) | > c ˜ ν ( A ) for every A ⊆ X with 0 < ˜ ν ( A ) ≤ ˜ ν ( X ) . This should be thought of as a dynamicalanalogue of edge-expansion for graphs.Our first technical result is a generalisation of the proof of the equivalence betweenedge-expansion and spectral expansion from the context of random walks on graphs tothat of general reversible Markov kernels. This result is probably known to experts, butwe were not able to find a proof in the literature. Theorem A (Theorem 3.7) . Let Π be a reversible Markov kernel on X with finite reversingmeasure m. Let λ be the infimum of the spectrum of the restriction of ∆ to the space offunctions with zero average, and let κ ≔ inf | ∂ Π ( A ) | / m ( A ) for A ⊆ X with < m ( A ) ≤ m ( X ) . Then κ ≤ − λ ≤ κ . As a corollary, we immediately obtain a characterisation of Markov expansion in termsof the spectrum of ∆ ∈ B ( L ( X , ˜ ν )) . Furthermore, it is relatively easy to show that, whenthe Radon–Nikodym derivatives are bounded, Markov expansion is equivalent to the orig-inal notion of expansion in measure (this is analogous to the equivalence between edge-ex-pansion and vertex-expansion for graphs of uniformly bounded degrees). This leads us tothe following: Corollary B (Corollary 3.17, Remark 3.18) . Let Θ ≥ be a constant. A measure-class-pre-serving action ρ : Γ y ( X , ν ) with / Θ ≤ r ( s , x ) ≤ Θ for every s ∈ S and x ∈ X isexpanding in measure if and only if 0 is a simple isolated point in the spectrum of ∆ ∈ B ( L ( X , ˜ ν )) .Remark . It is not hard to show that Corollary B and [20, Theorem 3.2] are in factequivalent. However, we find that our approach has various advantages:(1) We find that the Laplacian operator ∆ is more natural than T . It should be easierto handle ( e.g. , to control spectral gap), and it allows us to borrow several calcula-tions and results from the classical setting of random walks on graphs.(2) The spectral gap condition can be rephrased by saying that the restriction of theMarkov operator P to the space functions with zero-average has operator normstrictly less than 1. It can be useful to know that P n converges in the operatornorm to the projection onto constant functions (see also Section 4.3).(3) It allows for a finer control of the expansion constants.(4) The spectral characterisation of Markov expansion holds true also for actions withunbounded Radon–Nikodym derivatives (this should be of independent interest).We restricted the previous discussion to the case of actions of finitely generated groupsfor the sake of simplicity. However, the machinery of Markov kernels is very flexible, andall the results mentioned above will actually be proved for actions of arbitrary discretecountable groups. Furthermore, we will also study restrictions of actions to subsets of X which are not necessarily invariant. As a sample application, we note that Corollary Bimplies the following (see Section 2 for the relevant definitions and Corollary 3.17): It would be also possible to extend this theory to include general countable measurable equivalencerelations.
MARKOVIAN AND ROE-ALGEBRAIC APPROACH TO ASYMPTOTIC EXPANSION 5
Corollary C ([16]) . A measure-preserving action Γ y ( X , ν ) has local spectral gap withrespect to Y ⊆ X if and only if Y is a domain of expansion.
Being able to work with subsets of X is a necessary requirement to use the structuretheorems established in [26], which characterise asymptotic expansion in terms of exhaus-tions (see Section 2.5). Combining those results with the Markov machinery developedabove, we are able to prove an additional structure result which will play a key role in therest of the paper: Theorem D (Theorem 3.21) . A measure-class-preserving action Γ y ( X , ν ) on a prob-ability space is asymptotically expanding in measure if and only if every subset Y ⊆ Xadmits an exhaustion by domains of Markov expansion.Remark . The above theorem remains true when replacing “probability” by “ σ -finite”and “asymptotically expanding in measure” by “strongly ergodic”.1.2. Warped cones and finite propagation approximations.
Our next aim is to studyasymptotically expanding actions via analytic properties of certain projection operators.This is done by using the warped cone construction as a bridge between the metric anddynamical setting, and then utilizing Markov expansion. The end result is a dynamicalanalogue of the theory developed in [24] to characterise asymptotic expanders using aver-aging projections.The notion of warped cone was firstly introduced by Roe in [40] to explore more ex-amples with/without Yu’s property A and coarse embeddings into Hilbert spaces. Thegeometry of warped cones was subsequently studied by a number of people, e.g. , [12, 14,33, 42, 43, 44, 45, 51, 52, 55]. Roughly speaking, given a continuous action Γ y ( X , d ) on a compact metric space with diameter at most 2, the associated unified warped cone isthe metric space ( O Γ X , d Γ ) , where O Γ X = X × [ , ∞ ) as a set and d Γ is a metric on O Γ X defined in terms of the group action (see Section 4.1 for details).Given a probability measure ν on ( X , d ) , we consider the averaging projection P X on L ( X , ν ) , which is the rank-one orthogonal projection onto the space of constant functionson X . Denoting by λ the Lebesgue measure on [ , ∞ ) , the Dru¸tu–Nowak projection isdefined as G = P X ⊗ Id L ([ , ∞ )) ∈ B ( L ( O Γ X , ν × λ )) , which is the orthogonal projectiononto C ⊗ L ([ , ∞ ) , λ ) .The Dru¸tu–Nowak projection G was first introduced in [12, Section 6.c.] in theirstudy on the coarse Baum–Connes conjecture (more details will be provided later). Theyshowed that if an action is measure-preserving and has a spectral gap, then the projection G is a norm limit of finite propagation operators in B ( L ( O Γ X , ν × λ )) . Recall that anoperator T ∈ B ( L ( O Γ X , ν × λ )) has finite propagation if there exists R > f , g ∈ C ( O Γ X ) with d Γ ( supp ( f ) , supp ( g )) > R we have f T g =
0, where f and g areregarded as diagonal operators on L ( O Γ X , ν × λ ) via the multiplication representation.In this paper, we study the converse of Dru¸tu–Nowak’s result and prove the followinganalytic characterisation for asymptotically expanding actions: Theorem E (Theorem 4.8 and Theorem 4.16) . Let ( X , d ) be a metric space with diameterat most equipped with a Radon probability measure ν , and ρ : Γ y X be a continuousmeasure-class-preserving action. The following are equivalent:(1) ρ is asymptotically expanding;(2) the Dru¸tu–Nowak projection G is quasi-local;(3) the Dru¸tu–Nowak projection G is a norm limit of operators with finite propaga-tion. KANG LI, FEDERICO VIGOLO, AND JIAWEN ZHANG
Remark . The notion of quasi-locality was introduced by Roe in [38]. It is weaker than the property of admitting an approximation by finite propagation operators, and it isrelatively easy to verify. For more details on quasi-locality, we refer readers to [13, 25,27, 50, 54].Theorem E is a dynamical analogue of [24, Theorem 6.1], and there are two mainingredients in its proof. Firstly, we introduce a dynamical notion of finite propagationapproximation and quasi-locality (see Section 4.2 and 4.4) as an intermediate bridge toconnect asymptotic expansion and analytic properties of the Dru¸tu–Nowak projection.Secondly, we apply the tool of Markov expansion to approximate dynamical quasi-localoperators with finite dynamical propagation ones. Due to some correspondence results(Proposition 4.7 and 4.15), we can then pass from the dynamical notions to their analyticanalogues for unified warped cones and obtain Theorem E.As a byproduct, we construct numerous projections which can be approximated byoperators with finite propagation (see Corollary 4.17). These projections will be importantin the next section, where we deal with the coarse Baum–Connes conjecture.1.3. Roe algebras and the coarse Baum–Connes conjecture.
Roe algebras are C ∗ -algebras associated with metric spaces. These C ∗ -algebras encode coarse geometric infor-mation of the metric spaces and play key roles in higher index theory (see, e.g. , [38, 39, 58]for more details). We conclude this paper by studying Roe algebras of warped cones as-sociated to asymptotically expanding actions and provide an application to the so-calledcoarse Baum–Connes conjecture.Given a continuous action Γ y ( X , d ) on a compact metric space with diameter at most2 and a non-atomic probability measure ν on ( X , d ) with full support, we consider themultiplication representation C ( O Γ X ) → B ( L ( O Γ X , ν × λ )) . The Roe algebra of theunified warped cone, denoted by C ∗ ( O Γ X ) , is the norm closure of all finite propagationlocally compact operators in B ( L ( O Γ X , ν × λ )) (see Section 5.1 for more details).Although the Dru¸tu–Nowak projection G can be approximated by finite propagationoperators, it is not locally compact because its restriction on L ([ , ∞ ) , λ ) is the identityoperator. In order to obtain non-trivial projections in the Roe algebra, Sawicki [43] sug-gested to consider the integral warped cone and the associated integral Dru¸tu–Nowakprojection (see Section 5.1). Since the integral warped cone is coarsely equivalent tothe original warped cone, they have ∗ -isomorphic Roe algebras. Based on [12], Sawicki[43, Proposition 1.3] showed that for a measure-preserving action with spectral gap, theintegral Dru¸tu–Nowak projection belongs to the associated Roe algebra.Theorem E allows us to both extend and provide a converse to Sawicki’s result: Theorem F (Theorem 5.2, Corollary 5.6) . Let ( X , d ) be a compact metric space withdiameter at most , ν a non-atomic Radon probability measure on X of full support, and ρ : Γ y ( X , d , ν ) a continuous measure-class-preserving action. Then ρ is asymptoticallyexpanding if and only if the integral Dru¸tu–Nowak projection belongs to the Roe algebraC ∗ ( O Γ X ) . Moreover, the integral Dru¸tu–Nowak projection is non-compact and ghost. The study of projections in Roe algebras is motivated by the computation of their K-theories. The coarse Baum–Connes conjecture asserts that K-theories of Roe algebrascan be computed in terms of homology information of underlying metric spaces. Whentrue, this establishes a connection between geometry, topology and analysis. One ground-breaking result on the subject is due to Yu [61], as he showed that the coarse Baum–Connes conjecture holds for all metric spaces with bounded geometry that are coarsely It is conjectured that quasi-locality should be strictly weaker than admitting such approximations.
MARKOVIAN AND ROE-ALGEBRAIC APPROACH TO ASYMPTOTIC EXPANSION 7 embeddable into Hilbert spaces. On the other hand, counterexamples to the conjecturewere subsequently discovered by Higson [17] (see also [18]) using expander graphs. Ina recent joint work with Khukhro, we found more counterexamples using asymptoticexpanders [24].Understanding which spaces satisfy the coarse Baum–Connes conjecture is still one ofthe major questions in higher index theory, as it has significant applications to other areasof mathematics, such as topology and geometry (see [19, 46, 49, 60] for more details).It is an open question whether warped cones arising from actions with spectral gap arecounterexamples to the coarse Baum–Connes conjecture. This question was the motiva-tion behind the introduction of the Dru¸tu–Nowak projection in [12]. Recently, Sawicki[43, Theorem 3.5] proved that sparse warped cones (see Section 5.2) do provide coun-terexamples to the coarse Baum–Connes conjecture. His proof follows a similar outlineof Higson’s original proof for expander graphs. Using our work on asymptotically ex-panding actions, we can generalise Sawicki’s result as follows:
Theorem G (Corollary 5.15) . Let ( X , d ) be a compact metric space of diameter at most equipped with a non-atomic probability measure ν of full support, and ρ : Γ y ( X , d , ν ) be a free Lipschitz measure-class-preserving asymptotically expanding action. Under either of the following conditions:(1) if Γ has property A and X is a manifold;(2) if the asymptotic dimension of Γ is finite and X is an ultrametric space;the coarse Baum–Connes conjecture for the sparse warped cone fails.Remark . We can produce examples whose violation of the coarse Baum–Connes con-jecture can be deduced from Theorem G, but not from any previously known results (seeExample 5.16).Under some extra conditions (ONL and bounded geometry), it follows by combiningTheorem G with Yu’s result [61] that warped cones arising from asymptotically expandingactions cannot coarsely embed into Hilbert spaces. Our last result shows that these extraconditions are in fact unnecessary (this partially generalises [33, Theorem 3.1]):
Proposition H (Proposition 5.18) . Let ( X , d ) be a compact metric space of diameterat most equipped with a non-atomic probability measure ν , and ρ : Γ y ( X , d , ν ) bea continuous measure-class-preserving and asymptotically expanding action. Then thewarped cone O Γ X does not admit a coarse embedding into any Hilbert space.
Structure of the paper.
Section 2 covers some preliminaries and further illustratesthe connections between this paper and other works. The first half of Section 3 can be readindependently from the rest of the paper and is devoted to introducing reversible Markovkernels/expansion and proving Theorem A. The second part connects this theory to thestudy of measure-class-preserving actions. Here we prove Corollary B, C and Theorem D.These results will be important to both of the following sections. In Section 4, we recallthe warped cone construction and study asymptotic expansion from the point of view ofwarped cones. Here we prove Theorem E. Section 5 is mostly devoted to the study ofRoe algebras of warped cones. In the first part, we prove Theorem F, and in the secondpart we provide new counterexamples to the coarse Baum–Connes conjecture by provingTheorem G. Finally, we conclude this section by proving Proposition H.
Acknowledgments.
We wish to thank Amine Marrakchi for pointing out [9, 31] to us andfor manifesting interest in our work. The first author wishes to thank Damian Sawicki forhelpful discussions on the coarse Baum–Connes conjecture. The second author wishes
KANG LI, FEDERICO VIGOLO, AND JIAWEN ZHANG to thank Uri Bader for his helpful conversations. The third author wishes to thank JanŠpakula for several useful discussions on dynamical quasi-locality.2. P
RELIMINARIES
Standing conventions.
Throughout the paper, Γ will always be a countable discretegroup. The group Γ will be made into a metric space by fixing a proper length function(see below). The letter S will always denote a finite subset in Γ . Such a set will often—butnot always—be symmetric ( i.e. , γ ∈ S implies that γ − ∈ S ) and containing the identityelement 1 ∈ Γ . We will not generally assume that S generates Γ .All the measure spaces will be σ -finite and all the actions will be measurable. Moreprecisely, we say that Γ y ( X , ν ) is an action as shorthand for saying that Γ is a countablediscrete group acting measurably on a σ -finite measure space ( X , ν ) . When we equip ametric space ( X , d ) with a measure ν , we will always assume that ν is defined on theBorel σ -algebra.2.2. Actions on measure spaces.
Let ( X , ν ) be a measure space. A measurable subset A ⊆ X of positive finite measure is called a domain . An exhaustion of ( X , ν ) is a sequenceof nested measurable subsets Y ⊆ Y ⊆ · · · such that S n ∈ N Y n = X up to measure zero.We denote exhaustions by Y n ր ( X , ν ) , or simply Y n ր X if the measure is clear from thecontext.A proper length function on Γ is a function ℓ : Γ → { } ∪ N which satisfies the follow-ing: • ℓ ( γ ) = γ = Γ ); • ℓ ( γ ) = ℓ ( γ − ) for every γ ∈ Γ ; • ℓ ( γ γ ) ≤ ℓ ( γ ) + ℓ ( γ ) for every γ , γ ∈ Γ ; • the number of γ ∈ Γ with ℓ ( γ ) ≤ k is finite for every k ∈ N .It is easy to show that every countable discrete group Γ admits a proper length function(see e.g. [35, Proposition 1.2.2]). For example, if Γ is a finitely generated group then wecan simply take the word length with respect to an arbitrary finite symmetric generatingset. Any proper length function ℓ induces a left-invariant metric d ℓ on Γ by d ℓ ( γ , γ ) ≔ ℓ ( γ − γ ) . This makes Γ into a proper discrete metric space. Choosing a different lengthfunction ℓ ′ will yield a coarsely equivalent metric on Γ (we will not need this fact).For each k ∈ N , we denote by B k the closed ball in ( Γ , ℓ ) with radius k and centred atthe identity: B k ≔ { γ ∈ Γ | ℓ ( γ ) ≤ k } . It follows from the definition of length function that each B k is finite and symmetric,1 ∈ B k and B k · B l ⊆ B k + l for every k , l ∈ N .We will be concerned with actions of Γ on ( X , ν ) . Given A ⊆ X and K ⊆ Γ , let K · A ≔ [ γ ∈ K γ · A . Since 1 ∈ B k , we note that A ⊆ B k · A for every A ⊆ X and every k ∈ N .Recall that an action Γ y ( X , ν ) is measure-class-preserving if it sends measure-zerosets to measure-zero sets. In this case, for every γ ∈ Γ there is an associated Radon–Nikodym derivative d γ − ∗ ν / d ν that is well-defined up to measure-zero sets. MARKOVIAN AND ROE-ALGEBRAIC APPROACH TO ASYMPTOTIC EXPANSION 9
Expansion in measure.
Let Γ y X be an action and S ⊆ Γ a finite symmetric set.For any measurable subset A ⊆ X we denote ∂ Γ S A ≔ S · A r A , which should be regardedas the “boundary of A with respect to the action by S ”. Definition 2.1 ([53]) . An action ρ : Γ y ( X , ν ) on a probability measure space ( X , ν ) iscalled expanding (in measure) if there exist a constant c > S ⊂ Γ such thatfor any measurable subset A ⊆ X with 0 < ν ( A ) ≤ , we have ν ( ∂ Γ S A ) > c ν ( A ) . In thiscase, we say that ρ is ( c , S ) -expanding or simply S-expanding .If an action is ( c , B k ) -expanding for some k ∈ N , we may also say that it is ( c , k ) -ex-panding. Note that every expanding action is ( c , k ) -expanding for some c > k ∈ N .In an independent work, Grabowski–Máthé–Pikhurko defined a “local” version of ex-pansion under the name of domain of expansion : Definition 2.2 ([16] ) . Let ρ : Γ y ( X , ν ) be an action. A domain Y ⊆ X is called a domain of expansion for ρ if there exist a constant c > S ⊆ Γ such that forevery measurable subset A ⊆ Y with 0 < ν ( A ) ≤ ν ( Y ) , we have ν (cid:0) ( S · A ) ∩ Y (cid:1) > ( + c ) ν ( A ) . In this case, we say that Y is a domain of ( c , S ) -expansion or simply of S-expansion . Asbefore, if S = B k we may say that Y ⊆ X is a domain of ( c , k ) -expansion.We note that when ν is finite, ρ : Γ y ( X , ν ) is expanding if and only if X is a domainof expansion for ρ . We end this subsection by recalling the following elementary fact,which will be used in the proof of Proposition 3.19: Lemma 2.3 ([26, Lemma 3.14]) . Let ρ : Γ y ( X , ν ) be an action and Y ⊆ X a domain.Assume that Y , Y ⊆ Y are domains of S-expansion. If ν ( Y ) > ν ( Y ) and ν ( Y ) > ν ( Y ) then the union Y ∪ Y is a domain of S-expansion as well. (Local) spectral gap. We will work with complex L p -spaces for p ∈ [ , ∞ ) . If wewish to stress that the L p -norm of a function on X is computed with respect to the measure ν , we denote it by k f k ν , p . Similarly, we will denote the inner product on the Hilbert space L ( X , ν ) by h f , g i ν .Given a measurable subset Y in a measure space ( X , ν ) , we denote the restriction of ν to Y by ν | Y . With a slight abuse of notation, we also use the symbol ν | Y to denote themeasure on X which gives measure 0 to X r Y and coincides with ν on all measurablesubsets of Y ( i.e. , ν | Y = χ Y · ν where χ Y is the indicator function of Y ). This will notcause confusion, as the meaning will be clear from the context.A measure-preserving action ρ : Γ y ( X , ν ) on a probability measure space ( X , ν ) hasa spectral gap if there exist a constant κ > S ⊆ Γ such that for every function f ∈ L ( X , ν ) with R X f d ν = k f k ≤ κ ∑ γ ∈ S k γ · f − f k , where γ · f ( x ) ≔ f ( γ − · x ) . It can be shown that the action ρ is expanding in measure ifand only if it has a spectral gap (see, e.g. , [53, Section 7]).In [7], Boutonnet–Ioana–Golsefidy introduced the following localised version of spec-tral gap: The authors of [16] only consider measure-preserving actions, but their definition makes sense forgeneral measurable actions as well.
Definition 2.4 ([7, Definition 1.2]) . Let ρ : Γ y ( X , ν ) be a measure-preserving actionand Y ⊆ X be a domain. The action ρ has local spectral gap with respect to Y if thereexist a constant κ > S ⊆ Γ such that(2.2) k f k ν | Y , ≤ κ ∑ γ ∈ S k γ · f − f k ν | Y , for every f ∈ L ( X , ν ) with R Y f d ν = ν is a probability measure, ρ has spectral gap if and only if it haslocal spectral gap with respect to the whole X .It is shown in [16, Lemma 5.2] that a measure-preserving action ρ : Γ y ( X , ν ) haslocal spectral gap with respect to a domain Y ⊆ X if and only if Y is a domain of expansionfor ρ . This fact can also be deduced from [20, Theorem 3.2] (or by adapting the argumentsof [53, Section 7]). Later on, we will provide an alternative proof based on our study ofMarkov kernels (see Corollary 3.17). Remark . Equations (2.1) and (2.2) make sense also if the action is not measure-pre-serving (although in this case it would be perhaps more appropriate to refer to them asPoincaré inequalities, rather than spectral gaps). It follows from [20, Theorem 3.2] that—as long as the Radon–Nikodym derivatives are bounded—the characterisation of (domainsof) expansion in measure in terms of (local) spectral gaps also holds for actions that donot necessarily preserve measures.2.5.
Asymptotic expansion in measure and structure theorems.
The following weak-ening of expansion in measure was defined in [26] in analogy with [24, 25]:
Definition 2.6 ([26, Definition 3.1]) . Let ρ : Γ y ( X , ν ) be an action on a space ( X , ν ) of finite measure. The action ρ is called asymptotically expanding (in measure) if thereexist functions ¯ c : ( , ] → R > and ¯ k : ( , ] → N such that for every α ∈ ( , ] we have(2.3) ν (cid:0) B ¯ k ( α ) · A (cid:1) > ( + ¯ c ( α )) ν ( A ) for every measurable subset A ⊆ X with αν ( X ) ≤ ν ( A ) ≤ ν ( X ) .For a finite S ⊆ Γ , we say that ρ is ( ¯ c , S ) -asymptotically expanding (in measure) (orsimply S-asymptotically expanding ) if for every α ∈ ( , ] and measurable subset A ⊆ X with αν ( X ) ≤ ν ( A ) ≤ ν ( X ) , we have ν ( S · A ) > ( + ¯ c ( α )) ν ( A ) . Remark . When the acting group Γ is finitely generated by a finite symmetric set S , ameasure-class-preserving action ρ : Γ y ( X , ν ) on a probability space is asymptoticallyexpanding if and only if it is S -asymptotically expanding (see [26, Lemma 3.16]). Wewill not need this fact in this paper.This notion turns out to be naturally related to strong ergodicity. Recall that a mea-sure-class-preserving action ρ : Γ y ( X , ν ) on a probability space ( X , ν ) is called stronglyergodic [10, 47] if any sequence of measurable subsets { C n } n ∈ N in X with lim n → ∞ ν ( C n △ γ C n ) = γ ∈ Γ , must satisfy lim n → ∞ ν ( C n )( − ν ( C n )) = . Note that for two equivalent finite measures ν , ν ′ on a space X and any sequence ofmeasurable subsets ( A n ) n ∈ N in X , ν ( A n ) → if and only if ν ′ ( A n ) →
0. Hence, strongergodicity only depends on the measure-class of the given measure. Therefore, the fol-lowing is well-posed:
MARKOVIAN AND ROE-ALGEBRAIC APPROACH TO ASYMPTOTIC EXPANSION 11
Definition 2.8 ([21]) . A measure-class-preserving action on a (possibly infinite) measurespace is strongly ergodic if ρ : Γ y ( X , ν ′ ) is strongly ergodic with respect to some (henceevery) probability measure ν ′ equivalent to ν . Proposition 2.9 ([26, Proposition 3.5]) . Let ρ : Γ y ( X , ν ) be a measure-class-preserv-ing action on a probability space. Then ρ is strongly ergodic if and only if it is asymptot-ically expanding in measure. In particular, explicit examples of strongly ergodic actions ( e.g. , those constructed in[1]) give rise to explicit examples of asymptotically expanding actions.The following is a localised version of Definition 2.6:
Definition 2.10 ([26, Definition 3.7]) . Let ρ : Γ y ( X , ν ) be an action. A domain Y ⊆ X is called a domain of asymptotic expansion for ρ if there exist functions ¯ c : ( , ] → R > and ¯ k : ( , ] → N such that for every α ∈ ( , ] and measurable A ⊆ Y with αν ( Y ) ≤ ν ( A ) ≤ ν ( Y ) , we have ν (cid:0) ( B ¯ k ( α ) · A ) ∩ Y (cid:1) > ( + ¯ c ( α )) ν ( A ) . For a finite S ⊆ Γ , we say that Y is a domain of ( ¯ c , S ) -asymptotic expansion (or simply domain of S-asymptotic expansion ) if for every α ∈ ( , ] and measurable subset A ⊆ X with αν ( X ) ≤ ν ( A ) ≤ ν ( X ) , we have ν (cid:0) ( S · A ) ∩ Y (cid:1) > ( + ¯ c ( α )) ν ( A ) .When ν is finite, an action ρ : Γ y ( X , ν ) is asymptotically expanding (in measure) ifand only if X is a domain of asymptotic expansion for ρ .The fact that Definitions 2.6 and 2.10 are only concerned with domains of measure atmost ν ( X ) / ν ( Y ) /
2) makes them easier to verify, but sometimes awkward to use.The following elementary result is helpful to bypass this issue:
Lemma 2.11 ([26, Lemma 3.8]) . Let Y ⊆ X be a domain of asymptotic expansion for anaction Γ y ( X , ν ) . Then there exist functions ¯ b : [ , ) → R > and ¯ h : [ , ) → N suchthat for every β ∈ [ , ) , we have ν (cid:0) ( B ¯ h ( β ) · A ) ∩ Y (cid:1) > ( + ¯ b ( β )) ν ( A ) for every measurable subset A ⊆ Y with ν ( Y ) ≤ ν ( A ) ≤ βν ( Y ) . For later use, we end this subsection by recalling some structure results established in[26, Section 4].
Proposition 2.12 ([26, Proposition 4.5 and 4.11]) . Let Γ y ( X , ν ) be an action, Y ⊆ Xbe a domain of asymptotic expansion and ( Z n ) n ∈ N be a sequence of nested subsets of Ywith ν ( Z n ) → . Then there exist N ∈ N , a sequence of finite subsets S n ⊆ Γ and anexhaustion Y n ր Y by domains of S n -expansion such that Y n ⊆ Y r Z n for every n > N . Theorem 2.13 ([26, Theorem 4.9]) . Let ρ : Γ y ( X , ν ) be a measure-class-preservingaction. Then the following are equivalent:(1) ρ is strongly ergodic;(2) every finite measure subset is a domain of asymptotic expansion;(3) ρ is ergodic and X admits a domain of expansion.Remark . For measure-preserving actions, the equivalence “ ( ) ⇔ ( ) ” of Theo-rem 2.13 had been previously proved in [31, Theorem A].
3. E
XPANSION AND REVERSIBLE M ARKOV KERNELS
In the first part of this section, we introduce the language of Markov kernels and toprove a general estimate for the Cheeger constant of a reversible Markov kernel in termsof the spectrum of the associated Laplacian operator. In the second part, we show thatmeasure-class preserving actions give rise to reversible Markov kernels. This allows usto define the notion of (domain of) Markov expansion and to characterise asymptoticexpansion in terms of exhaustions by domains of Markov expansion. This result will bepivotal in the subsequent sections.3.1.
Preliminaries on Markov kernels.
We begin by recalling a few elementary prop-erties of reversible Markov kernels. We refer to the first chapters of [37] for more back-ground and details.
Definition 3.1.
Let E be a σ -algebra on a set X . A Markov kernel on the measurablespace ( X , E ) is a function Π : X × E → [ , ] such that:(1) for every x ∈ X , the function Π ( x , - ) : E → [ , ] is a probability measure;(2) for every A ∈ E , the function Π ( - , A ) : X → [ , ] is E -measurable.If f : X → R is integrable with respect to the probability measure Π ( x , - ) , we denoteits integral by Z X f ( y ) Π ( x , d y ) ≔ Z X f ( y ) d Π ( x , - )( y ) (the integral is then naturally extended to complex-valued functions). The associated Markov operator P is a linear operator on the space of bounded E -measurable functions,defined by P f ( x ) ≔ Z X f ( y ) Π ( x , d y ) . Since Π ( - , A ) is measurable for every A ∈ E , we can define an operator ˇ P on the spaceof measures on ( X , E ) by letting ν ˇ P ( A ) ≔ Z X Π ( x , A ) d ν ( x ) for every measure ν on ( X , E ) . The operators P and ˇ P are dual to one another in thesense that(3.1) Z X P f ( x ) d ν ( x ) = Z X f ( x ) d ν ˇ P ( x ) , whenever the integrals are defined. Definition 3.2 ([22]) . Given a measure ν on ( X , E ) and an A ∈ E , the ( ν -)size of theboundary of A (with respect to Π ) is defined as | ∂ Π A | ν ≔ Z A Π ( x , X r A ) d ν ( x ) . Remark . Heuristically, a Markov kernel can be described as “moving mass across X ”without creating nor destroying it: the value Π ( x , A ) is the proportion of the mass that ismoved from the point x into the set A . The measure ν ˇ P is the distribution of mass on X that is obtained after moving the initial distribution ν according to the kernel Π . Thefunction P f assigns to a point x ∈ X the expected value of f when spreading x across X according to the kernel Π .The duality formula (3.1) on the indicator function f = χ A can be understood as sayingthat the total ν -mass that is moved into a set A by the kernel Π is equal to to ν -integral of MARKOVIAN AND ROE-ALGEBRAIC APPROACH TO ASYMPTOTIC EXPANSION 13 the likelihood that Π will take x into A . The size of the boundary of A ∈ E is the amountof ν -mass that is carried outside A by Π .We will only be concerned with some special Markov kernels: Definition 3.4.
A Markov kernel Π is called reversible if there exists a measure m on ( X , E ) such that Z X f ( x ) P g ( x ) d m ( x ) = Z X P f ( x ) g ( x ) d m ( x ) for every pair of measurable bounded functions f , g : X → R . The measure m is said to bea reversing measure for Π (note that m need not be unique in general). To specify whichreversing measure is being considered, we say that Π is a reversible Markov kernel on ( X , m ) .Let m be a measure on X . We define the measure µ on X × X by letting(3.2) µ ( A × B ) ≔ Z A Π ( x , B ) d m ( x ) = Z X χ A ( x ) P χ B ( x ) d m ( x ) for every A , B ∈ E . Then m is a reversing measure if and only if µ is symmetric , i.e. , µ ( A × B ) = µ ( B × A ) for every A , B ∈ E . In this case, we have(3.3) | ∂ Π ( A ) | m = µ (cid:0) A × ( X r A ) (cid:1) = µ (cid:0) ( X r A ) × A (cid:1) = | ∂ Π ( X r A ) | m . In other words, the m -size of the boundary of any measurable set is equal to the m -size ofthe boundary of its complement.For the rest of this section, let us fix a reversible Markov kernel Π on ( X , m ) . We notethat m ˇ P ( A ) = µ ( X × A ) = µ ( A × X ) = Z A Π ( x , X ) d m ( x ) = m ( A ) , i.e. , m is invariant under ˇ P . Hence, the Cauchy–Schwarz inequality yields: Z X | P f ( x ) | d m ( x ) ≤ Z X P | f | ( x ) d m ( x ) = Z X | f | ( x ) d m ˇ P ( x ) = Z X | f | ( x ) d m ( x ) = k f k m , . Therefore, the Markov operator P can be regarded as a bounded operator on L ( X , m ) with norm k P k ≤
1. Since m is reversing, the operator P is self-adjoint.Now, for any p ∈ [ , ∞ ) and any f ∈ L p ( X , m ) , we define its p-Dirichlet energy as E p ( f ) ≔ Z X × X | f ( x ) − f ( y ) | p d µ ( x , y ) . Since µ is symmetric, we note that Z X × X | χ A ( x ) − χ A ( y ) | p d µ ( x , y ) = µ (cid:0) A × ( X r A ) (cid:1) + µ (cid:0) ( X r A ) × A (cid:1) = µ (cid:0) A × ( X r A ) (cid:1) . Hence for every p ∈ [ , ∞ ) , we have(3.4) E p ( χ A ) = | ∂ Π ( A ) | m . Finally, we observe that for any f ∈ L ( X , m ) we have E ( f ) = Z X × X | f | ( x ) + | f | ( y ) − Re ( f ( x ) f ( y )) d µ ( x , y ) = k f k m , − h f , P f i m , where the last equality uses the reversibility.We define the Laplacian of Π as ∆ ≔ − P , then we have E ( f ) = h f , ∆ f i m for every f ∈ L ( X , m ) . In particular, the Laplacian ∆ is a positive self-adjoint operator whosespectrum is contained in [ , ] . Isoperimetric inequalities and spectra of Markov kernels.
It is a well-known re-sult that a sequence of finite graphs is a family of expanders if and only if the Markovoperators associated with the simple random walks have a uniform spectral gap [2, 3, 11].In this subsection, we will extend this result to the context of Markov kernels.Let Π be a reversible Markov kernel on ( X , m ) , where m is a finite measure . Then allconstant functions on X belong to L ( X , m ) and are fixed by P . It follows that k P k = P . Denote the orthogonal complement of the constantfunctions in L ( X , m ) by L ( X , m ) , i.e. , L ( X , m ) ≔ n f ∈ L ( X , m ) (cid:12)(cid:12) Z X f ( x ) d m ( x ) = o . Note that L ( X , m ) is P -invariant and that the spectrum of the restriction of P on L ( X , m ) is contained in [ − , ] . We denote the supremum of this spectrum by λ ∈ R . We makethe following definition: Definition 3.5.
A reversible Markov kernel on a finite measure space ( X , m ) is said tohave a spectral gap if λ < Π has a spectral gap if and onlyif P and the 1-eigenspace consists of constant functionson X . Equivalently, this happens if and only if ∆ = − P and the 0-eigenspace consists of constant functions. Obviously, we have that(3.5) 1 − λ = inf ( E ( f ) k f k m , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ∈ L ( X , m ) ) . In analogy with the notion of Cheeger constants for finite graphs, we define:
Definition 3.6.
The
Cheeger constant for a reversible Markov kernel Π on ( X , m ) is κ ≔ inf (cid:26) | ∂ Π ( A ) | m m ( A ) (cid:12)(cid:12)(cid:12)(cid:12) A ∈ E , < m ( A ) ≤ m ( X ) (cid:27) . We can now state the following theorem relating Cheeger constants and spectral gapsin the context of Markov kernels:
Theorem 3.7.
Let Π be a reversible Markov kernel on ( X , m ) where m is finite. Then κ ≤ − λ ≤ κ . As mentioned before, special cases of the above result are very well-known. Two suchinstances are given by simple random walks on finite or countably infinite graphs: the for-mer gives a spectral characterisation of expansion [2, 3, 11], while the latter characterisesnon-amenability [11, 23, 32]. In [22] Kaimanovich proved a version of Theorem 3.7 forreversible Markov kernels on infinite measure spaces (he actually proved much more re-fined results concerning p -capacities and Dirichlet norms). Lyon–Nazarov proved it forMarkov kernels arising from measure-preserving actions on probability spaces [29, Theo-rem 3.1] . Yet, we could not find an actual proof of Theorem 3.7 in the literature and thusprefer to provide one here. Our proof is modelled on the one for Markov processes on fi-nite state spaces [36, 59]: most of its key points generalise without difficulties (Lemma 3.8 [29, Theorem 3.1] also claims that the spectrum of the Markov operator is bounded away from −
1, butthis is not correct: there is a small mistake at the very end of their proof. It is also worth pointing out thattheir proof is based on an inequality which they claim holds true by “checking cases”. We are unable toverify such inequality.
MARKOVIAN AND ROE-ALGEBRAIC APPROACH TO ASYMPTOTIC EXPANSION 15 and Lemma 3.9), but the concluding argument is considerably more involved—especiallyfor the claimed bounds.
Lemma 3.8.
If g ∈ L ( X , m ) is a function that takes value in [ , ∞ ) and such that m ( { g > } ) ≤ m ( X ) , then E ( g ) ≥ κ k g k m , .Proof. By hypothesis, we have that Z X g ( x ) d m = Z ∞ m ( { g ≥ t } ) d t ≤ κ Z ∞ | ∂ Π ( { g ≥ t } ) | m d t , where κ denotes the Cheeger constant. Using (3.4) we deduce: Z X g ( x ) dm ≤ κ Z ∞ E ( χ { g ≥ t } ) d t . = κ Z ∞ Z X × X | χ { g ≥ t } ( x ) − χ { g ≥ t } ( y ) | d µ ( x , y ) d t = κ Z X × X (cid:18) Z ∞ | χ { g ≥ t } ( x ) − χ { g ≥ t } ( y ) | d t (cid:19) d µ ( x , y )= κ Z X × X | g ( x ) − g ( y ) | d µ ( x , y ) , thus proving the lemma. (cid:3) In turn, this is used to prove the estimate that lies at the heart of the proof of Theo-rem 3.7:
Lemma 3.9.
If g ∈ L ( X , m ) is a function that takes value in [ , ∞ ) and such that m ( { g > } ) ≤ m ( X ) , then E ( g ) ≥ κ k g k m , . Proof.
Firstly, we note that k g k m , = k g k m , . We can hence apply Lemma 3.8 to thefunction g to obtain(3.6) κ k g k m , ≤ E ( g ) . Using the Cauchy–Schwarz inequality, we can estimate the value E ( g ) as follows: E ( g ) = Z X × X | g ( x ) − g ( y ) | d µ ( x , y )= Z X × X | g ( x ) − g ( y ) | · | g ( x ) + g ( y ) | d µ ( x , y ) ≤ (cid:16) Z X × X | g ( x ) − g ( y ) | d µ ( x , y ) (cid:17) (cid:16) Z X × X | g ( x ) + g ( y ) | d µ ( x , y ) (cid:17) = (cid:0) E ( g ) (cid:1) (cid:16) k g k m , + h g , P g i m (cid:17) ≤ √ E ( g ) k g k m , . The proof is complete once we combine the above estimate with (3.6). (cid:3)
Finally, we are ready to prove the main theorem of this subsection:
Proof of Theorem 3.7.
Given a measurable A ⊆ X with 0 < m ( A ) ≤ m ( X ) , let f A ≔ χ A − m ( A ) m ( X ) be the projection of χ A to L ( X , m ) . Then k f A k m , = ( m ( X ) − m ( A )) m ( A ) m ( X ) ≥ m ( A ) . Using (3.4) we deduce that1 − λ ≤ E ( f A ) k f A k m , = | ∂ Π ( A ) | m k f A k m , ≤ | ∂ Π ( A ) | m m ( A ) , and hence 1 − λ ≤ κ .For the other direction, we need to show κ ≤ inf f ∈ L ( X , m ) h f , ∆ f i m k f k m , = inf f ∈ L ( X , m ) E ( f ) k f k m , = − λ . Since P is self-adjoint, the spectral theorem implies that there exists a sequence ofreal-valued functions f n ∈ L ( X , m ) with k f n k m , = k P f n − λ f n k m , → h f n , ∆ f n i → − λ . Write f n = f + n − f − n , where f + n ( x ) ≔ max { , f n ( x ) } and f − n ( x ) ≔ max { , − f n ( x ) } . Replacing f n with − f n if necessary, we can assume that m ( { f n ( x ) > } ) ≤ m ( X ) .If each f n was an eigenfunction for ∆ , we would immediately have(3.7) h f + n , ( ∆ f n ) + i m k f + n k m , = h f n , ∆ f n i m k f n k m , . In this case, the proof of the theorem would easily follow from Lemma 3.9. Yet, this neednot be the case for general Markov kernels. This is the place where our argument differsfrom the classical proof for finite-state processes.On the way to overcome this difficulty, we will first need to modify f n to ensure that k f + n k m , is bounded away from 0. If k f + n k m , does not tend to 0, we simply pass toa subsequence h n ≔ f k n so that k h + n k m , is bounded away from 0. Otherwise, we have k f + n k m , →
0. Since m is finite, we also have k f + n k m , →
0. On the other hand, k f + n k m , = k f − n k m , because f n ∈ L ( X , m ) . It follows that there exists a sequence c n > c n → m ( { f − n ( x ) ≥ c n } ) →
0. We then define h n ≔ − ( f n + c n ) and also note that k P h n − λ h n k m , ≤ k P f n − λ f n k m , + k P − λ k · k c n k → n → ∞ . For n large enough we have m ( { h + n ( x ) > } ) ≤ m ( X ) / k h + n k m , ≥ k f − n k m , − k c n k m , tends to 1, as 1 = k f + n k m , + k f − n k m , and k f + n k m , → h h + n , ( P h n ) + i m − λ k h + n k m , = h h + n , ( P h n ) + − λ h + n i m and by the Cauchy–Schwarz inequality, we have that |h h + n , ( P h n ) + − λ h + n i|k h + n k m , ≤ k ( P h n ) + − ( λ h n ) + k m , k h + n k m , ≤ k P h n − λ h n k m , k h + n k m , . Since k h + n k m , is bounded away from 0, the right hand side in the above inequality tendsto 0 and therefore 1 − λ = lim n → ∞ k h + n k m , − h h + n , ( P h n ) + i m k h + n k m , . MARKOVIAN AND ROE-ALGEBRAIC APPROACH TO ASYMPTOTIC EXPANSION 17
Finally, since h h + n , P ( h + n ) i m ≥ h h + n , ( P h n ) + i m , we deduce that1 − λ ≥ lim n → ∞ k h + n k m , − h h + n , P ( h + n ) i m k h + n k m , = lim n → ∞ h h + n , ∆ ( h + n ) i m k h + n k m , = lim n → ∞ E ( h + n ) k h + n k m , and the latter is greater or equal to κ by Lemma 3.9, as desired. (cid:3) Markov kernels from actions.
Let Γ y ( X , ν ) be a measure-class-preserving ac-tion. For every γ ∈ Γ and x ∈ X , let r ( γ , x ) ≔ d γ − ∗ ν d ν ( x ) be the Radon-Nikodym derivative.Note that r ( γ , x ) = r ( γ − , γ ( x )) − and for any measurable function f on X we have Z X f ( γ · x ) d ν ( x ) = Z X f ( x ) r ( γ − , x ) d ν ( x ) when the integrals exist. In particular, for every measurable Y ⊆ X we have(3.8) Z Y f ( γ · x ) r ( γ , x ) d ν ( x ) = Z γ ( Y ) f ( x ) r ( γ − , x ) d ν ( x ) when the integrals exist.Now fix a finite symmetric subset S ⊆ Γ containing the identity 1, and a measurablesubset Y ⊆ X (which might have infinite measure). For every x ∈ Y , let S Y , x ≔ { s ∈ S | s · x ∈ Y } and(3.9) σ Y , S ( x ) ≔ ∑ s ∈ S Y , x r ( s , x ) . Definition 3.10.
Let Γ y ( X , ν ) be a measure-class-preserving action. The normalisedlocal Markov kernel associated with Y and S is the Markov kernel on Y defined by Π Y , S ( x , - ) ≔ σ Y , S ( x ) ∑ s ∈ S Y , x r ( s , x ) δ s · x where δ y is the Dirac delta measure on the point y , and we denote the associated Markovoperator by P Y , S . We say that Π S ≔ Π X , S is the normalised Markov kernel associatedwith S .For later use, we record here an elementary but convenient integration formula: forevery measurable function G : S × Y → C we have(3.10) Z Y ∑ s ∈ S Y , x G ( s , x ) d ν ( x ) = Z Y ∑ s ∈ S χ { s − ( Y ) } ( x ) G ( s , x ) d ν ( x ) = ∑ s ∈ S Z Y ∩ s − ( Y ) G ( s , x ) d ν ( x ) (when the integral is defined).One of the key properties of normalised local Markov kernels is that they are reversible.In fact, consider the measure ˜ ν Y , S on Y defined byd ˜ ν Y , S ≔ σ Y , S · d ( ν | Y ) . In other words, ˜ ν Y , S is obtained by rescaling the restriction of ν to Y by the density func-tion σ Y , S . Then the following holds true: Proposition 3.11.
Let Γ y ( X , ν ) be a measure-class-preserving action and S be a finitesymmetric subset of Γ containing the identity . Then:(1) The measure ˜ ν Y , S is equivalent to the restriction ν | Y .(2) If ν ( Y ) is finite, then ν ( A ) ≤ ˜ ν Y , S ( A ) ≤ | S | p ν ( A ) ν ( Y ) for any measurable A ⊆ Y .In particular, in this case ˜ ν Y , S ( Y ) is also finite. (3) The measure ˜ ν Y , S is reversing for the normalised local Markov kernel Π Y , S . Theassociated measure µ on Y × Y —defined by (3.2)—is determined by the formula: (3.11) Z Y × Y F ( x , y ) d µ ( x , y ) = ∑ s ∈ S Z Y ∩ s − ( Y ) r ( s , x ) F ( x , s · x ) d ν ( x ) for every integrable function F on Y × Y .Proof. (1). Note that 0 < r ( s , x ) < ∞ for ν -almost every x ∈ X because the action ismeasure-class-preserving. Since S contains the identity 1, we know that S Y , x is non-emptyfor every x ∈ Y . It follows immediately that a measurable subset of Y is ν -null if and onlyif it is ˜ ν Y , S -null.(2). Since 1 ∈ S Y , x , we have ν ( A ) ≤ ˜ ν Y , S ( A ) for any measurable A ⊆ Y . On the otherhand, by (3.10) and the Cauchy–Schwarz inequality we have˜ ν Y , S ( A ) = ∑ s ∈ S Z A ∩ s − ( Y ) r ( s , x ) d ν ( x ) ≤ ∑ s ∈ S ν ( A ∩ s − ( Y )) (cid:0) Z A ∩ s − ( Y ) r ( s , x ) d ν ( x ) (cid:1) ≤ (cid:0) ∑ s ∈ S ν ( A ) (cid:1) · (cid:0) ∑ s ∈ S ν (( s · A ) ∩ Y ) (cid:1) ≤ | S | p ν ( A ) ν ( Y ) . (3). Let us first verify the formula for µ . By definition, for any measurable function F on Y × Y we have that Z Y × Y F ( x , y ) d µ ( x , y ) = Z Y Z Y F ( x , y ) Π Y , S ( x , d y ) d ˜ ν Y , S ( x )= Z Y σ Y , S ( x ) ∑ s ∈ S Y , x r ( s , x ) F ( x , s · x ) d ˜ ν Y , S ( x )= Z Y ∑ s ∈ S Y , x r ( s , x ) F ( x , s · x ) d ν ( x )= ∑ s ∈ S Z Y ∩ s − ( Y ) r ( s , x ) F ( x , s · x ) d ν ( x ) , where the last step follows from (3.10).In order to show that ˜ ν Y , S is reversing for Π Y , S , it suffices to prove: Z Y × Y F ( x , y ) d µ ( x , y ) = Z Y × Y F ( y , x ) d µ ( x , y ) for every measurable function F on Y × Y . From (3.11), we have that Z Y × Y F ( y , x ) d µ ( x , y ) = ∑ s ∈ S Z Y ∩ s − ( Y ) r ( s , x ) F ( s · x , x ) d ν ( x )= ∑ s ∈ S Z sY ∩ Y r ( s − , x ) F ( x , s − · x ) d ν ( x )= ∑ s ∈ S Z Y ∩ s − ( Y ) r ( s , x ) F ( x , s · x ) d ν ( x )= Z Y × Y F ( x , y ) d µ ( x , y ) , where we use (3.8) for the second equation, and use S = S − for the third one. (cid:3) Remark . The assumption that 1 ∈ S is only used to ensure that S Y , x is always non-emptyfor every x ∈ Y . This assumption can be dropped if one already knows, a priori , that S Y , x is MARKOVIAN AND ROE-ALGEBRAIC APPROACH TO ASYMPTOTIC EXPANSION 19 non-empty ( e.g. , Y is Γ -invariant). On the contrary, the condition that S = S − is essentialfor the proof of reversibility in Proposition 3.11.Having introduced the reversible normalised local Markov kernel Π Y , S , we would liketo apply the techniques developed in previous subsections to asymptotically expandingactions. Firstly, let us give the following definition: Definition 3.13.
Let Γ y ( X , ν ) be a measure-class-preserving action and Y ⊆ X be adomain. Let S ⊆ Γ be a finite symmetric subset with 1 ∈ S , then Y is called a domainof Markov S-expansion (for the action) if the associated normalised local Markov kernel Π Y , S has strictly positive Cheeger constant (see Definition 3.6). Y is called a domain ofMarkov expansion if it is a domain of Markov S -expansion for some finite symmetric S ⊆ Γ with 1 ∈ S .By Theorem 3.7, Y is a domain of Markov S -expansion if and only if the normalisedlocal Markov kernel Π Y , S has spectral gap. In other words, 1 is isolated in the spectrumof the Markov operator P Y , S and the 1-eigenspace consists of constant functions on X .When this is the case, the following lemma holds true. Lemma 3.14.
The Markov kernel Π Y , S has spectral gap if and only if the restriction ofthe Markov operator P Y , S on L ( Y , ˜ ν Y , S ) has norm strictly less than .Proof. The proof follows from direct calculations and a classical argument using the uni-form convexity of Hilbert spaces and is hence left to the reader (note that the spectrum of P Y , S is bounded away from − ∈ S ). (cid:3) The following result provides the connection between expansion in measure (Definition2.2) and Markov expansion (Definition 3.13) under an assumption of bounded Radon–Nikodym derivatives:
Lemma 3.15.
Let Γ y ( X , ν ) be a measure-class-preserving action, Y ⊆ X be a domainand S be a finite symmetric subset of Γ containing the identity. If there is a constant Θ ≥ such that / Θ ≤ r ( s , x ) ≤ Θ for every x ∈ Y and s ∈ S Y , x , then Y is a domain ofS-expansion if and only if it is a domain of Markov S-expansion.Proof. By definition and (3.11), for any measurable subset A ⊆ Y we have that | ∂ Π Y , S ( A ) | ˜ ν Y , S = Z Y × Y χ A ( x ) χ Y r A ( y ) d µ ( x , y ) = ∑ s ∈ S Z Y ∩ s − ( Y ) r ( s , x ) χ A ( x ) χ Y r A ( s · x ) d ν ( x )= ∑ s ∈ S Z ( A r s − ( A )) ∩ s − ( Y ) r ( s , x ) d ν ( x ) = ∑ s ∈ S Z ( s · A r A ) ∩ Y r ( s − , x ) d ν ( x ) , where the last equality uses (3.8).Hence, it follows from the assumption on r that(3.12) 1 √ Θ | ∂ Π Y , S ( A ) | ˜ ν Y , S ≤ ∑ s ∈ S ν (cid:0) ( s · A r A ) ∩ Y (cid:1) ≤ √ Θ | ∂ Π Y , S ( A ) | ˜ ν Y , S . Moreover, it follows by the definition of ˜ ν Y , S that(3.13) ν ( A ) ≤ ˜ ν Y , S ( A ) ≤ | S | √ Θ · ν ( A ) for any measurable subset A ⊆ Y .Now we assume that Y is a domain of ( c , S ) -expansion for some constant c >
0. Fix ameasurable subset A ⊆ Y with 0 < ˜ ν Y , S ( A ) ≤ ˜ ν Y , S ( Y ) . In particular, both ν ( A ) > ν ( Y r A ) > If ν ( A ) ≤ ν ( Y ) , it follows from Definition 2.2 and (3.12) that c ν ( A ) < ν (( S · A r A ) ∩ Y ) ≤ ∑ s ∈ S ν (( s · A r A ) ∩ Y ) ≤ √ Θ | ∂ Π Y , S ( A ) | ˜ ν Y , S . Together with (3.13), we conclude that | ∂ Π Y , S ( A ) | ˜ ν Y , S > c | S | Θ ˜ ν Y , S ( A ) . If ν ( A ) > ν ( Y ) , we can apply the same argument to Y r A and deduce from (3.3) that | ∂ Π Y , S ( A ) | ˜ ν Y , S = | ∂ Π Y , S ( Y r A ) | ˜ ν Y , S > c | S | Θ ˜ ν Y , S ( Y r A ) ≥ c | S | Θ ˜ ν Y , S ( A ) . Thus, Y is a domain of Markov S -expansion as desired.The proof of the converse implication is similar. Let κ > Π Y , S and fix any measurable subset A ⊂ Y with0 < ν ( A ) ≤ ν ( Y ) .If ˜ ν Y , S ( A ) ≤ ˜ ν Y , S ( Y ) , then (3.12) implies that ν ( ∂ Γ S A ∩ Y ) ≥ | S | ∑ s ∈ S ν (cid:0) ( s · A r A ) ∩ Y (cid:1) ≥ | S | √ Θ | ∂ Π Y , S ( A ) | ˜ ν Y , S ≥ κ | S | √ Θ ˜ ν Y , S ( A ) , where ∂ Γ S A = S · A r A . Together with (3.13) we obtain that ν ( ∂ Γ S A ∩ Y ) ≥ κ | S | √ Θν ( A ) . If ˜ ν Y , S ( A ) > ˜ ν Y , S ( Y ) , then ∂ Γ S A ∩ Y ⊇ ∂ Γ S ( Y r S · A ) ∩ Y implies that ν ( ∂ Γ S A ∩ Y ) ≥ ν (cid:0) ∂ Γ S ( Y r S · A ) ∩ Y (cid:1) ≥ κ | S | √ Θν ( Y r S · A ) . Moreover, using 1 ∈ S we note that ν ( Y r S · A ) = ν ( Y ) − ν ( A ) − ν ( ∂ Γ S A ∩ Y ) ≥ ν ( A ) − ν ( ∂ Γ S A ∩ Y ) . So it is easy to conclude that ν ( ∂ Γ S A ∩ Y ) ≥ κ | S | √ Θ + κ · ν ( A ) . This shows that Y is a domain of S -expansion for the action. (cid:3) Remark . The statement of Lemma 3.15 is an analogue of the fact that for graphswith bounded degrees, there are bounds between edge-expansion and vertex-expansion.More precisely, the Cheeger constant of the normalised (local) Markov kernel should beregarded as the “measured” Cheeger constant of the edge-expansion, while the notion ofexpansion in measure is clearly an analogue of the (exterior) vertex-expansion for graphs.The assumption that the Radon-Nikodym derivatives are bounded corresponds to that thegraphs have bounded degree.Consequently, we obtain an alternative and direct proof for [16, Lemma 5.2]:
Corollary 3.17 ([16, Lemma 5.2]) . Let Γ y ( X , ν ) be a measure-preserving action andY ⊆ X a domain. Then Y is a domain of expansion if and only if the action has localspectral gap with respect to Y .
MARKOVIAN AND ROE-ALGEBRAIC APPROACH TO ASYMPTOTIC EXPANSION 21
Proof.
Using indicator functions, it is easy to see that the existence of a local spectralgap implies that Y is a domain of expansion. Hence, we only focus on the converseimplication.Let Y be a domain of ( c , k ) -expansion and let S ≔ B k . Then the normalised localMarkov kernel Π Y , S has a spectral gap by Theorem 3.7 and Lemma 3.15. Using (3.11) inProposition 3.11, we obtain that for every g ∈ L ( Y , ˜ ν Y , S ) :(3.14) ( − λ ) · k g k ν Y , S , ≤ E ( g ) = ∑ s ∈ S Z Y ∩ s − ( Y ) | g ( x ) − g ( s · x ) | d ν ( x ) . where 1 − λ is bounded away from zero (Definition 3.5).Now we fix an f ∈ L ( X , ν ) with R Y f d ν =
0. Then f | Y ∈ L ( Y , ˜ ν Y , S ) and g ≔ f | Y − R Y f | Y d ˜ ν Y , S ∈ L ( Y , ˜ ν Y , S ) by construction and Proposition 3.11(2). Thus, it follows from S = S − and (3.14) that ∑ s ∈ S k s · f − f k ν | Y , ≥ (cid:16) ∑ s ∈ S Z Y | f ( x ) − f ( s · x ) | d ν ( x ) (cid:17) ≥ ( E ( g )) ≥ p ( − λ ) · k g k ˜ ν Y , S , . Moreover, since R Y f d ν = k g k ν Y , S , ≥ k g k ν | Y , = k f k ν | Y , + ν ( Y ) (cid:16) Z Y f | Y d ˜ ν Y , S (cid:17) ≥ k f k ν | Y , . Combining the above inequalities we conclude that ∑ s ∈ S k s · f − f k ν | Y , ≥ p ( − λ ) · k f k ν | Y , , as required. (cid:3) Remark . Note that the proof of Corollary 3.17 holds also for non-measure-preservingactions as long as the action is measure-class-preserving and there is a uniform upperbound Θ ≥ r ( s , x ) . This can be used to provide analternative proof for [20, Theorem 3.2].3.4. Markov expansion and the structure of strongly ergodic actions.
Now we arein the position to prove the Markovian analogue of the structure theorem for stronglyergodic actions (Theorem 2.13). Let us start with the following local version (comparewith Proposition 2.12):
Proposition 3.19.
Let ρ : Γ y ( X , ν ) be a measure-class-preserving action. If Y ⊆ X isa domain of asymptotic expansion, then Y admits an exhaustion by domains Y n of MarkovS ( n ) -expansion such that for each n ∈ N there is a constant Θ n ≥ such that / Θ n ≤ r ( s , y ) ≤ Θ n for every y ∈ Y n and s ∈ S ( n ) Y n , y .Moreover, if Y ⊆ X is a domain of S-asymptotic expansion, then Y admits an exhaustionby domains Y n of Markov S-expansion.Proof. It follows from Proposition 2.12 that there exists an exhaustion Y ( k ) ր Y by do-mains of S ( k ) -expansion in measure. Without loss of generality, we can assume that S ( k ) is symmetric, 1 ∈ S ( k ) and S ( k ) ⊆ S ( k + ) for every k ∈ N . Let Z ( k ) m ≔ (cid:8) y ∈ Y ( k ) (cid:12)(cid:12) r ( s , y ) < m or r ( s , y ) > m for some s ∈ S ( k ) Y ( k ) , y (cid:9) . Since each Y ( k ) has finite measure and the action is measure-class-preserving, for every k ∈ N we have ν ( Z ( k ) m ) → m → ∞ . Hence, we can choose for every n ∈ N a sequence of integers ( m ( n ) k ) k ∈ N such that ∑ k ∈ N ν (cid:0) Z ( k ) m ( n ) k (cid:1) ≤ n . Let e Z n ≔ [ k ∈ N Z ( k ) m ( n ) k , then we have ν ( e Z n ) → n → ∞ . We can further assume that m ( n + ) k ≥ m ( n ) k for every n ∈ N so that e Z n + ⊆ e Z n .Now for every k , it follows from Proposition 2.12 that Y ( k ) admits an exhaustion Y ( k ) l ր Y ( k ) by domains of S ( k ) -expansion such that Y ( k ) l ∩ e Z l = /0. By a diagonal argument, thereexists a sequence ( l k ) k ∈ N such that Y ( k ) l k converges in measure to Y . Let Y n ≔ n [ k = Y ( k ) l k . Ignoring finitely many k if necessary, we can assume that ν ( Y ( ) l ) ≥ ν ( Y ) . Then weconclude from Lemma 2.3 that each Y n is a domain of S ( n ) -expansion. Since Y n ∩ e Z n = /0,it follows from Lemma 3.15 that it is also a domain of Markov S ( n ) -expansion. The secondstatement is obtained by the special case where S ( k ) = S for all k ∈ N . (cid:3) Remark . The main technical difficulty in the previous proof is to obtain increasing sequences of domains of Markov expansion. This is largely due to the fact that it is hardto control Markov S -expansion as the set S varies. More precisely, choosing different S could yield widely different measures ˜ ν Y , S and this would in turn influence the Cheegerconstant of Π Y , S .An alternative approach to Proposition 3.19 would be to go through the proof of Propo-sition 2.12 and reprove it using the language of Markov expansion.It is now simple to prove a structure result for strongly ergodic actions in terms ofMarkovian expansion: Theorem 3.21.
Let ρ : Γ y ( X , ν ) be a measure-class-preserving action. Then ρ isstrongly ergodic if and only if every domain Y ⊆ X admits an exhaustion by domainsof Markov expansion.Proof. Necessity : This follows from Theorem 2.13 “(1) ⇒ (2)” and Proposition 3.19. Sufficiency : It follows from the same argument as in the proof of [26, Theorem 4.9“(5) ⇒ (6)”] that ρ must be ergodic. By Theorem 2.13 “(3) ⇒ (1)”, it is hence enough toshow that X admits a domain of expansion.We can choose a domain Y ⊆ X for which there exist constants C ( γ ) ≥ γ ∈ Γ such that C ( γ ) − ≤ r ( γ , y ) ≤ C ( γ ) for every y ∈ Y (such a domain can be constructedusing an argument similar to that in the proof of Proposition 3.19). By the hypothesis,there is an exhaustion Y n ր Y by domains of Markov expansion. Finally, Lemma 3.15implies that any such Y n produces the desired domain of expansion in measure. (cid:3)
4. W
ARPED CONES AND FINITE ( DYNAMICAL ) PROPAGATION APPROXIMATIONS
The aim of this section is to introduce warped cones associated with group actionson metric measure spaces and to study the effects of asymptotic expansion on the ana-lytic properties of said warped cones. More precisely, adapting the techniques in [24]
MARKOVIAN AND ROE-ALGEBRAIC APPROACH TO ASYMPTOTIC EXPANSION 23 to the context of group actions and using the structure results in Subsection 3.4, we cancharacterise asymptotic expansion in terms of finite propagation approximations of theDru¸tu–Nowak projections. As an intermediate bridge, we introduce dynamical versionsof quasi-locality and finite propagation approximation to connect actions and projectionson warped cones. In turn, this allows us to construct a multitude of non-compact ghostprojections which will be used in Section 5 to construct counterexamples to the coarseBaum–Connes conjecture.4.1.
Preliminaries on warped cones.
Recall that the countable group Γ is equipped witha proper length function ℓ . Let ( X , d ) be a metric space and ρ : Γ y X be a continuousaction. For every t ≥ d t be the rescaling of d by t , i.e. , d t ( x , y ) ≔ td ( x , y ) . Definition 4.1.
The warped cone associated with the action Γ y X is the family of metricspaces W C ( Γ y X ) ≔ { ( X , d t Γ ) | t ∈ [ , ∞ ) } , where d t Γ is the largest metric such that d t Γ ≤ d t and d t Γ ( x , γ · x ) ≤ ℓ ( γ ) for every x ∈ X and γ ∈ Γ .If the diameter of ( X , d ) is at most 2, we can also define the unified warped cone as themetric space ( O Γ X , d Γ ) , where O Γ X = X × [ , ∞ ) as a set and d Γ (cid:0) ( x , t ) , ( x , t ) (cid:1) ≔ d t ∧ t Γ ( x , x ) + | t − t | where t ∧ t = min { t , t } . The requirement on the diameter is necessary to ensure that d Γ is a metric.We will also need the following: Lemma 4.2.
Let Γ y ( X , d ) be a continuous action and R > fixed. Given A ⊆ X , letN R ( A ; d t Γ ) ⊆ X be the closed R-neighbourhood of A with respect to the metric d t Γ . Then \ t ≥ N R ( A ; d t Γ ) = B R · A . Proof.
It is clear that B R · A is contained in N R ( A ; d t Γ ) for every t ≥
1. For the converse, itsuffices to prove it for A closed. If R <
1, we see that N R ( A ; d t Γ ) = N R ( A ; d t ) = N R / t ( A ; d ) ,because ℓ only takes integer values. So the result holds trivially. If R ≥
1, first note thatfor every fixed γ ∈ Γ we have \ t ≥ γ · N R ( A ; d t Γ ) = γ ( A ) . Assume now that 1 ≤ R < N R ( A ; d t Γ ) ⊆ N R ( A ; d t ) ∪ N R − (cid:0) B · N R − ( A ; d t ) ; d t Γ (cid:1) . For every fixed t >
0, the set B · N R − ( A ; d t ) is closed. Since R − <
1, we deduce asbefore that \ t ≥ N R − (cid:0) B · N R − ( A ; d t ) ; d t Γ (cid:1) ⊆ \ t > t N R − (cid:0) B · N R − ( A ; d t ) ; d t (cid:1) = B · N R − ( A ; d t ) and therefore \ t ≥ N R − (cid:0) B · N R − ( A ; d t ) ; d t Γ (cid:1) ⊆ \ t > B · N R − ( A ; d t ) = B · A . This shows the the right hand side of (4.1) shrinks down to B · A as t goes to infinity.Thus, we have proved the claim for R <
2. An obvious inductive generalisation of thisargument proves the statement for every fixed R > (cid:3) For more details and elementary facts on the geometry of warped cones, we refer to[40, 44, 51, 56].4.2. (Dynamical) quasi-local characterisations for asymptotic expansion.
In this sub-section, we will introduce a notion of dynamical quasi-locality and explain its relationwith the ordinary quasi-locality for operators on warped cones. Using the dynamicalquasi-locality, we will study the Dru¸tu–Nowak projection associated to a warped cone,and show that the ordinary quasi-locality of this projection characterises asymptotic ex-pansion in measure.Let ρ : Γ y X be a continuous action on a metric space ( X , d ) of diameter at most 2.Let ν be a probability measure on ( X , d ) and λ be the Lebesgue measure on [ , ∞ ) . Equipthe unified warped cone O Γ X = X × [ , ∞ ) with the product measure ν × λ .For any measurable non-null Y ⊆ X , denote by P Y ∈ B ( L ( X , ν )) the averaging pro-jection on Y , which is the orthogonal projection onto the one-dimensional subspace in L ( X , ν ) spanned by χ Y . In other words, P Y f ≔ h f , ν ( Y ) · χ Y i χ Y , where f ∈ L ( X , ν ) . The Dru¸tu–Nowak projection (see [12, Section 6.c.]) is defined as G = P X ⊗ Id L ([ , ∞ )) ∈ B ( L ( O Γ X , ν × λ )) . In other words, it is the orthogonal projectiononto C ⊗ L ([ , ∞ ) , λ ) .Recall from [38, 39] that an operator T ∈ B ( L ( O Γ X , ν × λ )) is quasi-local if for every ε >
0, there exists an R > A , C ⊆ O Γ X with d Γ ( A , C ) > R we have k χ A T χ C k < ε . Analogously, a family of operators { T t } t ∈ [ , ∞ ) in B ( L ( X , ν )) is uniformly quasi-local on W C ( Γ y X ) if for every ε > R > t ∈ [ , ∞ ) and every pair of measurable subsets A , C ⊆ X with d t Γ ( A , C ) > R , we have k χ A T t χ C k < ε .Now we introduce the following dynamical analogue of quasi-locality for operators in B ( L ( X , ν )) where ( X , ν ) is a probability space with a Γ -action: Definition 4.3.
Let ρ : Γ y ( X , ν ) be an action on a probability space ( X , ν ) . An operator T ∈ B ( L ( X , ν )) is called ρ -quasi-local if for every ε > k ∈ N such thatfor any measurable subsets A , C ⊆ X with ν (( B k · A ) ∩ C ) =
0, we have k χ A T χ C k < ε (recall that B k = { γ ∈ Γ | ℓ ( γ ) ≤ k } ).Similarly to [25, Lemma 3.8], quasi-locality of the averaging projection P X can bedetected by the following calculation: Lemma 4.4.
For every measurable subsets A , C in X , we have that k χ A P X χ C k B ( L ( X , ν )) = p ν ( A ) ν ( C ) . Proof.
By direct calculations, we have that k χ A P X χ C k = sup k v k = k w k = |h χ A P X χ C v , w i| = sup k v k = k w k = |h P X χ C v , P X χ A w i| = sup k v k = k w k = (cid:12)(cid:12)(cid:10) h χ C v , i , h χ A w , i (cid:11)(cid:12)(cid:12) = sup k v k = k w k = |h v , χ C ih w , χ A ih , i|≤ p ν ( A ) ν ( C ) , where the last inequality follows from the Cauchy–Schwarz inequality. On the other hand,if we let v and w be the normalised characteristic functions of C , A respectively then wehave that h P X χ C v , P X χ A w i = p ν ( A ) ν ( C ) . (cid:3) MARKOVIAN AND ROE-ALGEBRAIC APPROACH TO ASYMPTOTIC EXPANSION 25
The following corollary is a dynamical analogue of [25, Proposition 3.9] and it is animmediate consequence of Lemma 4.4:
Corollary 4.5.
Let ρ : Γ y ( X , ν ) be an action on a probability space ( X , ν ) and P X bethe associated averaging projection on X . Then P X is ρ -quasi-local if and only iflim k → + ∞ sup (cid:8) ν ( A ) ν ( C ) (cid:12)(cid:12) A , C ⊆ X measurable with ν (( B k · A ) ∩ C ) = (cid:9) = . We are now ready to show that asymptotic expansion in measure can be characterisedby ρ -quasi-locality of the associated averaging projections. This is an analogue of [25,Theorem 3.11]. Proposition 4.6.
Let ρ : Γ y ( X , ν ) be an action on a probability space ( X , ν ) and P X be the associated averaging projection on X . Then ρ is asymptotically expanding if andonly if P X is ρ -quasi-local.Proof. Necessity: Suppose P X is not ρ -quasi-local, then by Corollary 4.5 we have: α ≔
12 lim k → + ∞ sup (cid:8) ν ( A ) ν ( C ) (cid:12)(cid:12) A , C ⊆ X measurable with ν (( B k · A ) ∩ C ) = (cid:9) > . In particular, ≤ − α <
1. Thus, we can choose a sequence ( A n , C n ) n ∈ N , where A n , C n ⊆ X are measurable subsets with ν (( B n · A n ) ∩ C n ) = ν ( A n ) ν ( C n ) ≥ α . Since ν ( A n ) ≤ ν ( C n ) ≤
1, both ν ( A n ) and ν ( C n ) are at least α . Furthermore, ν (( B n · A n ) ∩ C n ) = ν ( A n ∩ C n ) =
0. In particular, both ν ( A n ) and ν ( C n ) are notgreater than 1 − α for each n ∈ N .If the action was asymptotically expanding, then Definition 2.6 and Lemma 2.11 wouldimply that there exist constants b > h ∈ N such that for every measurable subset A ⊆ X with α ≤ ν ( A ) ≤ − α , we have ν ( B h · A ) > ( + b ) ν ( A ) . Let k ≔ mh , where m ≔ ⌈ log + b ( − α ) ⌉ . Then either ν ( B k · A ) > − α or we deduce by induction on m that ν ( B k · A ) > ( + b ) m ν ( A ) ≥ − α . Note that A n satisfies α ≤ ν ( A n ) ≤ − α for all n ∈ N . Hence for n ≥ k , we have ν ( B n · A n ) > − α . This is a contradiction to ν ( C n ) ≥ α and ν (( B n · A n ) ∩ C n ) = Sufficiency:
Assume that ρ is not asymptotically expanding. Then there exists α ∈ ( , ] such that for every n ∈ N there exists a measurable subset A n ⊆ X with α ≤ ν ( A n ) ≤ and ν ( B n · A n ) ≤ ν ( A n ) . For every n we have ν ( X r ( B n · A n )) = − ν ( B n · A n ) ≥ − ν ( A n ) ≥ . Hence, we have that ν ( A n ) · ν ( X r ( B n · A n )) ≥ α > , which implies that the limitlim n → + ∞ sup (cid:8) ν ( A ) ν ( C ) (cid:12)(cid:12) A , C ⊆ X measurable with ν (( B n · A ) ∩ C ) = (cid:9) ≥ α > . Hence, P X is not ρ -quasi-local by Corollary 4.5. (cid:3) We will now show that the dynamical quasi-locality completely determines the ordinaryquasi-locality for those operators of the (unified) warped cone that arise as transformationsof the base space. To be precise, consider the following ∗ -homomorphism:(4.2) Φ : B ( L ( X , ν )) → B ( L ( O Γ X , ν × λ )) , T T ⊗ Id L ([ , ∞ )) (note that the Dru¸tu–Nowak projection G equals to Φ ( P X ) ). We can then prove the fol-lowing: Proposition 4.7.
Let ( X , d ) be a metric space with diameter at most equipped with aprobability measure ν , and ρ : Γ y X be a continuous action. For any T ∈ B ( L ( X , ν )) ,we consider the following conditions:(1) T is ρ -quasi-local;(2) Φ ( T ) is quasi-local;(3) the family of operators T t ≡ T for t ∈ [ , ∞ ) is uniformly quasi-local on W C ( Γ y X ) .Then we have (1) ⇒ (2) ⇒ (3). Furthermore, if ν is Radon then they are all equivalent.Proof. (1) ⇒ (2): Fix an ε >
0. Since T is ρ -quasi-local, there exists k ∈ N such that forany measurable subsets A ′ , C ′ ⊆ X with ν (( B k · A ′ ) ∩ C ′ ) =
0, we have k χ A ′ T χ C ′ k < ε .Given a pair of measurable subsets A , C ⊆ O Γ X = X × [ , ∞ ) with d Γ ( A , C ) > k , we canwrite A = F t ∈ [ , ∞ ) A t × { t } and C = F t ∈ [ , ∞ ) C t × { t } , where A t , C t are measurable subsetsin X .For every ( x , t ) ∈ O Γ X = X × [ , ∞ ) and every γ ∈ Γ , we have d Γ (( γ · x , t ) , ( x , t )) ≤ ℓ ( γ ) .Since d Γ ( A , C ) > k , it follows that ν (( B k · A t ) ∩ C t ) = t ∈ [ , ∞ ) . Hence, weconclude that k χ A t T χ C t k < ε for every t ∈ [ , ∞ ) .For every ξ ∈ L ( O Γ X , ν × λ ) , we set ξ t ( x ) ≔ ξ ( x , t ) so that ξ t ∈ L ( X , ν ) for almostevery t ∈ [ , ∞ ) . Using Fubini’s Theorem, we obtain that k χ A Φ ( T ) χ C ξ k = Z O Γ X (cid:12)(cid:12)(cid:0) χ A ( T ⊗ Id L ([ , ∞ )) ) χ C ξ (cid:1) ( x , t ) (cid:12)(cid:12) d ( ν × λ )( x , t )= Z ∞ Z X (cid:12)(cid:12) ( χ A t T χ C t ξ t )( x ) (cid:12)(cid:12) d ν ( x ) d t = Z ∞ (cid:13)(cid:13) χ A t T χ C t ξ t (cid:13)(cid:13) d t ≤ Z ∞ ε k ξ t k d t = ε k ξ k . It follows that Φ ( T ) is quasi-local. (2) ⇒ (3): For any measurable subsets A , C ⊆ X , we note that d t Γ ( A , C ) = d Γ ( A × [ t , t + ] , C × [ t , t + ]) . For every f ∈ L ( X , ν ) and t ∈ [ , ∞ ) , we construct a F t ∈ L ( O Γ X , ν × λ ) by letting F t ( x , s ) = f ( x ) if t ≤ s ≤ t + k f k ν = k F t k ν × λ and k χ A T t χ C f k = k χ A T χ C f k = k χ A × [ t , t + ] Φ ( T ) χ C × [ t , t + ] F t k . Now the rest of the proofis obvious. (3) ⇒ (1): Fix an ε >
0. Then by the assumption that there exists an R > t ∈ [ , ∞ ) and measurable subsets A , C ⊆ X with d t Γ ( A , C ) > R we have k χ A T χ C k < ε . We will verify that k χ A T χ C k < ε for measurable subsets A , C ⊆ X with ν (( B R · A ) ∩ C ) = A , C ⊆ X are compact subsets such that ( B R · A ) ∩ C = /0. It followsfrom Lemma 4.2 that \ t ≥ N R ( A ; d t Γ ) ∩ C = ( B R · A ) ∩ C = /0 . MARKOVIAN AND ROE-ALGEBRAIC APPROACH TO ASYMPTOTIC EXPANSION 27
Since C is compact and all N R ( A ; d t Γ ) are closed, we deduce that N R ( A ; d t Γ ) ∩ C = /0 forsome t large enough. This means that d t Γ ( A , C ) > R and hence k χ A T χ C k < ε by thehypothesis.For general measurable subsets A , C ⊆ X with ν (( B R · A ) ∩ C ) =
0, replacing C by C r ( B R · A ) if necessary (which only differ by a null set) we may assume that ( B R · A ) ∩ C = /0. Since the measure ν is Radon and finite, there exist increasing sequences of compactsubsets { A n ⊆ A } n ∈ N and { C n ⊆ C } n ∈ N such that lim n → ∞ ν ( A r A n ) = n → ∞ ν ( C r C n ) = A fortiori , we have ( B R · A n ) ∩ C n = /0 and it follows from the discussion in the secondparagraph that k χ A n T χ C n k < ε for all n ∈ N . Thus, k χ A T χ C k = sup n k χ A n T χ C n k < ε . (cid:3) Combining Proposition 4.6 with Proposition 4.7 implies the desired characterisation ofasymptotic expansion in measure in terms of quasi-locality:
Theorem 4.8.
Let ( X , d ) be a metric space with diameter at most equipped with a Radonprobability measure ν , and ρ : Γ y ( X , d ) be a continuous action. If P X is the associatedaveraging projection on X and G = P X ⊗ Id L ([ , ∞ )) is the Dru¸tu–Nowak projection, thenthe following are equivalent:(1) ρ is asymptotically expanding;(2) P X is ρ -quasi-local;(3) G is quasi-local;(4) the family of operators ( P X ) t ≡ P X for t ∈ [ , ∞ ) is uniformly quasi-local on W C ( Γ y X ) . Projections approximated by finite dynamical propagation operators.
In the pre-vious subsection, we showed that asymptotic expansion in measure can be characterisedby (dynamical) quasi-locality of certain projections (Theorem 4.8). In the same spirit of[24, Section 6], we would like to connect (dynamical) quasi-locality with finite (dynami-cal) propagation operators.In doing so, we will show that unified warped cones arising from asymptotically ex-panding actions admit plenty of projections which can be approximated by finite propa-gation operators. This greatly generalise [12, Theorem 6.6]. It will turn out that all ofthese projections lie outside the image of the coarse Baum–Connes assembly map. Wewill return to these aspects in Section 5.We once again introduce a dynamical analogue of an analytic property of operators,namely, the dynamical propagation (see Subsection 4.4 for the notion of ordinary finitepropagation operators):
Definition 4.9.
Let ρ : Γ y ( X , ν ) be an action on a probability space ( X , ν ) . We saythat an operator T ∈ B ( L ( X , ν )) has finite ρ -propagation if there is a k ∈ N such that χ A T χ C = A , C ⊆ X with ν (( B k · A ) ∩ C ) =
0. The smallest k satisfying the above condition is called the ρ -propagation of T .Throughout the rest of this subsection, let Γ y ( X , ν ) be a measure-class-preservingaction, Y ⊆ X be a domain and S ⊆ Γ be a finite symmetric set containing the iden-tity. Recall from Proposition 3.11 that such action induces a normalised local Markovkernel Π Y , S on Y (Definition 3.10). This kernel is reversible with a reversing measure˜ ν Y , S , where d ˜ ν Y , S = σ Y , S d ( ν | Y ) for the function σ Y , S defined in (3.9). We denote by P Y , S ∈ B ( L ( Y , ˜ ν Y , S )) and ∆ Y , S = − P Y , S ∈ B ( L ( Y , ˜ ν Y , S )) the Markov and Laplacianoperators associated with Π Y , S , respectively.We will present two different ways to produce projections of finite ρ -propagation usingMarkov S -expansion. One is normalised to better accommodate the associated Markov kernel, while the other is non-normalised and more related to the original averaging pro-jection P X . In either case, our construction relies heavily on the techniques developed inSection 3.4.3.1. Normalised projections.
Let ˜ P Y , S ∈ B ( L ( Y , ˜ ν Y , S )) be the orthogonal projectiononto constant functions on Y (this need not coincide with P Y , as the projection is takenwith respect to the inner product h· , ·i ˜ ν Y , S ). Let us consider the isometric embedding b I Y , S : L ( Y , ˜ ν Y , S ) ֒ → L ( X , ν ) defined by pointwise multiplication by the function √ σ Y , S on Y and then extending by 0on X r Y . This induces the following adjoint ∗ -homomorphism:(4.3) c Ad : B ( L ( Y , ˜ ν Y , S )) → B ( L ( X , ν )) , by T b I Y , S ◦ T ◦ ( b I Y , S ) ∗ . Note that b I Y , S ( ) = √ σ Y , S , where 1 is the constant function 1 in L ( Y , ˜ ν Y , S ) and σ Y , S isdefined to be 0 on every x ∈ X r Y . It follows that b P Y , S ≔ c Ad ( ˜ P Y , S ) ∈ B ( L ( X , ν )) is the orthogonal projection onto the 1-dimensional subspace of L ( X , ν ) spanned by thevector √ σ Y , S . We also transfer the Markov operator P Y , S ∈ B ( L ( Y , ˜ ν Y , S )) to b P Y , S ≔ c Ad ( P Y , S ) ∈ B ( L ( X , ν )) .Now the techniques developed in Section 3 can be used to prove the following: Proposition 4.10.
Let ρ : Γ y ( X , ν ) be a measure-class-preserving action and Y ⊆ X bea domain of Markov S-expansion (Definition 3.13). Then the associated projection b P Y , S ∈ B ( L ( X , ν )) is a norm limit of operators ( b P Y , S ) n , which all have finite ρ -propagation.Proof. Since the operator b P Y , S has ρ -propagation at most max { ℓ ( s ) | s ∈ S } , all of itspowers ( b P Y , S ) n have finite ρ -propagation as well. By Theorem 3.7 and Lemma 3.14, theMarkov operator P Y , S on L ( Y , ˜ ν Y , S ) has spectrum contained in [ − + ε , − ε ] ∪ { } forsome ε >
0. It follows that the sequence ( P Y , S ) n converges (as n → ∞ ) in the operatornorm to the projection onto the 1-eigenspace, which is exactly the projection ˜ P Y , S . Since c Ad is a ∗ -homomorphism, k ( b P Y , S ) n − b P Y , S k → n → ∞ . This finishes the proof. (cid:3)
Theorem 3.21 shows that strongly ergodic actions provide plenty of domains of Markovexpansion. We can hence use Proposition 4.10 as an abundant source of projections whichcan be approximated by finite ρ -propagation operators.As a corollary to Proposition 4.10 we also recover the following result by Dru¸tu andNowak: Corollary 4.11 ([12, Theorem 6.6] ) . Let ρ : Γ y ( X , ν ) be a measure-preserving ac-tion on a probability space ( X , ν ) . Suppose that ρ has spectral gap, then the averagingprojection P X is a norm limit of operators with finite ρ -propagation.Proof. Since the action is measure preserving, we have ˜ ν X , S = | S | · ν . It follows that b P X , S = P X for any choice of S ⊆ Γ . If ρ has spectral gap, then X is a domain of expansion(see e.g. Corollary 3.17). Hence, X is a domain of Markov expansion by Lemma 3.15 andwe can apply Proposition 4.10 to conclude the proof. (cid:3) Strictly speaking, [12, Theorem 6.6] concerns the operator G = P X ⊗ Id L ([ , ∞ )) and also shows that itis “ghost”. We will recover these facts in Section 5. MARKOVIAN AND ROE-ALGEBRAIC APPROACH TO ASYMPTOTIC EXPANSION 29
Non-normalised projections.
Now we move on to the second construction, wherewe show that the averaging projections P Y are norm limits of operators with finite ρ -prop-agation as well. Unlike the previous construction, these projections will not be limitsof powers of a fixed Markov operator. Instead, we will apply our structure theory forasymptotically expanding actions to produce appropriate sequences of operators.We define a different embedding I Y , S : L ( Y , ˜ ν Y , S ) ֒ → L ( X , ν ) simply by extending each function in L ( Y , ˜ ν Y , S ) by 0 on X r Y . In general, I Y , S is notisometric and may even be unbounded.Assume now that there exists Θ ≥ / Θ ≤ r ( s , y ) ≤ Θ for every y ∈ Y and s ∈ S Y , y . Under this assumption, it is clear that I Y , S is bounded. So it induces the followingadjoint map:Ad : B ( L ( Y , ˜ ν Y , S )) → B ( L ( X , ν )) , by T I Y , S ◦ T ◦ ( I Y , S ) ∗ . Note that— while being a bounded linear map preserving ∗ -operations—the adjoint mapAd might not be multiplicative.As before, let ˜ P Y , S ∈ B ( L ( Y , ˜ ν Y , S )) be the orthogonal projection onto constant func-tions, while P Y ∈ B ( L ( X , ν )) is the orthogonal projection onto the one-dimensional sub-space in L ( X , ν ) spanned by χ Y . Since ( I Y , S ) ∗ ( g ) = σ Y , S g | Y for g ∈ L ( X , ν ) , we havethat(4.4) Ad ( ˜ P Y , S ) = ν ( Y ) ˜ ν Y , S ( Y ) P Y . We prove the following:
Lemma 4.12.
Let ρ : Γ y ( X , ν ) be a measure-class-preserving action and Y ⊆ X be adomain of Markov S-expansion. Assume further that there exists Θ ≥ such that / Θ ≤ r ( s , y ) ≤ Θ for every y ∈ Y and s ∈ S Y , y . Then the averaging projection P Y ∈ B ( L ( X , ν )) is a norm limit of operators with finite ρ -propagation.Proof. By Theorem 3.7 and Lemma 3.14, the Markov operator P Y , S on L ( Y , ˜ ν Y , S ) hasspectrum contained in [ − + ε , − ε ] ∪ { } for some ε >
0. Hence, ( P Y , S ) n converges inthe operator norm to the projection ˜ P Y , S in B ( L ( Y , ˜ ν Y , S )) as n → ∞ .Since the embedding I Y , S is bounded, we obtain that I Y , S ◦ ( P Y , S ) n ◦ ( I Y , S ) ∗ = Ad (( P Y , S ) n ) → Ad ( ˜ P Y , S ) = ν ( Y ) ˜ ν Y , S ( Y ) P Y in B ( L ( X , ν )) , where the last equality comes from (4.4). Since each I Y , S ◦ ( P Y , S ) n ◦ ( I Y , S ) ∗ has ρ -propagationbounded by n · max { ℓ ( s ) | s ∈ S } , so the conclusion holds. (cid:3) Unlike Proposition 4.10, Lemma 4.12 concerns projections that do not depend on the fi-nite symmetric set S . This allows us to prove a result for domains of asymptotic expansionas well: Proposition 4.13.
Let ρ : Γ y ( X , ν ) be a measure-class-preserving action. Then forany domain Y ⊆ X of asymptotic expansion, the averaging projection P Y is a norm limitof operators with finite ρ -propagation.Proof. From Proposition 3.19, it follows that there is an exhaustion Y n ր Y by domainsof Markov S ( n ) -expansion such that for every n ∈ N there is a Θ n ≥ / Θ n ≤ r ( s , y ) ≤ Θ n for every y ∈ Y n and s ∈ S ( n ) Y n , y . Now it follows from Lemma 4.12 that each P Y n is a norm limit of operators with finite ρ -propagation. Since Y n increasingly converges to Y in measure and Y has finite measure,then P Y n converges to P Y in the operator norm. Hence, a diagonal argument will concludethe proof. (cid:3) It follows easily from the definitions that norm limits of operators with finite ρ -prop-agation are ρ -quasi-local. Hence, combining Proposition 4.6 with Proposition 4.13 weimmediately obtain the following: Corollary 4.14.
Let ρ : Γ y ( X , ν ) be a measure-class-preserving action on a probabilityspace ( X , ν ) . Then ρ is asymptotically expanding if and only if P X is a norm limit ofoperators with finite ρ -propagation. Characterising asymptotic expansion by finite propagation approximations.
Fi-nally, we conclude this section by combining results in Subsections 4.2 and 4.3 to provethat an action is asymptotically expanding if and only if the Dru¸tu–Nowak projection canbe approximated by operators with finite propagation.Let ( X , d ) be a metric space of diameter at most 2, ρ : Γ y X be a continuous action and O Γ X the associated unified warped cone. If X is equipped with a probability measure ν ,we give O Γ X the product measure ν × λ and say that an operator T ∈ B ( L ( O Γ X , ν × λ )) has finite propagation if there exists an R > A , C ⊆ O Γ X with d Γ ( A , C ) > R , we have χ A T χ C = Proposition 4.15.
Let ( X , d ) be a metric space with diameter at most equipped with aprobability measure ν , and ρ : Γ y X be a continuous action. If T ∈ B ( L ( X , ν )) hasfinite ρ -propagation, then Φ ( T ) has finite propagation. If in addition ν is Radon, theconverse implication holds as well.Proof. The argument is identical to that of Proposition 4.7 with ε = (cid:3) Since norm limits of operators with finite propagation are quasi-local, we can combineProposition 4.15 and Corollary 4.14 with Theorem 4.8 ( ) ⇒ ( ) to obtain a dynamicalcounterpart of [24, Theorem C]: Theorem 4.16.
Let ( X , d ) be a metric space with diameter at most equipped with aprobability measure ν , and ρ : Γ y X be a continuous measure-class-preserving action.The following are equivalent:(1) ρ is asymptotically expanding;(2) the averaging projection P X is a norm limit of operators with finite ρ -propagation;(3) the Dru¸tu–Nowak projection G is a norm limit of operators with finite propaga-tion. For later use, we record that we can apply Proposition 4.15 to the projections con-structed in Proposition 4.10 and Proposition 4.13 and obtain the following:
Corollary 4.17.
Let ( X , d ) be a metric space with diameter at most equipped with aprobability measure ν , and ρ : Γ y X be a continuous and measure-class-preservingaction. Let P ∈ B ( L ( X , ν )) be one of the following rank-one projection:(1) P = b P Y , S for a domain Y ⊆ X of Markov S-expansion;(2) P = P Y for a domain Y ⊆ X of asymptotic expansion.Then the projection Φ ( P ) = P ⊗ Id L ([ , ∞ )) ∈ B ( L ( O Γ X , ν × λ )) is a norm limit of oper-ators with finite propagation. MARKOVIAN AND ROE-ALGEBRAIC APPROACH TO ASYMPTOTIC EXPANSION 31
5. T
HE COARSE B AUM –C ONNES C ONJECTURE
In this section, we will use the projections constructed in Section 4.3 to provide newcounterexamples to the coarse Baum–Connes conjecture. These arise from certain warpedcones associated with asymptotically expanding actions. We will follow the outline of [43,Section 3] (the origin of this method goes back to [17] and [57]).Throughout this section, ( X , d ) will be a compact metric space with diameter at most 2endowed with a non-atomic probability measure ν of full support ( i.e. , every singleton hasmeasure zero and every open set has positive measure). As usual, Γ is a countable discretegroup with a proper length function ℓ . Furthermore, Γ y ( X , d , ν ) will be a continuousmeasure-class-preserving action.5.1. Roe algebras and projections.
Let us begin by recalling some basic notions con-cerning Roe algebras.Let ( Y , d ) be any proper metric space. In particular, Y is locally compact and σ -compact. Let C ( Y ) be the C ∗ -algebra of continuous functions on Y vanishing at infinity.A non-degenerate ∗ -representation C ( Y ) → B ( H ) on some separable Hilbert space H is called ample if no non-zero element of C ( Y ) acts as a compact operator on H . Anoperator a ∈ B ( H ) has finite propagation if there is r > f ag = f , g ∈ C ( Y ) satisfy d ( supp ( f ) , supp ( g )) > r . Moreover, an operator a ∈ B ( H ) is called locally compact if f a and a f are compact for all f ∈ C ( Y ) .The algebraic Roe algebra C [ Y ] of Y is the ∗ -algebra of locally compact finite propa-gation operators in B ( H ) , and the Roe algebra C ∗ ( Y ) of Y is the norm-closure of C [ Y ] in B ( H ) . Note that the Roe algebra C ∗ ( Y ) does not depend on the choice of the non-degenerate ample ∗ -representation of C ( Y ) , but only up to non-canonical ∗ -isomorphism(see e.g. [58, Remark 5.1.13]). On the other hand, the K -theory groups K ∗ ( C ∗ ( Y )) donot depend on the choice of such representations up to canonical ∗ -isomorphism (see e.g. [58, Theorem 5.1.15]). It is well-known that the isomorphism class of C ∗ ( Y ) is a coarseinvariant for the metric space Y .Let now ( X , d , ν ) be a metric measure space as outlined at the beginning of Section 5.Since ν has full support and is non-atomic, the multiplication representation of C ( X ) on L ( X , ν ) is non-degenerate and ample. Hence the multiplication representation of C ( O Γ X ) on L ( X × [ , ∞ ) , ν × λ ) is also non-degenerate and ample. We can thus use itto form the Roe algebra C ∗ ( O Γ X ) .As explained by Sawicki in [43, Proposition 1.1], the original Dru¸tu–Nowak projection G ∈ B ( L ( O Γ X , ν × λ )) is not locally compact because its image contains a copy of L ( R , λ ) . In particular, G cannot belong to the Roe algebra. One way to overcome thisissue is to consider the subspace ( X × N , d Γ ) of the unified warped cone O Γ X instead. Wewill call this the integral warped cone . Since the embedding ( X × N , d Γ ) ֒ → ( O Γ X , d Γ ) is a quasi-isometry, their Roe algebras are isomorphic. We will hence abuse the notationand denote also the integral warped cone by O Γ X .Similarly, we also define the following integral analogue of the ∗ -homomorphism Φ defined in (4.2) (still denoted by Φ ): Φ : B ( L ( X , ν )) → B ( L ( O Γ X , ν × λ N )) , T T ⊗ Id ℓ ( N ) , where λ N denotes the counting measure on N . It is elementary to check that Theorem 4.8and Proposition 4.15 still hold in the integral setting. It follows that the integral analogues It is easy to check that for a proper metric space ( X , d ) , T ∈ B ( L ( X , ν )) has finite propagation ifand only if there exists an R > χ A T χ C = A , C ⊆ X are measurable subsets with d ( A , C ) > R . In particular, this definition is equivalent to the one given in Subsection 4.4. of Theorem 4.16 and Corollary 4.17 hold true as well. We will henceforth use theirintegral versions without further notice.Let us now focus on the projections considered in Corollary 4.17. More precisely, wedenote by P the set of rank one projections in B ( L ( X , ν )) as follows: P ∈ P ⇔ either P = b P Y , S for a domain Y ⊆ X of Markov S -expansionor P = P Y for a domain Y ⊆ X of asymptotic expansion . For the averaging projection P X , the associated projection Φ ( P X ) = P X ⊗ Id ℓ ( N ) (still de-noted by G ) is called the integral Dru¸tu–Nowak projection (see [43, Proposition 1.3]). Itfollows from Corollary 4.17 that the projection Φ ( P ) can be approximated by finite prop-agation operators for every P ∈ P . Actually, we can even show the following strongerstatement: Proposition 5.1.
For every P ∈ P , the projection Φ ( P ) is non-compact and belongs tothe Roe algebra C ∗ ( O Γ X ) of the integral warped cone O Γ X . In particular, when the actionis asymptotically expanding the integral Dru¸tu–Nowak projection G belongs to C ∗ ( O Γ X ) .Proof. Clearly, each Φ ( P ) = P ⊗ Id ℓ ( N ) is non-compact for P ∈ P . We only show that Φ ( P ) belongs to C ∗ ( O Γ X ) when P = b P Y , S for a domain Y ⊆ X of Markov S -expansion,as the other case is similar and almost identical to the proof of [43, Proposition 1.3]. Re-call that b P Y , S is the orthogonal projection onto the one-dimensional subspace of L ( X , ν ) spanned by the vector √ σ Y , S defined in (3.9).Since X is compact, there exists a Borel partition V = { V i | i ∈ I } of O Γ X such that each V i has diameter at most 1 and is contained in some level set X × { n } , and for each n ∈ N only finitely many V i are contained in X × { n } . For each i ∈ I , we write V i = U i × { n ( i ) } for Borel U i ⊆ X and n ( i ) ∈ N . We consider the closed subspace W ⊆ L ( O Γ X , ν × λ N ) spanned by (cid:8) ( χ U i · √ σ Y , S ) ⊗ χ { n ( i ) } (cid:12)(cid:12) i ∈ I (cid:9) . Let R ∈ B ( L ( O Γ X , ν × λ N )) be the orthogonal projection onto W . It is clear that Φ ( P ) isa subprojection of R , so Φ ( P ) = R ◦ Φ ( P ) ◦ R . Moreover, the projection R has propagationat most one.By Corollary 4.17, Φ ( P ) is a norm limit of finite propagation operators T n ∈ B ( L ( O Γ X , ν × λ N )) . In particular, we have Φ ( P ) = lim n → ∞ RT n R . Since each RT n R has finite propaga-tion, it suffices to show that it is also locally compact. If φ ∈ C ( O Γ X ) is a function ofcompact support, then its range is contained in L ( X × { , , . . . , N } ) for some N ∈ N .This implies that R φ is of finite rank. Since the set of compact operators is norm-closed,we have that R ψ is compact for every ψ ∈ C ( O Γ X ) . Since R is self-adjoint, ψ R is com-pact as well. Hence, we conclude that both RT n R ψ and ψ RT n R are compact for every ψ ∈ C ( O Γ X ) , as desired. (cid:3) Proposition 5.1 allows us to use Theorem 4.8 to deduce the main theorem of this sub-section. Namely, the following dynamical version of [24, Theorem C]:
Theorem 5.2.
Let ( X , d ) be a compact metric space with diameter at most equippedwith a non-atomic Radon probability measure ν of full support, and ρ : Γ y ( X , d , ν ) bea continuous and measure-class-preserving action.If P X is the associated averaging projection on X and G = P X ⊗ Id ℓ ( N ) is the integralDru¸tu–Nowak projection, then the following are equivalent:(1) ρ is asymptotically expanding;(2) P X is ρ -quasi-local; MARKOVIAN AND ROE-ALGEBRAIC APPROACH TO ASYMPTOTIC EXPANSION 33 (3) G is quasi-local;(4) G belongs to the Roe algebra C ∗ ( O Γ X ) of the integral warped cone O Γ X .
Another important feature of the projections Φ ( P ) for P ∈ P is that they are ghostoperators . This notion was originally introduced by Yu (unpublished) in his study of thecoarse Baum–Connes conjecture. We will use the following: Definition 5.3 ([12, Definition 6.5]) . Given a metric measure space ( Z , d , ν ) , an operator T ∈ B ( L ( Z , ν )) is called ghost if for every R , ε >
0, there exists a bounded subset C ⊆ Z such that for any φ ∈ L ( Z , ν ) with k φ k = ( φ ) ⊆ B R ( x ; d ) for some x ∈ Z r C we have k T φ k ≤ ε .Firstly, we observe the following easy fact: Lemma 5.4.
A non-atomic probability measure ν on a metric space ( Z , d ) is necessarilyupper uniform ( [12, Definition 6.1] ) in the sense that lim r → sup z ∈ Z ν ( B r ( z ; d )) = .Proof. If there exist an ε > z n ∈ Z such that ν ( B / n ( z n ; d )) ≥ ε > n ∈ N , then there must be some point ¯ z ∈ Z that belongs to B / n ( z n ; d ) forinfinitely many n . To see this, it is sufficient to note that ν ( T N ∈ N S n > N B / n ( z n ; d )) = lim N → ∞ ν ( S n > N B / n ( z n ; d )) ≥ ε as the probability measure ν is continuous from above.On the other hand, such a ¯ z must be an atom for ν so that ν cannot be non-atomic. (cid:3) The following lemma is a generalisation of [12, Theorem 6.6]:
Lemma 5.5.
If T ∈ B ( L ( X , ν )) is any orthogonal rank one projection, then Φ ( T ) ∈ B ( L ( O Γ X , ν × λ N )) is ghost. In particular, Φ ( P ) is a ghost projection for every P ∈ P .Proof. Since X is compact, the action Γ y X is uniformly continuous. Then the proofof Lemma 4.2 can be adapted to show that the balls B R ( x ; d n Γ ) are contained in N δ n ( B ⌊ R ⌋ · x ; d ) ⊆ X for some positive δ n independent of x and such that δ n →
0, where B ⌊ R ⌋ denotesthe ball in Γ . Since N δ n ( B ⌊ R ⌋ · x ; d ) ⊆ S γ ∈ B ⌊ R ⌋ B δ n ( γ · x ; d ) and B ⌊ R ⌋ is finite, we easilydeduce from the upper uniformity of ν (Lemma 5.4) that lim n → ∞ sup x ∈ X ν ( B R ( x ; d n Γ )) = ε , R > C N ≔ O Γ X ∩ (cid:0) X × [ , N ) (cid:1) for N ∈ N . We note that C N is a bounded subset of O Γ X and any point in O Γ X r C N is of the form ( x , n ) for some n ≥ N and x ∈ X . It is well-known that every rank one projection T ∈ B ( L ( X , ν )) isof the form T η = h η , ξ i ξ for some unit vector ξ ∈ L ( X , ν ) . In order to show that Φ ( T ) ∈ B ( L ( O Γ X , ν × λ N )) is ghost, we fix any φ ∈ L ( O Γ X , ν × λ N ) with k φ k = ( φ ) ⊆ B R (( x , n ) ; d Γ ) for some ( x , n ) ∈ O Γ X r C N . So we have that k Φ ( T )( φ ) k = ∑ m ∈ N k ξ k · | Z X φ ( y , m ) ξ ( y ) d ν ( y ) | ≤ n + R ∑ m = n − R (cid:0) Z B R ( x ; d m Γ ) | φ ( y , m ) ξ ( y ) | d ν ( y ) (cid:1) ≤ n + R ∑ m = n − R (cid:0) Z X | φ ( y , m ) | d ν ( y ) · Z B R ( x ; d m Γ ) | ξ ( y ) | d ν ( y ) (cid:1) ≤ n + R ∑ m = n − R Z B R ( x ; d m Γ ) | ξ ( y ) | d ν ( y ) , where the last inequality uses the fact that R X | φ ( y , m ) | d ν ( y ) ≤ k φ k = m ∈ N .Since ξ ∈ L ( X , ν ) and sup x ∈ X ν ( B R ( x ; d n Γ )) → n → ∞ , it follows that k Φ ( T )( φ ) k → n → ∞ . We can hence choose N large enough so that k Φ ( T )( φ ) k ≤ ε for every φ with supp ( φ ) ⊆ B R (( x , n ) ; d Γ ) for some ( x , n ) ∈ O Γ X r C N , as desired. (cid:3) Combining Proposition 5.1 with Lemma 5.5, we obtain the following:
Corollary 5.6.
Let ( X , d ) be a compact metric space with diameter at most endowedwith a non-atomic probability measure ν of full support, and Γ y ( X , d , ν ) a measure-class-preserving continuous action. Then each Φ ( P ) ∈ B ( L ( O Γ X , ν × λ N )) for P ∈ P is a non-compact ghost projection in the Roe algebra C ∗ ( O Γ X ) of the integral warpedcone O Γ X .
Counterexamples to the coarse Baum–Connes conjecture.
In this subsection, wewill consider the subset Q X ≔ X × { n | n ∈ N } of X × [ , ∞ ) and the associated subspace Q Γ X (which we will call sparse warped cone ) of the unified warped cone O Γ X . The maingoal is to show that under certain mild assumptions all non-compact ghost projections inthe Roe algebra C ∗ ( Q Γ X ) lie outside the image of the coarse Baum–Connes assemblymap. In particular, they all violate the coarse Baum–Connes conjecture.As before, we define a ∗ -homomorphism Φ Q as follows: Φ Q : B ( L ( X , ν )) → B ( L ( Q Γ X , ν × λ N )) , T T ⊗ Id ℓ ( { n | n ∈ N } ) . It is easy to see that Corollary 5.6 still holds in this setting: under the same assumption,each Φ Q ( P ) with P ∈ P is a non-compact ghost projection in the Roe algebra C ∗ ( Q Γ X ) . We call G Q = Φ Q ( P X ) the sparse Dru¸tu–Nowak projection .The idea of the proof is to construct two “trace” maps τ d and τ u on K ( C ∗ ( Q Γ X )) ,whose restrictions to the image of the coarse assembly map coincide and yet take differentvalues on every non-compact ghost projection in C ∗ ( Q Γ X ) . The following argument is acombination of those in [17, 43, 57]. We have decided to provide here a fair amount ofdetails, because it also requires a few (minor) adaptations and extensions. Remark . The choice of 2 n in the definition of Q Γ X is rather arbitrary and made forthe sake of concreteness. We could equally set Q X = X × { a n | n ∈ N } for any othersequence { a n } n ∈ N ⊆ [ , ∞ ) as long as lim n , m → ∞ | a n − a m | = ∞ .5.2.1. The trace τ d . For each n ∈ N , we denote by Q n ∈ B ( L ( Q Γ X , ν × λ N )) the orthog-onal projection onto L ( X × { n } , ν ) . For T ∈ B ( L ( Q Γ X , ν × λ N )) with propagation atmost 2 n − , we have Q n T = T Q n and define T n ≔ Q n T Q n ∈ C ∗ ( X × { n } ) . Hence, themap C [ Q Γ X ] ∋ T ( T n ) n ∈ N ∈ ∏ n C ∗ ( X × { n } ) L n C ∗ ( X × { n } ) is multiplicative, contractive and ∗ -preserving on the algebraic Roe algebra C [ Q Γ X ] .Thus, it yields a ∗ -homomorphism on the entire Roe algebra C ∗ ( Q Γ X ) .As each X × { n } is compact, C ∗ ( X × { n } ) is ∗ -isomorphic to the C ∗ -algebra of com-pact operators K ( L ( X × { n } )) . Hence, the canonical trace map Tr on K ( L ( X × { n } )) induces Tr ∗ : K ( C ∗ ( X × { n } )) → Z . As in [57, Section 6] and [43, Section 3], we definethe trace map τ d : K ( C ∗ ( Q Γ X )) → ∏ R L R as the composition of the trace Tr ∗ : K ( C ∗ ( X × { n } )) → Z ⊆ R with the map K ( C ∗ ( Q Γ X )) → K (cid:18) ∏ n C ∗ ( X × { n } ) L n C ∗ ( X × { n } ) (cid:19) MARKOVIAN AND ROE-ALGEBRAIC APPROACH TO ASYMPTOTIC EXPANSION 35 induced by T ( T n ) n ∈ N under the identification K (cid:18) ∏ n C ∗ ( X × { n } ) L n C ∗ ( X × { n } ) (cid:19) (cid:27) K ( ∏ n C ∗ ( X × { n } )) K ( L n C ∗ ( X × { n } )) (cid:27) ∏ n K ( C ∗ ( X × { n } )) L n K ( C ∗ ( X × { n } )) . The proof of the following lemma is almost identical to the proof of [57, Theorem 6.1],we include here a short proof for the convenience of the reader.
Lemma 5.8.
Let p ∈ C ∗ ( Q Γ X ) be any projection, then τ d ([ p ]) = p iscompact. In particular, we have τ d ([ Φ Q ( P )]) , for every P ∈ P .Proof. Firstly, we note that for every T ∈ C ∗ ( Q Γ X ) , we have [ T , Q n ] → n → ∞ . Inparticular, for every projection p ∈ C ∗ ( Q Γ X ) we have that Q n pQ n gets arbitrarily close tosome honest projections q n in C ∗ ( X × { n } ) as n → ∞ . In other words, [( Q n pQ n ) n ∈ N ] =[( q n ) n ∈ N ] in ∏ n C ∗ ( X × { n } ) / L n C ∗ ( X × { n } ) .By the definition of τ d , we have that τ d ([ p ]) = [( Tr ( q ) , Tr ( q ) , . . . )] = [( dim ( q ) , dim ( q ) , . . . )] , where dim ( q n ) denotes the dimension of the range of q n . On the other hand, as Q n pQ n → q n it follows that the projection p is compact if and only if dim ( q n ) = τ d ([ p ]) = p is compact. (cid:3) The trace τ u . In order to construct the other trace map τ u , we need some extraassumptions and preliminaries.Following [45], we equip Q X = X × { n | n ∈ N } with the open cone metric d Q (( x , t ) , ( x , t )) ≔ ( t ∧ t ) · d ( x , x ) + | t − t | so that Q X and Q Γ X coincide as sets but are equipped with different metrics. We canthen define a metric d Γ × Q on the product Γ × Q X as the largest metric such that • d Γ × Q (( γ , ( x , t )) , ( γ , ( x , t ))) ≤ d Q (( x , t ) , ( x , t )) ; • d Γ × Q (( γ , ( x , t )) , ( ηγ , η · ( x , t )) ≤ ℓ ( η ) for every γ , η ∈ Γ and ( x , t ) , ( x , t ) ∈ Q X . The projection to the second coordinate gives a natural quotient map π : Γ × Q X → Q Γ X and the metric d Γ × Q is defined so that the quotient metric on Q Γ X coincides withthe warped metric d Γ . Since X is compact, it is shown in [45, Proposition 3.10] that theaction on X is free if and only if π is asymptotically faithful. Recall that a surjectivemap between metric spaces π : ( Y , d Y ) → ( Z , d Z ) is called asymptotically faithful if forevery R > C R ⊆ Z such that the restriction of π to every R -ballcentred at a point outside of π − ( C R ) is an isometry [45, 57]. Asymptotic faithfulnesswill play an important role later on, we thus need to restrict our attention to free actions.In order to estimate operator norms of finite propagation operators in B ( L ( Γ × Q X )) ,we assume that the metric space ( Γ × Q X , d Γ × Q ) has the operator norm localisationproperty (ONL) (see [8]). Namely, if we equip Γ × Q X with the product measure λ Γ × ν × λ N (here λ Γ is the counting measure on the discrete group Γ ), we say that ( Γ × Q X , d Γ × Q ) has ONL if for every c ∈ ( , ) and r > R > T ∈ B ( L ( Γ × Q X )) of propagation at most r there exists a unit vector ξ ∈ L ( Γ × Q X ) with diam ( supp ξ ) ≤ R satisfying k T ξ k ≥ c k T k . We remark that the metric d Γ × Q is denoted by d ′ in [43] and it is isometric—but not equal—to themetric d used in [45, Definition 3.6]. The latter is also denoted by d in [43]. Remark . It follows from [8, Proposition 2.4] that the above definition of ONL isequivalent to the original definition in [8, Definition 2.3]. It follows from the work ofSako [41] that—for metric spaces that are proper and have bounded geometry— ONL isalso equivalent to property A in the sense of [40, Definition 2.1] (see [43, Corollary 2.5]for a proof).
Remark . For a Lipschitz action Γ y X on a compact space X , the metric space ( Γ × Q X , d Γ × Q ) has ONL under either of the following conditions:(1) if Γ has property A and X is a manifold;(2) if the asymptotic dimension of Γ is finite and X is an ultrametric space.We refer to [43, Corollary 2.11] for a more general statement.As in [43, Section 3.2], let ρ : Γ y X be a free action so that π : Γ × Q X → Q Γ X is asymptotically faithful. Let T ∈ C ∗ ( Q Γ X ) be an operator with propagation at most r and let n be large enough so that for every n > n the quotient map π restricts to anisometry on every ball of radius 3 r in Γ × X × { n } ⊆ Γ × Q X . This allows us to define,for every n > n , a canonical Γ -equivariant lift T ′ n ∈ C [ Γ × X × { n } ] Γ of the operator T n = Q n T Q n ∈ C ∗ ( X × { n } ) . Specifically, given ξ , η ∈ L ( Γ × X × { n } ) with supportof diameter at most r we define h T ′ n ξ , η i ≔ ( h T n ( ξ ◦ σ ) , η ◦ σ i , if d Γ × Q ( supp ξ , supp η ) ≤ r , , otherwise , where σ is the inverse of the restriction of π to supp ( ξ ) ∪ supp ( η ) . Note that the subspacespanned by vectors with diameter of supports at most r is dense in L ( Γ × X × { n } ) ,hence T ′ n is well-defined. It is verified in [43, Lemma 3.1] that each T ′ n is bounded, and itis clear that each T ′ n has propagation at most r and is locally compact and invariant underconjugations.Moreover, [43, Lemma 3.2] shows that if ( Γ × Q X , d Γ × Q ) has ONL, then k T ′ n k ≤ c k T k for every n ∈ N and some uniform constant c > T [( T ′ n ) n ∈ N ] induces an algebraic ∗ -homomorphism: Ψ : C [ Q Γ X ] −→ ∏ n C ∗ ( Γ × { n } × X ) Γ L n C ∗ ( Γ × { n } × X ) Γ which can be extended to a C ∗ -homomorphism on the whole C ∗ ( Q Γ X ) . As a matter offact, it is possible to obtain a slightly improved control on the norm of Ψ ( T ) (the proof isomitted as it is equal to the proof of [58, Lemma 13.3.11]): Lemma 5.11.
Let Γ y X be a free action and assume that ( Γ × Q X , d Γ × Q ) has ONL.Then for every T ∈ C [ Q Γ X ] we have that k Ψ ( T ) k = sup R ≥ lim n → ∞ sup {k T n ξ k | ξ ∈ L ( X × { n } ) , k ξ k = , and diam ( supp ξ ) ≤ R } . Lemma 5.11 allows us to identify the kernel of Ψ with the closed ideal consisting of allghost operators in C ∗ ( Q Γ X ) (compare with [58, Corollary 13.3.14]): Corollary 5.12.
Let Γ y X be a free action and assume that ( Γ × Q X , d Γ × Q ) has ONL.Given T ∈ C ∗ ( Q Γ X ) , then Ψ ( T ) = T is a ghost operator.Proof.
By continuity of Ψ , it follows from Lemma 5.11 that the formula k Ψ ( T ) k = sup R ≥ lim n → ∞ sup {k T n ξ k | ξ ∈ L ( X × { n } ) , k ξ k = , and diam ( supp ξ ) ≤ R } MARKOVIAN AND ROE-ALGEBRAIC APPROACH TO ASYMPTOTIC EXPANSION 37 also holds for every T ∈ C ∗ ( Q Γ X ) . So Ψ ( T ) = R , ε >
0, thereexists an N ∈ N such that for every n > N and every unit vector ξ ∈ L ( X × { n } ) withdiam ( supp ξ ) ≤ R , we have k T n ξ k ≤ ε . The latter condition holds if and only if T is aghost operator. (cid:3) We resume the construction of the trace τ u following [43, Section 3.2]. It can be shownthat for every n ∈ N we have a ∗ -isomorphism C ∗ (( Γ × X × { n } ) , d Γ × Q X ) Γ (cid:27) C ∗ r ( Γ ) ⊗ K ( L ( X × { n } )) , where K ( L ( X × { n } )) denotes the compact operators. The latter admits a trace τ comingfrom the canonical traces on both tensor factors. More precisely, we let τ ( p ) ≔ Tr ( χ p χ ) , where χ is the characteristic function of { } × X × { n } , and Tr is the canonical traceon K ( L ( X × { n } )) . Finally, we define the trace τ u on K ( C ∗ ( Q Γ X )) as the followingcomposition: K ( C ∗ ( Q Γ X )) Ψ ∗ −→ K (cid:18) ∏ n C ∗ ( Γ × X × { n } , d Γ × Q ) Γ L n C ∗ ( Γ × X × { n } , d Γ × Q ) Γ (cid:19) (cid:27) ∏ n K ( C ∗ ( Γ × X × { n } , d Γ × Q ) Γ ) L n K ( C ∗ ( Γ × X × { n } , d Γ × Q ) Γ ) τ ∗ −→ ∏ R L R . Consequently, Corollary 5.12 together with Corollary 5.6 prove the following:
Proposition 5.13.
Let ( X , d ) be a compact metric space of diameter at most equippedwith a non-atomic probability measure ν of full support, and Γ y ( X , d , ν ) be a freemeasure-class-preserving continuous action. Assume that ( Γ × Q X , d Γ × Q ) has ONL.Then for every ghost projection p ∈ C ∗ ( Q Γ X ) , we have τ u ([ p ]) = . In particular, forany projection P ∈ P we have τ u ([ Φ Q ( P )]) = . Comparing the two traces.
The concluding argument goes exactly as in [17, 43,57]. The key idea is to use Atiyah Γ -index Theorem [4] to show that whenever p is aprojection in the Roe algebra such that [ p ] belongs to the range of the coarse assemblymap, then τ d ([ p ]) = τ u ([ p ]) ∈ ∏ R L R . This argument first appeared in [17, Proposition 5.6]. The detailed proof (in the case ofgraphs) can be found in [57, Lemma 6.5] (see also [43, Theorem 3.3] for the case ofcompact metric spaces).Together with Lemma 5.8 and Proposition 5.13, we deduce that every K -theory classof a non-compact ghost projection in the Roe algebra is not in the image of the coarseassembly map (see [57, Theorem 6.1] for the case of graphs). In particular this applies toany projection Φ ( P ) with P ∈ P . We record this fact as a theorem, as it is the main resultof Section 5 and it generalises [43, Theorem 3.5] from measure-preserving actions with aspectral gap to measure-class-preserving asymptotically expanding actions: Theorem 5.14.
Let ( X , d ) be a compact metric space of diameter at most equippedwith a non-atomic probability measure ν of full support, and ρ : Γ y ( X , d , ν ) be a freecontinuous measure-class-preserving action. Further assume that ( Γ × Q X , d Γ × Q ) hasONL.If p = Φ Q ( P Y ) for any domain Y ⊆ X of asymptotic expansion or p = Φ Q ( b P Y , S ) forany domain Y ⊆ X of Markov S-expansion, then [ p ] does not belong to the image of thecoarse assembly map. In particular, if the action ρ is asymptotically expanding then the class of the sparseDru¸tu–Nowak projection G Q violates the coarse Baum–Connes conjecture for the sparsewarped cone Q Γ X .
The following corollary follows immediately from Theorem 5.14 and Remark 5.10:
Corollary 5.15.
Let ( X , d ) be a compact metric space of diameter at most equippedwith a non-atomic probability measure ν of full support, and let ρ : Γ y ( X , d , ν ) be afree Lipschitz measure-class-preserving asymptotically expanding action under either ofthe following conditions:(1) if Γ has property A and X is a manifold;(2) if the asymptotic dimension of Γ is finite and X is an ultrametric space.Then the coarse Baum–Connes conjecture for the sparse warped cone Q Γ X fails.Example . Given a chain of finite index subgroups Γ > Γ > Γ > · · · , we considerthe inverse limit X = lim ←− Γ / Γ i . This space is homeomorphic to a Cantor set, and theuniform measures on Γ / Γ i induce a natural probability measure ν on X ( ν is obviouslynon-atomic and with full-support). Further, X can also be given an ultrametric by letting d (( γ i Γ i ) i ∈ N , ( γ ′ i Γ i ) i ∈ N ) = − n where n is the smallest index such that γ n Γ n , γ ′ n Γ n . Clearly, Γ acts X by left multiplication and the action is isometric and measure-preserving. Further,if T i ∈ N Γ i = { } then the action is free. Such an action is called a profinite action .Abért–Elek constructed in [1, Theorem 5] a free profinite action F k y ( X , d , ν ) of anyfinitely generated non-abelian free group F k that is strongly ergodic (and hence asymptot-ically expanding) but does not have a spectral gap.Since the free group has asymptotic dimension 1, we can hence apply Corollary 5.15to deduce that Q F k X violates the coarse Baum–Connes conjecture. This fact does notdirectly follow from [43, Theorem 3.5], as the action does not have spectral gap. More-over, X is an ultrametric space, so Q F k X is not coarsely equivalent to a disjoint union offinite graphs ([43, Proposition 4.6 (1)]). This implies that this result cannot be deducedby combining the approximating space construction from [26] with results in [24]. Remark . It is not hard to check that the sparse and unified warped cones arising froma free profinite action have bounded geometry if and only if there is a uniform upperbound on the indices [ Γ i : Γ i + ] for i ∈ N .It follows that the sparse warped cone in Example 5.16 does not have bounded geome-try in general: the construction of Abért–Elek requires chains of subgroups with indicesgrowing very quickly (this is important in the proof of [1, Lemma 6.2]). It would be in-teresting to know if it is possible to find a chain Γ > Γ > · · · with uniformly boundedindices [ Γ i : Γ i + ] such that the induced profinite action is strongly ergodic but has nospectral gap.5.3. Non-coarse embeddability.
In this subsection we prove that warped cones arisingfrom asymptotically expanding actions do not coarsely embed into any Hilbert space. Oneof the ground-breaking results by Yu was to verify the coarse Baum–Connes conjecturefor every proper bounded geometry metric space which coarsely embeds into some Hilbertspace [61]. It follows from Theorem 5.14 that the sparse warped cone Q Γ X coming froman asymptotically expanding action of Γ on a compact metric space X cannot coarselyembed into Hilbert spaces provided that ( Γ × Q X , d Γ × Q ) has ONL and Q Γ X has boundedgeometry (it is not hard to show that if ( X , d ) is proper, so is ( O Γ X , d Γ ) ). Below we will A metric space ( X , d ) has bounded geometry if for every ε , R > N ∈ N such that any ε -separated subset of an R -ball of X has at most N elements. MARKOVIAN AND ROE-ALGEBRAIC APPROACH TO ASYMPTOTIC EXPANSION 39 strengthen this result and show both ONL of ( Γ × Q X , d Γ × Q ) and bounded geometry of Q Γ X are redundant.Recall that a map F : ( X , d X ) → ( Z , d Z ) is a coarse embedding between metric spacesif there exist non-decreasing unbounded functions ρ ± : [ , ∞ ) → [ , ∞ ) such that ρ − ( d X ( x , x ′ )) ≤ d Z ( F ( x ) , F ( x ′ )) ≤ ρ + ( d X ( x , x ′ )) , for all x , x ′ ∈ X .The following proposition is a (partial) extension of [33, Theorem 3.1] from the set-ting of measure-preserving actions with a spectral gap to asymptotically expanding mea-sure-class-preserving actions. The proof combines the idea in [33, Theorem 3.1] withProposition 3.19: Proposition 5.18.
Let ( X , d ) be a compact metric space of diameter at most equippedwith a non-atomic probability measure ν , and ρ : Γ y ( X , d , ν ) be a continuous measure-class-preserving and asymptotically expanding action.If A ⊆ [ , ∞ ) is any unbounded subset and d Γ is the warped cone metric on O Γ X , then ( X × A , d Γ ) does not admit a coarse embedding into any Hilbert space.Proof. From Proposition 3.19, there exist a finite symmetric subset 1 ∈ S ⊆ Γ and adomain Y ⊆ X of Markov S -expansion such that there is a constant Θ ≥ / Θ ≤ r ( s , x ) ≤ Θ for every x ∈ Y and s ∈ S Y , x = { s ∈ S | s · x ∈ Y } . Hence, we have1 − λ > κ ≔ ( − λ ) >
0. For every g ∈ L ( Y , ν ) , it follows from Proposition 3.11(1)-(2) that g ∈ L ( Y , ˜ ν Y , S ) . Hence, it fol-lows from Proposition 3.11(3) (see also (3.14)) that k g k ν , ≤ κ ∑ s ∈ S Z Y ∩ s − ( Y ) r ( s , x ) / | g ( x ) − g ( s · x ) | d ν ( x ) ≤ κ √ Θ ∑ s ∈ S Z Y ∩ s − ( Y ) | g ( x ) − g ( s · x ) | d ν ( x ) . Assume now that ( X × A , d Γ ) admits a coarse embedding into Hilbert space ℓ ( N ) .If F : ( Y × A , d Γ ) → ℓ ( N ) denotes the restriction of the coarse embedding, then we let F t : Y → ℓ ( N ) be the restriction of F to the level set Y × { t } for t ∈ A . For each t ∈ A , theassociated coefficient function of F t is denoted by F ( n ) t for n ∈ N . Every F ( n ) t is a functionin L ( Y , ν ) , and we have ∑ n ∈ N ∑ s ∈ S Z Y ∩ s − ( Y ) | F ( n ) t ( x ) − F ( n ) t ( s · x ) | d ν ( x ) = ∑ s ∈ S Z Y ∩ s − ( Y ) k F t ( x ) − F t ( s · x ) k ℓ ( N ) d ν ( x ) ≤ ∑ s ∈ S Z Y ∩ s − ( Y ) ρ + ( d t Γ ( x , s · x )) d ν ( x ) ≤ ρ + ( M ) | S | , where M ≔ max s ∈ S ℓ ( s ) .After translating F t if necessary, we may assume that F ( n ) t ∈ L ( Y , ν ) . This implies thatfor each t ∈ A we have k F t k ν , = ∑ n ∈ N k F ( n ) t k ν , ≤ κ √ Θρ + ( M ) | S | < ∞ . On the other hand, since F ( n ) t ∈ L ( Y , ν ) we also have " Y × Y k F t ( x ) − F t ( y ) k ℓ ( N ) d ν ( x ) ν ( y ) = k F t k ν , . We will thus reach a contradiction by showing that(5.1) " Y × Y k F t ( x ) − F t ( y ) k ℓ ( N ) d ν ( x ) ν ( y ) → ∞ , as t → ∞ . Since ν is non-atomic and Γ is countable, the set N ≔ { ( x , y ) ∈ Y × Y | Γ · x = Γ · y } is measurable and has measure zero by Fubini’s theorem. On the other hand, for any x , y ∈ Y lying in different Γ -orbits the distance d Γ (( x , t ) , ( y , t )) = d t Γ ( x , y ) → ∞ as t → ∞ .Since F is a coarse embedding, it follows that k F t ( x ) − F t ( y ) k ℓ ( N ) → ∞ . Hence, wededuce (5.1) and obtain the desired contradiction. (cid:3) Remark . Let 1 < p < ∞ . By interpolation, if Y is a domain of Markov S -expansionthen the Markov operator P Y , S has norm strictly less than one also when regarded as anoperator on L p ( Y , ˜ ν Y , S ) . An easy modification of the proof of Proposition 5.18 showsthat the warped cone does not coarsely embed into L p for any 1 < p < ∞ . Moreover, thewarped cone cannot coarsely embed into L -spaces as well because it is shown in [34,Proposition 4.1] that every L -space coarsely embeds into a Hilbert space. Example . [33, Theorem 3.1] does not apply to the warped cones arising from theprofinite actions F k y ( X , d , ν ) of Abért–Elek (Example 5.16). However, we may useProposition 5.18 to conclude that the sparse warped cone Q F k X as well as the unifiedwarped O F k X cannot be coarsely embedded into any Hilbert space.Note also that the non-embeddability of Q F k X does not immediately follow from thefact that it violates the coarse Baum–Connes conjecture. In fact, Yu’s argument only ap-plies to bounded geometry proper metric spaces, while the warped cones in Example 5.16have unbounded geometry (Remark 5.17).R EFERENCES [1] Miklós Abért and Gábor Elek. Dynamical properties of profinite actions.
Ergodic Theory and Dynam-ical Systems , 32(6):1805–1835, 2012.[2] Noga Alon. Eigenvalues and expanders.
Combinatorica , 6(2):83–96, 1986.[3] Noga Alon and Vitali D Milman. λ
1, isoperimetric inequalities for graphs, and superconcentrators.
Journal of Combinatorial Theory, Series B , 38(1):73–88, 1985.[4] Michael Francis Atiyah. Elliptic operators, discrete groups and von Neumann algebras.
Astérisque ,32(33):43–72, 1976.[5] Yves Benoist and Nicolas de Saxcé. A spectral gap theorem in simple Lie groups.
Inventiones mathe-maticae , 205(2):337–361, 2014.[6] Jean Bourgain and Alex Gamburd. On the spectral gap for finitely-generated subgroups of SU(2).
Inventiones mathematicae , 171(1):83–121, 2007.[7] Rémi Boutonnet, Adrian Ioana, and Alireza Salehi Golsefidy. Local spectral gap in simple Lie groupsand applications.
Inventiones mathematicae , 208(3):715–802, 2017.[8] Xiaoman Chen, Romain Tessera, Xianjin Wang, and Guoliang Yu. Metric sparsification and operatornorm localization.
Advances in Mathematics , 218(5):1496–1511, 2008.[9] Ionut Chifan and Adrian Ioana. Ergodic subequivalence relations induced by a Bernoulli action.
Geo-metric and Functional Analysis , 20(1):53–67, 2010.[10] Alain Connes and Benjamin Weiss. Property (T) and asymptotically invariant sequences.
Israel Jour-nal of Mathematics , 37(3):209–210, 1980.[11] Jozef Dodziuk. Difference equations, isoperimetric inequality and transience of certain random walks.
Transactions of the American Mathematical Society , 284(2):787–794, 1984.[12] Cornelia Dru¸tu and Piotr W Nowak. Kazhdan projections, random walks and ergodic theorems.
Jour-nal für die reine und angewandte Mathematik (Crelles Journal) , 754:49–86, 2019.[13] Alexander Engel. Index theory of uniform pseudodifferential operators. arXiv:1502.00494v3 , 2018.
MARKOVIAN AND ROE-ALGEBRAIC APPROACH TO ASYMPTOTIC EXPANSION 41 [14] David Fisher, Thang Nguyen, and Wouter van Limbeek. Rigidity of warped cones and coarse geometryof expanders.
Advances in Mathematics , 346:665–718, 2019.[15] Alex Gamburd, Dmitry Jakobson, and Peter Sarnak. Spectra of elements in the group ring of SU ( ) . Journal of the European Mathematical Society (JEMS) , 1(1):51–85, 1999.[16] Lukasz Grabowski, András Máthé, and Oleg Pikhurko. Measurable equidecompositions for groupactions with an expansion property. arXiv:1601.02958 , 2016.[17] Nigel Higson. Counterexamples to the coarse Baum-Connes conjecture.
Available on the author’swebsite , 1999.[18] Nigel Higson, Vincent Lafforgue, and Georges Skandalis. Counterexamples to the Baum-Connes con-jecture.
Geometric and Functional Analysis , 12(2):330–354, 2002.[19] Nigel Higson and John Roe. On the coarse Baum-Connes conjecture.
Novikov conjectures, index the-orems and rigidity , 2:227–254, 1995.[20] Cyril Houdayer, Amine Marrakchi, and Peter Verraedt. Strongly ergodic equivalence relations: spec-tral gap and type III invariants.
Ergodic Theory and Dynamical Systems , 39:1904–1935, 2019.[21] Adrian Ioana. Strong ergodicity, property (T), and orbit equivalence rigidity for translation actions.
Journal für die reine und angewandte Mathematik (Crelles Journal) , 2017(733):203–250, 2017.[22] Vadim A Kaimanovich. Dirichlet norms, capacities and generalized isoperimetric inequalities forMarkov operators.
Potential Analysis , 1(1):61–82, 1992.[23] Harry Kesten. Full Banach mean values on countable groups.
Mathematica Scandinavica , pages 146–156, 1959.[24] Ana Khukhro, Kang Li, Federico Vigolo, and Jiawen Zhang. On the structure of asymptotic expanders. arXiv:1910.13320v2 , 2019.[25] Kang Li, Piotr Nowak, Ján Špakula, and Jiawen Zhang. Quasi-local algebras and asymptotic ex-panders. arXiv:1908.07814, to appear in Groups, Geometry and Dynamics , 2020.[26] Kang Li, Federico Vigolo, and Jiawen Zhang. Asymptotic expansion in measure and strong ergodicity. arXiv:2005.05697, 2020 .[27] Kang Li, Zhijie Wang, and Jiawen Zhang. A quasi-local characterisation of L p -Roe algebras. Journalof Mathematical Analysis and Applications , 474(2):1213–1237, 2019.[28] Alexander Lubotzky, Ralph Phillips, and Peter Sarnak. Hecke operators and distributing points on thesphere I.
Communications on Pure and Applied Mathematics , 39(S1):S149–S186, 1986.[29] Russell Lyons and Fedor Nazarov. Perfect matchings as iid factors on non-amenable groups.
EuropeanJournal of Combinatorics , 32(7):1115–1125, 2011.[30] Gregory A Margulis. Some remarks on invariant means.
Monatshefte für Mathematik , 90(3):233–235,1980.[31] Amine Marrakchi. Strongly ergodic actions have local spectral gap.
Proceedings of the AmericanMathematical Society , 146(9):3887–3893, 2018.[32] Bojan Mohar. Isoperimetric inequalities, growth, and the spectrum of graphs.
Linear algebra and itsapplications , 103:119–131, 1988.[33] Piotr Nowak and Damian Sawicki. Warped cones and spectral gaps.
Proceedings of the AmericanMathematical Society , 145(2):817–823, 2017.[34] Piotr W. Nowak. Coarse embeddings of metric spaces into Banach spaces.
Proceedings of the Ameri-can Mathematical Society , 133(9):2589–2596, 2005.[35] Piotr W. Nowak and Guoliang Yu.
Large scale geometry . EMS Textbooks in Mathematics. EuropeanMathematical Society (EMS), Zürich, 2012.[36] Gábor Pete. Probability and geometry on groups. Lecture notes for a graduate course, 2019,http://math.bme.hu/ gabor/PGG.pdf.[37] D. Revuz.
Markov chains , volume 11 of
North-Holland Mathematical Library . North-Holland Pub-lishing Co., Amsterdam, second edition, 1984.[38] John Roe. An index theorem on open manifolds. I, II.
J. Differential Geom. , 27(1):87–113, 115–136,1988.[39] John Roe.
Index theory, coarse geometry, and topology of manifolds , volume 90 of
CBMS RegionalConference Series in Mathematics . Published for the Conference Board of the Mathematical Sciences,Washington, DC; by the American Mathematical Society, Providence, RI, 1996.[40] John Roe. Warped cones and property A.
Geometry & Topology , 9(1):163–178, 2005.[41] Hiroki Sako. Property A and the operator norm localization property for discrete metric spaces.
Jour-nal für die Reine und Angewandte Mathematik (Crelles Journal) , 690:207–216, 2014. [42] Damian Sawicki. Super-expanders and warped cones. arXiv:1704.03865, to appear in Annales del’Institut Fourier , 2017.[43] Damian Sawicki. Warped cones violating the coarse Baum-Connes conjecture. , 2017.[44] Damian Sawicki.
On the geometry of metric spaces defined by group actions: from circle rotationsto super-expanders . PhD thesis, Instytut Matematyczny Polskiej Akademii Nauk, 2018, to appear inAnnales de l’Institut Fourier.[45] Damian Sawicki and Jianchao Wu. Straightening warped cones. arXiv:1705.06725, to appear in Jour-nal of Topology and Analysis , 2017.[46] Thomas Schick. The topology of positive scalar curvature. In
Proceedings of the International Con-gress of Mathematicians—Seoul 2014. Vol. II , pages 1285–1307. Kyung Moon Sa, Seoul, 2014.[47] Klaus Schmidt. Asymptotically invariant sequences and an action of SL ( , Z ) on the 2-sphere. IsraelJournal of Mathematics , 37(3):193–208, 1980.[48] Klaus Schmidt. Amenability, Kazhdan’s property (T), strong ergodicity and invariant means for er-godic group-actions.
Ergodic Theory Dynamical Systems , 1(2):223–236, 1981.[49] Georges Skandalis, Jean-Louis Tu, and Guoliang Yu. The coarse Baum–Connes conjecture andgroupoids.
Topology , 41(4):807–834, 2002.[50] Ján Špakula and Aaron Tikuisis. Relative commutant pictures of Roe algebras.
Communications inMathematical Physics , 365(3):1019–1048, 2019.[51] Federico Vigolo.
Geometry of actions, expanders and warped cones . PhD thesis, University of Oxford,2018.[52] Federico Vigolo. Discrete fundamental groups of warped cones and expanders.
Mathematische An-nalen , 373(1-2):355–396, 2019.[53] Federico Vigolo. Measure expanding actions, expanders and warped cones.
Transactions of the Amer-ican Mathematical Society , 371(3):1951–1979, 2019.[54] Ján Špakula and Jiawen Zhang. Quasi-locality and Property A.
Journal of Functional Analysis ,278(1):108299, 2020.[55] Qin Wang and Zhen Wang. Warped cones and proper affine isometric actions of discrete groups onBanach spaces. arXiv preprint arXiv:1705.08090 , 2017.[56] Rufus Willett. Some notes on property A. In
Limits of graphs in group theory and computer science ,pages 191–281. EPFL Press, Lausanne, 2009.[57] Rufus Willett and Guoliang Yu. Higher index theory for certain expanders and Gromov monstergroups, I.
Advances in Mathematics , 229(3):1380–1416, 2012.[58] Rufus Willett and Guoliang Yu.
Higher Index Theory . 189. Cambridge University Press, 2020.[59] Wolfgang Woess.
Denumerable Markov chains . EMS Textbooks in Mathematics. European Mathe-matical Society (EMS), Zürich, 2009. Generating functions, boundary theory, random walks on trees.[60] Guoliang Yu. Zero-in-the-spectrum conjecture, positive scalar curvature and asymptotic dimension.
Inventiones mathematicae , 127(1):99–126, 1997.[61] Guoliang Yu. The coarse Baum-Connes conjecture for spaces which admit a uniform embedding intoHilbert space.
Inventiones Mathematicae , 139(1):201–240, 2000.I
NSTITUTE OF M ATHEMATICS OF THE P OLISH A CADEMY OF S CIENCES , UL . ´S NIADECKICH
8, 00-656, W
ARSAW , P
OLAND
E-mail address : [email protected] F ACULTY OF M ATHEMATICS AND C OMPUTER S CIENCE , W
EIZMANN I NSTITUTE , 7610001, R E - HOVOT , I
SRAEL
E-mail address : [email protected] S CHOOL OF M ATHEMATICS , U
NIVERSITY OF S OUTHAMPTON , H
IGHFIELD , SO17 1BJ, U
NITED K INGDOM
E-mail address ::