A quasi-local characterisation of L p -Roe algebras
aa r X i v : . [ m a t h . F A ] M a r A QUASI-LOCAL CHARACTERISATION OF L p -ROE ALGEBRAS KANG LI , ZHIJIE WANG , AND JIAWEN ZHANG A bstract . Very recently, ˇSpakula and Tikuisis provide a new characterisation of(uniform) Roe algebras via quasi-locality when the underlying metric spaces havestraight finite decomposition complexity. In this paper, we improve their methodto deal with the L p -version of (uniform) Roe algebras for any p ∈ [1 , ∞ ). Due tothe lack of reflexivity on L -spaces, some extra work is required for the case of p = Mathematics Subject Classification (2010): 20F65, 46H35, 47L10.1. I ntroduction (Uniform) Roe algebras are C ∗ -algebras associated to metric spaces, which reflectcoarse properties of the underlying metric spaces. These algebras have beenwell-studied and have fruitful applications, among which the most importantones would be the (uniform) coarse Baum-Connes conjecture and the Novikovconjecture (e.g., [32, 33, 40, 41, 42, 43]). Meanwhile, they also provide a linkbetween coarse geometry of metric spaces and the theory of C ∗ -algebras (e.g.,[1, 11, 15, 16, 19, 20, 23, 29, 30, 32, 37, 39]), and turn out to be useful in the study oftopological phases of matter (e.g., [17, 10]) as well as the theory of limit operatorsin the study of Fredholmness of band-dominated operators (e.g., [14, 21, 31, 35]).By definition, the (uniform) Roe algebra of a proper metric space X is thenorm closure of all bounded locally compact operators T with finite propagation in the sense that there exists R > f , g ∈ C b ( X ) acting on L ( X )by pointwise multiplication, we have f Tg = R -separated (i.e., the distance between the supports of f and g is at least R ). Sincegeneral elements in (uniform) Roe algebras may not have finite propagation, itis usually di ffi cult to tell what operators exactly belong to them. On the otherhand, Roe [27] defined an asymptotic version of finite propagation as follows:An operator T on L ( X ) has finite ε -propagation for ε >
0, if there is R > f , g ∈ C b ( X ), we have k f Tg k ≤ ε k f k · k g k provided their supports Key words and phrases.
Quasi-local operators, L p -Roe algebras, straight finite decompositioncomplexity. Supported by the Danish Council for Independent Research (DFF-5051-00037), DeutscheForschungsgemeinschaft (SFB 878, Groups, Geometry and Actions) and the European ResearchCouncil (ERC-677120). Supported by NSFC (No. 11501249). Supported by the Sino-British Trust Fellowship by Royal Society, International Exchanges2017 Cost Share (China) grant EC \ NSFC \ , ZHIJIE WANG , AND JIAWEN ZHANG are R -separated. Operators with finite ε -propagation for all ε > quasi-local in [26]. It is clear that limits of finite ε -propagation operators still havefinite ε -propagation. Consequently, all operators in (uniform) Roe algebras arequasi-local.A natural question is that whether the converse holds as well, i.e., does everylocally compact quasi-local operator belong to the (uniform) Roe algebra? Ana ffi rmative answer to this question would provide a new approach to detect whatoperators belong to these algebras in a more practical way by estimating thenorms of operator-blocks far from strips around the diagonal, and it has severalimmediate consequences including the followings.The first one has its root in Engel’s work [8, Section 2], where he studiedthe index theory of pseudo-di ff erential operators. He showed that the indicesof uniform pseudo-di ff erential operators on Riemannian manifolds are quasi-local, while it is unclear to him whether they live in Roe algebras, which arewell-understood. Another application is in the work of White and Willett [38] onCartan subalgebras of uniform Roe algebras. They showed that if two uniform Roealgebras of bounded geometry metric spaces with Property A are ∗ -isomorphic,then the underlying metric spaces are bijectively coarsely equivalent providedthat every quasi-local operator belongs to the uniform Roe algebras.Historically, this question has been studied and partially addressed by manypeople including Lange and Rabinovich for X = Z n [18] (in fact they workedin a more general context, see the next paragraph), Engel for X is a manifold ofbounded geometry with polynomial volume growth [9], ˇSpakula and Tikuisis [34]for X has straight finite decomposition complexity in the sense of [7]. To our bestknowledge, this question is still open for general metric spaces.Based on the original definitions, various versions of Roe algebras are proposedand studied by di ff erent purposes. In fact, in recent years there has been an uptickin interest in the L p -version of (uniform) Roe algebras for p ∈ [1 , ∞ ), from thecommunities of both limit operator theory and coarse geometry (e.g. [31, 21, 35,14, 3, 44]). And it is natural and important to study the same question in thiscontext, i.e., does every locally compact and quasi-local operator belong to the L p -version of (uniform) Roe algebras for p ∈ [1 , ∞ )?In this paper, we improve the method of ˇSpakula and Tikuisis [34] in orderto generalise their result from the case of p = p ∈ [1 , ∞ ). The mainpart of our result is the following (see Theorem 3.3 for the complete version),which answers the L p -version of the question above under the condition that theunderlying metric space has straight finite decomposition complexity. Theorem A.
For a proper metric space with straight finite decomposition complexity andp ∈ [1 , + ∞ ) , quasi-locality is equivalent to being in the associated L p -Roe-like algebra. Here the notion of L p -Roe-like algebra is the L p -analogue of Roe-like algebrasˇSpakula and Tikuisis introduced for p = QUASI-LOCAL CHARACTERISATION OF L p -ROE ALGEBRAS 3 main result is established. However, we would like to point out that our definitionof L p -Roe-like algebras are more general than ˇSpakula and Tikuisis’ definition evenfor p =
2, as we drop a commutant condition in [34, Definition 2.3], which is usedin the proof of their main theorem. However, we observe that this condition isredundant for the proof of the main theorem if we replace it with Lemma 3.5below. The reason we drop this condition is inspired by the fact that it is notfulfilled for general L -Roe-like algebras, and an obvious advantage of doing thisis to allow more examples especially in the case of p = p ∈ (1 , ∞ ), except that the L p -Roe-like algebras need not possess abounded involution and von Neumann algebra techniques are invalid. Instead,we have to deal with asymmetric situation as in the proof of the implication “(iii) ⇒ (i)” in Theorem 3.3 and provide a direct and concrete proof of Lemma 4.4.The case of p = ffi culty comes from the lack of reflexivity on L -spaces,and the trick of the proof is to consider an artificial space L ( X ), which lies between C ( X ) and L ∞ ( X ). It is worth pointing out that Proposition 4.1 is based on a crucialintermediate result established in a more general setup of Banach spaces, and wehope that there might be some other applications in the future. The paper is organised as follows: we establish the settings of the paper by re-calling some background in Banach algebra theory and coarse geometry in Section2, where various examples of L p -Roe-like algebras are also provided. In Section3, we provide a complete version of our main result Theorem A, and prove therelatively easier part, where the assumption of straight finite decomposition com-plexity is not required. In Section 4, we prove the technical tool, Proposition 4.1,and finish the remaining proof of the main theorem. Conventions:
Let X be a Banach space. We denote the closed unit ball of X by X . For any a , b ∈ X and ε >
0, we denote k a − b k ≤ ε by a ≈ ε b . We also denote thebounded linear operators on X by B ( X ), and the compact operators on X by K ( X ).Moreover, for a Banach algebra A we define A ∞ : = ℓ ∞ ( N , A ) .n ( a n ) n ∈ N ∈ ℓ ∞ ( N , A ) : lim n →∞ k a n k = o , which is a Banach algebra with respect to the quotient norm. Throughout the paper, we fix a proper metric space ( X , d ) (i.e., every bounded subsetis pre-compact). Note that such a space is always locally compact and σ -compact. We also fix a Radon measure µ on ( X , d ) with full support (i.e., µ is a regular Borel After we finish this paper, ˇSpakula and the third-named author informed us that the maintheorem of this paper remains true if we only require Property A rather than straight finite de-composition complexity [36]. Their arguments include an essential application of Proposition 4.1.
KANG LI , ZHIJIE WANG , AND JIAWEN ZHANG measure on X taking finite values on compact subsets, and for each x ∈ X , thereexists a neighbourhood U of x such that µ ( U ) > reliminaries In this section, we provide the background settings of this paper by collectingseveral basic notions from Banach algebra theory and coarse geometry. Through-out the section, let E be a (complex) Banach space and ( X , d , µ ) be a proper metricspace with a Radon measure µ on X of full support.Denote C b ( X ) the space of bounded continuous functions on X , C ( X ) the spaceof continuous functions on X vanishing at infinity, and C c ( X ) the space of contin-uous functions on X with compact supports.2.1. Banach space valued L p -spaces. In this subsection, we recall some basicnotions and facts on Banach space valued L p -spaces. Definition 2.1.
Let p ∈ [1 , ∞ ]. For a Bochner measurable function (i.e., it equals µ -almost everywhere to a pointwise limit of a sequence of simple functions) ξ : ( X , µ ) → E , its p-norm is defined by k ξ k p : = (cid:16) Z X k ξ ( x ) k pE d µ ( x ) (cid:17) p , and its infinity-norm is defined by k ξ k ∞ : = ess sup {k ξ ( x ) k E : x ∈ X } . For p ∈ [1 , ∞ ], the space of E -valued L p -functions on ( X , µ ) is defined as follows: L p ( X , µ ; E ) : = n ξ : X → E (cid:12)(cid:12)(cid:12) ξ is Bochner measurable and k ξ k p < ∞ o. ∼ , where ξ ∼ η if and only if they are equal µ -almost everywhere. Equipped with the p -norm, L p ( X , µ ; E ) becomes a Banach space, which is called the L p -Bochner space .We also need the following closed linear subspace of L ∞ ( X , µ ; E ): L ( X , µ ; E ) : = n [ ξ ] ∈ L ∞ ( X , µ ; E ) (cid:12)(cid:12)(cid:12) ∀ ε > , ∃ compact K ⊆ X , s.t. k ξ | X \ K k ∞ < ε o , equipped with the norm k ξ k : = k ξ k ∞ . Clearly, L ( X , µ ; E ) contains C ( X ) but ismore flexible, as it also contains all characteristic functions of bounded subsets ofthe proper metric space ( X , d ). On the other hand, L ( X , µ ; E ) inherits some nicebehaviours of C ( X ), for example a representative can always be chosen for eachelement in L ( X , µ ; E ) such that its norm goes to zero when the variable goes toinfinity.In order to simplify notations, we regard ξ as an element in L p ( X , µ ; E ) and write L p ( X ; E ) instead if there is no ambiguity. If X is discrete and equipped with thecounting measure, we simply write ℓ p ( X ; E ). It follows from Pettis measurability theorem that Bochner measurability agrees with weakmeasurability when the Banach space E is separable. QUASI-LOCAL CHARACTERISATION OF L p -ROE ALGEBRAS 5 If p ∈ (1 , ∞ ), let q be the conjugate exponent to p (i.e., p + q =
1) and if p =
1, weset q = q = ∞ . It is worth noticing that the duality L p ( X ; E ) ∗ (cid:27) L q ( X ; E ∗ )does not hold in general (see e.g. [2, 5, 6]), but we still have the following lemma. Lemma 2.2.
When p ∈ (1 , ∞ ) , set q to be its conjugate exponent and when p = , setq = . Then there is an isometric embedding L q ( X ; E ∗ ) → L p ( X ; E ) ∗ defined by η ( ξ ) : = Z X η ( x )( ξ ( x ))d µ ( x ) where η ∈ L q ( X ; E ∗ ) and ξ ∈ L p ( X ; E ) . On the other hand, there is another isometricembedding L p ( X ; E ) → L q ( X ; E ∗ ) ∗ defined by ξ ( ζ ) : = Z X ζ ( x )( ξ ( x ))d µ ( x ) where ξ ∈ L p ( X ; E ) and ζ ∈ L q ( X ; E ∗ ) .Proof. For the first statement, when p > L q ( X ; C ) embeds isometrically into L p ( X ; C ) ∗ (which areindeed isomorphic). And for p =
1, we have the following maps L ( X ; E ∗ ) ⊆ L ∞ ( X ; E ∗ ) ֒ → L ( X , E ) ∗ where the second isometric embedding follows from the same argument showingthe classical result that L ∞ ( X ; C ) embeds isometrically into L ( X ; C ) ∗ .For the second statement, it su ffi ces to show that for any ξ ∈ L p ( X ; E ), we have k ξ k p = sup {| ξ ( ζ ) | : ζ ∈ L q ( X ; E ∗ ) and k ζ k q ≤ } . It is clear that the right hand side does not exceed the left. Conversely we mayassume, by the inner regularity of µ , that ξ is non-zero and ξ = P ni = y i χ Ω i forsome y i ∈ E and mutually disjoint compact subsets Ω i in X . Note that k ξ k pp = P ni = k y i k pE µ ( Ω i ). For each y i , choose a y ∗ i ∈ ( E ∗ ) such that y ∗ i ( y i ) = k y i k E . Define ζ : = n X i = k y i k p − E k ξ k p − p y ∗ i χ Ω i . Note that when p = ζ can be written simply as P ni = y ∗ i χ Ω i . It is straightforwardto check that ζ ∈ L q ( X ; E ∗ ) with k ζ k q = ξ ( ζ ) = k ξ k p (note that when p =
1, weset q = (cid:3) Finally we recall L p -tensor products (more details can be found in [4, Chapter7], [24, Theorem 2.16] and [22]), which will be used in Section 2.3 without furtherreference.For p ∈ [1 , ∞ ), there is a tensor product of L p -spaces with σ -finite measures suchthat there is a canonical isometric isomorphism L p ( X , µ ) ⊗ L p ( Y , ν ) (cid:27) L p ( X × Y , µ × ν ),which identifies the element ξ ⊗ η with the function ( x , y ) ξ ( x ) η ( y ) on X × Y forevery ξ ∈ L p ( X , µ ) and η ∈ L p ( Y , ν ). Moreover, the following properties hold: • Under the identification above, the linear spans of all ξ ⊗ η are dense in L p ( X × Y , µ × ν ). KANG LI , ZHIJIE WANG , AND JIAWEN ZHANG • || ξ ⊗ η || p = || ξ || p || η || p for all ξ ∈ L p ( X , µ ) and η ∈ L p ( Y , ν ). • The tensor product is commutative and associative. • If a ∈ B ( L p ( X , µ ) , L p ( X , µ )) and b ∈ B ( L p ( Y , ν ) , L p ( Y , ν )), then thereexists a unique element c ∈ B ( L p ( X × Y , µ × ν ) , L p ( X × Y , µ × ν ))such that under the identification above, c ( ξ ⊗ η ) = a ( ξ ) ⊗ b ( η ) for all ξ ∈ L p ( X , µ ) and η ∈ L p ( Y , ν ). We will denote this operator by a ⊗ b .Moreover, || a ⊗ b || = || a || · || b || . • The above tensor product of operators is associative, bilinear, and satisfies( a ⊗ b )( a ⊗ b ) = a a ⊗ b b .If A ⊆ B ( L p ( X , µ )) and B ⊆ B ( L p ( Y , ν )) are closed subalgebras, we define A ⊗ B ⊆ B ( L p ( X × Y , µ × ν )) to be the closed linear span of all a ⊗ b with a ∈ A and b ∈ B .2.2. Block cutdown maps.
Now we introduce block cutdown maps, providingan approach to cut an operator into the form of block diagonals.First, let us recall some more notions. For p ∈ { } ∪ [1 , ∞ ], the multiplicationrepresentation ρ : C b ( X ) → B ( L p ( X ; E )) is defined by pointwise multiplications:( ρ ( f ) ξ )( x ) = f ( x ) ξ ( x ), where f ∈ C b ( X ), ξ ∈ L p ( X ; E ) and x ∈ X . Without ambiguity,we write f T and T f instead of ρ ( f ) T and T ρ ( f ), for f ∈ C b ( X ) and T ∈ B ( L p ( X ; E )),respectively. It is worth noticing that µ has full support if and only if ρ is injective.We also recall that a net { T α } converges in strong operator topology (SOT) to T in B ( L p ( X ; E )) if and only if k T α ( ξ ) − T ( ξ ) k p → ξ ∈ L p ( X ; E ). Definition 2.3.
Given an equicontinuous family ( e j ) j ∈ J of positive contractionsin C b ( X ) with pairwise disjoint supports, define the block cutdown map θ ( e j ) j ∈ J : B ( L p ( X ; E )) → B ( L p ( X ; E )) by(2.1) θ ( e j ) j ∈ J ( a ) : = X j ∈ J e j ae j , where the sum converges in (SOT) by Lemma 2.4 below. We say that a closedsubalgebra B ⊆ B ( L p ( X ; E )) is closed under block cutdowns , if θ ( e j ) j ∈ J ( B ) ⊆ B forevery equicontinuous family ( e j ) j ∈ J of positive contractions in C b ( X ) with pairwisedisjoint supports. Lemma 2.4.
Let ( e j ) j ∈ J and ( f j ) j ∈ J be two equicontinuous families of positive contractionsin C b ( X ) with pairwise disjoint supports, and a ∈ B ( L p ( X ; E )) . Then the sum P j ∈ J f j ae j converges in (SOT) to an operator in B ( L p ( X ; E )) . Furthermore, we have: (cid:13)(cid:13)(cid:13) X j ∈ J f j ae j (cid:13)(cid:13)(cid:13) = sup j ∈ J k f j ae j k . Proof.
First of all, we prove in the case of p ∈ (1 , ∞ ) and let q be the conjugateexponent to p . Let Y j : = supp( e j ) and Z j : = supp( f j ). For any ξ ∈ L p ( X ; E ), any QUASI-LOCAL CHARACTERISATION OF L p -ROE ALGEBRAS 7 finite subset F ⊆ J and any η ∈ L q ( X ; E ∗ ) with k η k q ≤
1, we have that (cid:12)(cid:12)(cid:12) η (cid:16) X j ∈ F f j ae j ξ (cid:17)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) X j ∈ F ( ηχ Z j )( f j ae j χ Y j ξ ) (cid:12)(cid:12)(cid:12) ≤ X j ∈ F k η | Z j k q · k f j ae j ( ξ | Y j ) k p ≤ (cid:16) X j ∈ F k η | Z j k qq (cid:17) q · (cid:16) X j ∈ F k f j ae j ( ξ | Y j ) k pp (cid:17) p ≤ sup j ∈ J k f j ae j k · k ξ | ⊔ j ∈ F Y j k p . Hence, it follows from Lemma 2.2 that (cid:13)(cid:13)(cid:13)(cid:16) X j ∈ F f j ae j (cid:17) ξ (cid:13)(cid:13)(cid:13) p ≤ sup j ∈ J k f j ae j k · k ξ | ⊔ j ∈ F Y j k p . Since k ξ | ⊔ j ∈ J Y j k p ≤ k ξ k p < ∞ , we know n ξ | ⊔ j ∈ F Y j o F is a Cauchy net. Hence, P j ∈ J f j ae j converges in (SOT) and k P j ∈ J f j ae j k ≤ sup j ∈ J k f j ae j k . On the other hand, it is clearthat k P j ∈ J f j ae j k ≥ sup j ∈ J k f j ae j k . Hence we finish the proof for p >
1. Since theproof for the case of p = (cid:3) Remark . Note that the multiplication by C b ( X ) commutes with the block cut-downs, i.e., for any a ∈ B ( L p ( X ; E )) and f ∈ C b ( X ), we have f θ ( e j ) j ∈ J ( a ) = θ ( e j ) j ∈ J ( f a ) and θ ( e j ) j ∈ J ( a ) f = θ ( e j ) j ∈ J ( a f ) . Definition 2.6.
Suppose X is a metric family of subsets in X (recall that a metricfamily is a set of metric spaces), each of the subset is equipped with the inducedmetric and a ∈ B ( L p ( X ; E )). We say that a is block diagonal with respect to X , ifthere exist an equicontinuous family ( e j ) j ∈ J of positive contractions in C b ( X ) withpairwise disjoint supports and { Y j } j ∈ J ⊆ X , such that a = θ ( e j ) j ∈ J ( a ) , and supp( e j ) ⊆ Y j . In this case, we shall denote a Y j : = e j ae j , which is called the Y j -block of a .2.3. L p -Roe-like algebras. Now we introduce L p -Roe-like algebras, which are ourmain objects in this paper. Definition 2.7.
Let R ≥ a ∈ B ( L p ( X ; E )). We say that • a has propagation at most R , if for any f , f ′ ∈ C b ( X ) with d (supp( f ) , supp( f ′ )) > R , then f a f ′ = • a has ε -propagation at most R for some ε >
0, if for any f , f ′ ∈ C b ( X ) with d (supp( f ) , supp( f ′ )) > R , then k f a f ′ k < ε . • a is quasi-local , if it has finite ε -propagation for every ε > Definition 2.8.
Let ( X , d ) be a proper metric space equipped with a Radon measure µ whose support is X , and p ∈ [1 , + ∞ ). Suppose E is a Banach space and B ⊆ KANG LI , ZHIJIE WANG , AND JIAWEN ZHANG B ( L p ( X ; E )) is a Banach subalgebra such that C b ( X ) BC b ( X ) = B and is closed underblock cutdowns. Define:(i) Roe( X , B ) to be the norm-closure of all the operators in B with finite propa-gations. Roe( X , B ) is called the L p -Roe-like algebra of ( X , d , µ );(ii) K ( X , B ) to be the norm-closure of C ( X ) BC ( X ) in B ( L p ( X ; E )). Remark . The definition of L -Roe-like algebras come from [34, Definition 2.3],in which the following extra condition is also imposed:(2.2) [ C ( X ) , B ] ⊆ K ( X , B ) . This condition is used in the proof of their main theorem, [34, Theorem 2.8, “(i) ⇒ (iii)”]. However, it turns out to be redundant if we apply our Lemma 3.5 below.On the other hand, this condition is fulfilled by most of the well-known L p -Roe-like algebras for p ∈ (1 , ∞ ) (as we will see in the following examples), but not for p = ff ecting the main theorem. In this way, our main result(Theorem 3.3) is a slight generalisation of [34, Theorem 2.8].We notice that in the case of p =
2, it has been pointed out in [34, Remark 2.4]that K ( X , B ) is an ideal in Roe( X , B ) under the additional condition (2.2). Now weshow that it still holds in our settings. Lemma 2.10.
For any p ∈ [1 , + ∞ ) , K ( X , B ) is a closed two-sided ideal in Roe( X , B ) .Proof. It su ffi ces to show that for any b = f b g ∈ C c ( X ) BC c ( X ) and a ∈ B withfinite propagation at most R , ba ∈ K ( X , B ). Take a function g ∈ C c ( X ) such that g is 1 on the compact subset N R (supp( g )). It follows that g a (1 − g ) =
0, whichimplies that g a = g ag . Hence, we have ba = f b g a = f ( b g a ) g . Recall that C b ( X ) BC b ( X ) = B , so we have b g ∈ B and a ∈ B , which implies that ba ∈ C c ( X ) BC c ( X ). Similarly, ab ∈ C c ( X ) BC c ( X ) as well. So we finish the proof. (cid:3) Before we illustrate several examples of L p -Roe-like algebras, let us recall thefollowing notion related to matrix algebras. Definition 2.11.
Let ( X , d ) be a discrete proper metric space and p ∈ [1 , + ∞ ).Denote M pX : = C c ( X ) B ( ℓ p ( X )) C c ( X ) B ( ℓ p ( X )) , i.e., for any fixed point x ∈ X M pX = [ n ∈ N M pB n ( x ) where M pB n ( x ) = B ( ℓ p ( B n ( x ))) ⊆ B ( ℓ p ( X )), which is the matrix algebra over theclosed ball of radius n and centered in x . In other words, operators in M pX areexactly those that can be approximated by finite matrices. QUASI-LOCAL CHARACTERISATION OF L p -ROE ALGEBRAS 9 Phillips studied the relation between M pX and compact operators K ( ℓ p ( X )) in[25]. He showed that when p > M pX = K ( ℓ p ( X )) ([25, Lemma 1.7, Corollary 1.9]);and when p = M X ( K ( ℓ ( X )) in general as illustrated in [25, Example 1.10] (seealso Example 2.12).Now we are ready to provide various of examples of L p -Roe-like algebras,which include ℓ p -uniform Roe algebras, band-dominated operator algebras, L p -Roe algebras, ℓ p -uniform algebras and stable ℓ p -uniform Roe algebras. Example ℓ p -Uniform Roe Algebra ) . Let ( X , d ) be a discrete proper metricspace and p ∈ [1 , + ∞ ). Take E = C to be the complex number and B = B ( ℓ p ( X )),which is clearly closed under block cutdowns, and satisfies C b ( X ) BC b ( X ) = B . Inthis case, Roe( X , B ) is called the ℓ p -uniform Roe algebra of X , which is defined in[3] and denoted by B pu ( X ), and K ( X , B ) is M pX introduced above. It may be worthnoting that M X is structurally di ff erent from M pX for p > • p >
1: As pointed out above, M pX = K ( ℓ p ( X )). And condition (2.2) follows fromthe fact that C ( X ) B ⊆ K ( X , B ) and BC ( X ) ⊆ K ( X , B ). • p =
1: The algebra K ( X , B ) is in general properly contained in K ( ℓ ( X )) (seeExample 1.10 in [25]). For example, taking X to be the natural number N , considerthe operator T : ℓ ( N ) → ℓ ( N ) defined by T ( ξ ) : = (cid:16) X n ∈ N ξ ( n ) (cid:17) δ , where ξ ∈ ℓ ( N ) and δ ∈ ℓ ( N ) is the function taking value 1 at the origin point0, and 0 elsewhere. Since T has rank 1, it belongs to K ( ℓ ( N )). However, itis not hard to see that T < K ( N , B ( ℓ ( N ))) = M N . Furthermore, the operator T also illuminates that condition (2.2) does not hold in general, since [ δ , T ] < K ( N , B ( ℓ ( N ))). Example
Band-Dominated Operator Algebra ) . Let ( X , d ) be a uniformlydiscrete metric space of bounded geometry (in the sense that for a given R >
0, allclosed balls B ( x , R ) have a uniform bound on cardinalities for all x ∈ X ), p ∈ (1 , + ∞ )and E be a Banach space. Take B = B ( ℓ p ( X ; E )), which is clearly closed under blockcutdowns and satisfies C b ( X ) BC b ( X ) = B . Elements in B can be represented in thematrix form b = ( b x , y ) x , y ∈ X ∈ B ( ℓ p ( X ; E )) , where b x , y ∈ B ( E ) . In this case, Roe( X , B ) = A pE ( X ), which is the algebra of band-dominated operators(see [35, Definition 2.6]) and it is clear that K ( X , B ) = K pE ( X ), which is the set of all P -compact operators on ℓ p ( X ; E ), defined in [35, Definition 2.8]. Example L p -Roe Algebra ) . Let ( X , d ) be a proper metric space equipped witha Radon measure µ with support X , and p ∈ [1 , + ∞ ). We say that an operator b in B ( L p ( X ; ℓ p ( N ))) (cid:27) B ( L p ( X × N )) is locally compact if for any f ∈ C ( X ), f b and b f belong to K ( L p ( X × N )). , ZHIJIE WANG , AND JIAWEN ZHANG Now take E = ℓ p ( N ) and B to be the set of all locally compact operatorsin B ( L p ( X ; ℓ p ( N ))), which is clearly closed under block cutdowns and satisfies C b ( X ) BC b ( X ) = B . The corresponding L p -Roe-like algebra Roe( X , B ) is called the L p -Roe algebra of X , denoted by B p ( X ). It is, by definition, the norm closure of all locallycompact and finite propagation operators in B ( L p ( X ; ℓ p ( N ))). Analogous to thearguments in Example 2.12, one can check that when p > K ( X , B ) = K ( L p ( X × N ))and it does not hold in general when p = X is discrete, the L p -Roe algebra B p ( X ) coincides with the ℓ p -Roe algebradefined in [3] and in the case of p =
2, the L -Roe algebra is the classical Roealgebra in the literature. Remark . As explained in [3], there is another version of locally compactness:we say that an operator b in B ( L p ( X ; ℓ p ( N ))) is locally compact if for any f ∈ C ( X ), f b and b f belong to K ( L p ( X )) ⊗ M p N ⊆ B ( L p ( X × N )). Note that the subalgebra K ( L p ( X )) ⊗ M p N is isomorphic to the norm closure of S n ∈ N M pn ( K ( L p ( X ))). Therefore,we can alternatively define another version of the L p -Roe algebra of X to be thenorm closure of all locally compact (in this new sense) and finite propagationoperators in B ( L p ( X ; ℓ p ( N ))). When p >
1, it coincides with B p ( X ) defined inExample 2.14 as K ( L p ( X )) ⊗ M p N (cid:27) K ( L p ( X × N )). However, it is strictly containedin B ( X ) when p = Example ℓ p -Uniform Algebra ) . Let ( X , d ) be a discrete metric space withbounded geometry and p ∈ [1 , + ∞ ). Set E = ℓ p ( N ), and B to be the closure ofthe set of all b = ( b x , y ) x , y ∈ X ∈ B ( ℓ p ( X ; ℓ p ( N ))) for which the rank of b x , y ∈ B ( ℓ p ( N ))is uniformly bounded. Clearly, B is closed under block cutdowns, and satisfies C b ( X ) BC b ( X ) = B . In this case, Roe( X , B ) = UB p ( X ), the ℓ p -uniform algebra of X ,introduced in [3]. When p >
1, we have that K ( X , B ) = K ( l p ( X × N )). But it doesnot hold in general when p = Example
Stable ℓ p -Uniform Roe Algebra ) . Let ( X , d ) be a discrete metricspace with bounded geometry and p ∈ [1 , + ∞ ). Set E = ℓ p ( N ), and B to be theclosure of the set of all b = ( b x , y ) x , y ∈ X ∈ B ( ℓ p ( X ; ℓ p ( N ))) for which there existsa finite-dimensional subspace E b ⊆ ℓ p ( N ) such that b x , y ∈ B ( E b ) ⊆ B ( ℓ p ( N )).Clearly, B is closed under block cutdowns and satisfies C b ( X ) BC b ( X ) = B . Inthis case, Roe( X , B ) = B ps ( X ), the stable ℓ p -uniform Roe algebra of X , introducedin [3]. Moreover, B ps ( X ) (cid:27) B pu ( X ) ⊗ K ( ℓ p ( N )), which explains the terminology.Analogous to the arguments in Example 2.12, one can check that when p > K ( X , B ) = K ( l p ( X × N )) and it does not hold in general when p = Remark . As explained in [3], there is another version of the stable ℓ p -uniformRoe algebra of X , defined to be the norm closure of finite propagation operators b = ( b x , y ) x , y ∈ X ∈ B ( ℓ p ( X ; ℓ p ( N ))) for which there exists some k ∈ N such that b x , y ∈ M k ( C ) ⊆ B ( ℓ p ( N )) (here M k ( C ) is embedded as a subalgebra of B ( ℓ p ( N ))in a fixed way, independent of the points in X ). It is clear that this algebra is QUASI-LOCAL CHARACTERISATION OF L p -ROE ALGEBRAS 11 isomorphic to B pu ( X ) ⊗ M p N for all p ∈ [1 , ∞ ). As before for p >
1, it coincides with B ps ( X ) defined in Example 2.17. Remark . In general, we have that for discrete space X , B pu ( X ) ⊆ B ps ( X ) ⊆ UB p ( X ) ⊆ B p ( X ) . It is worth noticing that UB ( X ) is not contained in the weak version of the L -Roe algebra defined in Remark 2.15. Indeed, Example 2.12 provides a rank oneoperator T ∈ B ( ℓ ( N )) which does not sit in M N . Define the diagonal operator b ∈ B ( ℓ ( X ; ℓ ( N ))) by b x , x : = T for x ∈ X , and b x , y = x , y . Clearly, b is suchan example as desired.2.4. Straight finite decomposition complexity.
In this subsection, we explainthe notion of straight finite decomposition complexity, which will be used in thesequel.Straight finite decomposition complexity (sFDC) was introduced in [7] as aweak version of the original notion of finite decomposition complexity (FDC),which was introduced and studied by Guentner, Tessera and Yu in their studyof topological rigidity in [12]. In general, finite asymptotic dimension impliesfinite decomposition complexity [13, Theorem 4.1], which consequently impliesstraight finite decomposition complexity [7, Proposition 2.3]. Moreover, it wasalso shown in [7, Theorem 3.4] that straight finite decomposition complexity doesimply Yu’s Property A. However, it is still unknown whether (FDC), (sFDC) andYu’s Property A are all equivalent or not.
Definition 2.20.
Let ( X , d ) be a proper metric space and Z , Z ′ ⊆ X . Let X , Y bemetric families of subsets in X , and R ≥ • X is uniformly bounded , if sup X ∈X diam( X ) < ∞ . • Denote the
R-neighbourhood of Z by N R ( Z ) : = { z ∈ X : d ( z , Z ) ≤ R } . Set N R ( X ) : = {N R ( X ) : X ∈ X} . • A metric family ( Y j ) j ∈ J of subsets of X is R-disjoint , if d ( Y j , Y ′ j ) > R for all j , j ′ . Write G R − disjoint Y j for their union to indicate that the family is R -disjoint. • Z can R-decompose over Y , if Z can be decomposed into Z = X ∪ X and X i = G R − disjoint X ij , i = , , such that X ij ∈ Y for all i , j . • X can R-decompose over Y , denoted by X R −→ Y , if every Y ∈ X can R -decompose over Y . , ZHIJIE WANG , AND JIAWEN ZHANG • X has straight finite decomposition complexity , if for any sequence 0 ≤ R < R < · · · , there exists m ∈ N and metric families { X } = X , X , . . . , X m , suchthat X i − R i −→ X i for i = , . . . , m , and the family X m is uniformly bounded.We remark here briefly that one way to define finite decomposition complexityis to use a “decomposition game”, which means a priori that the choices of R i might depend on the previous families X , X , . . . , X i − . Consequently, sFDC canbe obviously implied from FDC.3. T he main theorem In this section, we present our main result (Theorem 3.3), which gives severaldi ff erent pictures of how elements in L p -Roe-like algebras may look like. We alsoprove the relatively easier part where straight finite decomposition complexityis not required, while leaving the rest of the proof to the next section after moretechnical tools are developed.To state our main theorem, we need to introduce some notions as follows. Definition 3.1 ([28]) . Let ( X , d ) be a proper metric space. A function g ∈ C b ( X ) iscalled a Higson function (also called a slowly oscillating function ), if for every R > ε >
0, there exists a compact set A ⊆ X such that for any x , y ∈ X \ A with d ( x , y ) < R , then | g ( x ) − g ( y ) | < ε . The set of all Higson functions on X is denotedby C h ( X ). Definition 3.2. [34, Definition 2.6], Let ( X , d ) be a metric space. A boundedsequence ( f n ) n ∈ N in C b ( X ) is called very Lipschitz , if for every L >
0, there exists n ∈ N such that f n is L -Lipschitz for any n ≥ n . Let VL( X ) denote the set of allvery Lipschitz bounded sequences in C b ( X ). DefineVL ∞ ( X ) : = VL( X ) .n ( f n ) n ∈ N ∈ VL( X ) : lim n →∞ k f n k = o . It is known from [34] that VL( X ) is a C ∗ -subalgebra of ℓ ∞ ( N , C b ( X )) and VL ∞ ( X )is a C ∗ -subalgebra of ( C b ( X )) ∞ . In the following, we will view both VL ∞ ( X ) and B ⊆ B ( L p ( X ; E )) as Banach subalgebras of B ( L p ( X ; E )) ∞ , and consider the relativecommutant : B ∩ VL ∞ ( X ) ′ = { b ∈ B : b commutes with elements in VL ∞ ( X ) } . It is clear that any operator in B ( L p ( X ; E )) with finite propagation commutes withVL ∞ ( X ). Hence, by taking limits it follows that(3.1) Roe( X , B ) ⊆ B ∩ VL ∞ ( X ) ′ . The converse inclusion is also true provided the space X has straight finite de-composition complexity and this is included in our main theorem as follows (thisis the complete version of Theorem A in Section 1): QUASI-LOCAL CHARACTERISATION OF L p -ROE ALGEBRAS 13 Theorem 3.3.
Let ( X , d ) be a proper metric space equipped with a Radon measure µ whosesupport is X, and p ∈ [1 , + ∞ ) . Suppose E is a Banach space and B ⊆ B ( L p ( X ; E )) is aBanach subalgebra such that C b ( X ) BC b ( X ) = B and B is closed under block cutdowns.Then for b ∈ B, the following are equivalent: (i). [ b , f ] = for all f ∈ VL ∞ ( X ) ; (ii). b is quasi-local; (iii). [ b , g ] ∈ K ( X , B ) for any g ∈ C h ( X ) .If X has straight finite decomposition complexity, then these are also equivalent to: (iv). b ∈ Roe( X , B ) . Recall that we have already explained in Remark 2.9 that Theorem 3.3 is aslight generalisation of [34, Theorem 2.8] as condition (2.2) is not required here.Also notice that (3.1) implies that “(iv) ⇒ (i)” holds generally and the converseimplication is also true under the extra condition of straight finite decompositioncomplexity.In the remaining of this section, we prove that (i), (ii) and (iii) in Theorem 3.3are all equivalent, and leave the implication “(i) ⇒ (iv)” to the next section, afterwe develop some technical tools such as Proposition 4.1.3.1. “(i) ⇔ (ii)”. We start with the proof of Theorem 3.3, “(i) ⇔ (ii)”. The implica-tion “(i) ⇒ (ii)” follows exactly from the same arguments in [34], while the proofof “(ii) ⇒ (i)” is slightly di ff erent from that one given in [34] due to the absence ofinner products. Fortunately, since both proofs are relativity short, we include thedetails for the convenience of the reader.Let us begin with the following characterisation of the condition (i) in The-orem 3.3, which is proved in [34] when p = p . Lemma 3.4. [34, Lemma 3.1]
Let p ∈ [1 , ∞ ) , b ∈ B ( L p ( X ; E )) and ε > . Then k [ b , f ] k < ε for every f ∈ VL ∞ ( X ) if and only if there exists some L > such thatb ∈ Commut( L , ε ) , where Commut( L , ε ) : = n a ∈ B ( L p ( X ; E )) : k [ a , f ] k < ε, for any L-Lipschitz f ∈ C b ( X ) o . Proof of Theorem 3.3, “(i) ⇔ (ii)”. Assume b ∈ B ( L p ( X ; E )) such that [ b , VL ∞ ( X ) ] = ε >
0. By Lemma 3.4, there exists some L > b ∈ Commut( L , ε ).For any f , g ∈ C b ( X ) with L − -disjoint supports, we may choose an L -Lipschitz h ∈ C b ( X ) such that h | supp f ≡ h | supp g ≡
0. In particular, k [ b , h ] k < ε . Therefore, k f bg k = k f hbg k ≤ k [ h , b ] k + k f bhg k < ε + = ε. Hence, b is quasi-local as desired.On the other hand, we assume that for any ε > b has finite ε -propagation.Without loss of generality, we may assume that b is a contraction. Given ε > N such that 6 / N < ε/
2. By the hypothesis, b has ε/ (2 N )-propagation at most , ZHIJIE WANG , AND JIAWEN ZHANG R >
0. For any (2 RN ) − -Lipschitz f ∈ C b ( X ) , we claim that k [ b , f ] k < ε . In fact,take A : = f − ([0 , N ]) , and A i : = f − (( i − N , iN ]) , i = , . . . , N . These sets partition X , and A i is 2 R -disjoint from A j for | i − j | >
1. Now choosea partition of unity e , . . . , e N ∈ C b ( X ) such that e i is supported in N R / ( A i ). Thus, k e i be j k < ε/ (2 N ) for | i − j | >
1. Meanwhile, we have f ≈ / N N X i = iN e i . Hence, it follows that k [ f , b ] k ≤ N + (cid:13)(cid:13)(cid:13) N X i = [ iN e i , b ] (cid:13)(cid:13)(cid:13) = N + (cid:13)(cid:13)(cid:13)(cid:16) N X i = iN e i b (cid:17)(cid:16) N X j = e j (cid:17) − (cid:16) N X i = e i (cid:17)(cid:16) N X j = jN be j (cid:17)(cid:13)(cid:13)(cid:13) ≤ N + X | i − j | > k e i be j k + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X | i − j |≤ ( iN − jN ) e i be j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . Each term in the first sum is dominated by ε N , hence P | i − j | > k e i be j k < ε/
2. Thesecond sum can be broken into four sums: note that the terms vanish when i = j ;what remain are j = i + j = i −
1, and we break each of these further into evenand odd parts. By Lemma 2.4, each of these terms has norm at most N . Hence,we have that k [ f , b ] k < N + ε + N < ε. So we complete the proof by Lemma 3.4. (cid:3) “(i) ⇔ (iii)”. Now we move on to Theorem 3.3, “(i) ⇔ (iii)”. Here ourmajor work is focused on omitting condition (2.2), as well as providing a “non-symmetric” version of the argument given in [34] for p =
2. However, the mainbody of the proof is still very similar to that of the original p = ff erences we make here.First of all, we recall that the proof of “(i) ⇒ (iii)” given in [34] requires condi-tion (2.2): [ C ( X ) , B ] ⊆ K ( X , B ) . After a careful reading of the proof, we realise that it is unnecessary to assumethe entire B essentially commuting with C ( X ) but only a closed subalgebra of B as shown in the following lemma: Lemma 3.5.
Let p ∈ [1 , ∞ ) and B be a Banach subalgebra of B ( L p ( X ; E )) such thatC b ( X ) BC b ( X ) = B. If b ∈ B satisfies [ b , VL ∞ ( X )] = , then [ b , C ( X )] ⊆ K ( X , B ) . QUASI-LOCAL CHARACTERISATION OF L p -ROE ALGEBRAS 15 Proof.
Let b ∈ B such that [ b , VL ∞ ( X )] =
0. Since K ( X , B ) is closed, we only needto prove [ b , g ] ∈ K ( X , B ) for any g ∈ C c ( X ).Fix a base point x ∈ X . For each k ∈ N , we may choose a ( k − )-Lipschitzfunction f k ∈ C b ( X ) such that f k vanishes on supp( g ) and f k | B Rk ( x ) c = ffi ciently large R k >
0. Hence, the sequence ( f k ) k ∈ N ∈ VL ∞ ( X ) and k [ b , f k ] k → k → ∞ by assumption. Since g f k = k ∈ N , it follows that k [ b , g ] f k k = k bg f k − gb f k k = k gb f k k = k g [ b , f k ] k ≤ k g k · k [ b , f k ] k → , as k → ∞ . Similarly, we have that k f k [ b , g ] k → k f k [ b , g ] f k k → k → ∞ .Moreover, we have that k [ b , g ] − (1 − f k )[ b , g ](1 − f k ) k ≤ k [ b , g ] f k k + k f k [ b , g ] k + k f k [ b , g ] f k k → , as k → ∞ . Since supp(1 − f k ) ⊆ B R k ( x ) and C b ( X ) BC b ( X ) = B , it follows that(1 − f k )[ b , g ](1 − f k ) ∈ C c ( X ) BC c ( X ). Hence, [ b , g ] ∈ K ( X , B ). (cid:3) Replacing condition (2.2) by Lemma 3.5 in the original proof for p = ⇒ (iii)” without any further changes. Hencewe omit the details.Now we outline the proof for the other direction, “(iii) ⇒ (i)”. Since L p -Roe-likealgebras may not possess a bounded involution in general, the proof becomesslightly di ff erent. Fix a base point x ∈ X . For R >
0, we define e R ∈ C ( X ) by e R ( x ) : = max { , − d ( x , B R ( x )) / R } . Lemma 3.6 ([34], Lemma 5.4) . For ( f k ) ∞ k = ∈ VL( X ) , a subsequence ( f k i ) ∞ i = , and asequence of positive numbers ( R i ) ∞ i = with R i + ≥ R i for each i, defineg ( f ki ) , ( R i ) : = ∞ X i = f k i ( e R i − e R i − ) . Then g ( f ki ) , ( R i ) ∈ C h ( X ) .Sketch of the proof of Theorem 3.3 “(iii) ⇒ (i)”. Fix a point x ∈ X , and set B R : = B R ( x ). Recall b satisfies the condition that [ b , g ] ∈ K ( X , B ) for any g ∈ C h ( X ).Now assume b is a contraction, and that there exists some f = ( f k ) ∞ k = ∈ VL ∞ ( X )such that [ b , f ] ,
0. Take an ε : 0 < ε < k [ b , f ] k , and consider two cases. Case I.
There exists R > S >
0, there exists infinitely many k :either k χ B R [ b , f k ](1 − χ B S ) k > ε , or k (1 − χ B S )[ b , f k ] χ B R k > ε . In other words, there exists R > S < S < . . . tending to ∞ , such that for any n ∈ N , there exist infinitely many k such that k χ B R [ b , f k ](1 − χ B Sn ) k > ε ;2) or: there exists a sequence S < S < . . . tending to ∞ , such that for any n ∈ N ,there exist infinitely many k such that k (1 − χ B Sn )[ b , f k ] χ B R k > ε . , ZHIJIE WANG , AND JIAWEN ZHANG We only prove in the first situation, while the second is similar.Since ( f k ) is very Lipschitz, f k | B R tends towards constant as k → ∞ . So withoutloss of generality, we can assume that f k | B R ≡ γ k , for all k . Setting ˆ f k : = f k − γ k gives us another very Lipschitz sequence ( ˆ f k ) ∞ k = satisfying the same condition andˆ f k | B R ≡
0. Additionally, for all k , we have that χ B R [ b , ˆ f k ] = χ B R b ˆ f k . By assumption, there exists a sequence S < S < . . . tending to ∞ , such that forany n ∈ N , we can find infinitely many k such that k χ B R b ˆ f k (1 − χ B Sn ) k > ε .As the original proof for p = k < k < . . . and R , R , . . . satisfying k χ B R b ˆ f k i ( e R i − e R i − ) k > ε . Applying Lemma 3.6 to ( ˆ f k i ) and ( R i ), we obtain g ∈ C h ( X ) defined by: g : = g ( ˆ f ki ) , ( R i ) = ∞ X i = ˆ f k i ( e R i − e R i − ) . Hence for any S > R , choose an i such that 3 R i − > S , and we have k [ b , g ](1 − χ B S ) k ≥ k χ B R [ b , g ](1 − χ B S )( χ B Ri − χ B Ri − ) k = k χ B R b ˆ f k i ( e R i − e R i − ) k > ε , which contradicts with the hypothesis that [ b , g ] ∈ K ( X , B ). Case II.
For every R >
0, there exists S > k ∈ N , we have k χ B R [ b , f k ](1 − χ B S ) k ≤ ε k (1 − χ B S )[ b , f k ] χ B R k ≤ ε . Without loss of generality, we may assume that S > R .Suppose we are given R > K ∈ N , and let S be given as above. Then thereexists k ≥ K such that(3.2) k χ B R [ b , f k ](1 − χ B S ) k ≤ ε , and k (1 − χ B S )[ b , f k ] χ B R k ≤ ε . By assumption, we have k [ b , f k ] k > ε ; and since ( f k ) is very Lipschitz, we can alsoassume that f k | B S ≈ ε/ γ for some constant γ . Hence, we have:(3.3) k χ B S [ b , f k ] χ B S k ≤ · ε · k b k ≤ ε . Now cutting the space by B R , B cR and B S , B cS , we have the following decompositionfor the operator T = [ b , f k ] (recall we assume that S > R ): T = (1 − χ B R ) T (1 − χ B R ) + χ B R T χ B S + χ B R T (1 − χ B S ) + ( χ B S − χ B R ) T χ B R + (1 − χ B S ) T χ B R . QUASI-LOCAL CHARACTERISATION OF L p -ROE ALGEBRAS 17 From Inequalities (3.2), the norms of the third and fifth items are less than or equalto ε ; and from Equality (3.3), the norms of the second and fourth items are lessthan or equal to ε as well. Hence by triangle inequality, we have: k (1 − χ B R )[ b , f k ](1 − χ B R ) k ≥ k [ b , f k ] k − ( k χ B R [ b , f k ] χ B S k + k ( χ B S − χ B R )[ b , f k ] χ B R k ) − ( k χ B R [ b , f k ](1 − χ B S ) k + k (1 − χ B S )[ b , f k ] χ B R k ) > ε − ε − ε = ε . In conclusion, for every R > K ∈ N , there exists k ≥ K such that:(3.4) k (1 − χ B R )[ b , f k ](1 − χ B R ) k > ε . As the original proof for p = k < k <. . . and R , R , . . . satisfying R i ≥ R i − and k ( χ B Ri − χ B Ri − )[ b , f k i ]( χ B Ri − χ B Ri − ) k > ε . Applying Lemma 3.6 to ( f k i ) and ( R i ), we obtain g ∈ C h ( X ) defined by: g : = g ( f ki ) , ( R i ) = ∞ X i = f k i ( e R i − e R i − ) . By the choice of ( R i ) above, for any i , we have( χ B Ri − χ B Ri − )[ b , g ]( χ B Ri − χ B Ri − ) = ( χ B Ri − χ B Ri − )[ b , f k i ]( χ B Ri − χ B Ri − ) . Hence for any S >
0, choose an i such that 6 R i − > S , and we have k [ b , g ](1 − χ B S ) k ≥ k ( χ B Ri − χ B Ri − )[ b , g ]( χ B Ri − χ B Ri − ) k = k ( χ B Ri − χ B Ri − )[ b , f k i ]( χ B Ri − χ B Ri − ) k > ε , which contradicts with the hypothesis that [ b , g ] ∈ K ( X , B ). (cid:3)
4. P roof of “( i ) ⇔ ( iv )”In this section, we will prove the remaining case of “(i) ⇔ (iv)” in Theorem 3.3.Recall that as explained in Section 3, “(iv) ⇒ (i)” holds in general. So we will onlyfocus on the opposite implication “(i) ⇒ (iv)”.A key ingredient to prove “(i) ⇒ (iv)” is to approximate a bounded operatorvia its block cutdowns as indicated in [34, Corollary 4.3] for the case of p = p ,
2. Hence we need to search for a substitution of [34,Corollary 4.3], and we figure out the following crucial result, which might be ofindependent interest to experts in Banach space theory. , ZHIJIE WANG , AND JIAWEN ZHANG Proposition 4.1.
Let ( X , d ) be a proper metric space equipped with a Radon measure µ whose support is X and p ∈ [1 , + ∞ ) . Suppose E is a Banach space, a ∈ B ( L p ( X ; E )) anda ∈ Commut( L , ε ) for some L , ε > . Let ( e j ) j ∈ J be an equicontinuous family of positivecontractions in C b ( X ) with / L-disjoint supports, and define e : = P j ∈ J e j . Then, we have (cid:13)(cid:13)(cid:13) eae − X j ∈ J e j ae j (cid:13)(cid:13)(cid:13) ≤ ε. The proof of the above proposition is technical and relatively long, so we decideto postpone it to Section 4.1 for the convenience of the reader, and first show howto use the proposition to prove “(i) ⇒ (iv)”. Let us start with the following lemma,which is a consequence of Proposition 4.1 by the same proof of [34, Lemma 4.6].It may be worth reminding the reader that for any L , ε >
0, we denoteCommut( L , ε ) = n a ∈ B ( L p ( X ; E )) : k [ a , f ] k < ε, for any L -Lipschitz f ∈ C b ( X ) o . Lemma 4.2.
Let X and Y be metric families of X such that X L − + −−−−−→ Y for some L > ,and a ∈ B ( L p ( X ; E )) be block diagonal with respect to X for p ∈ [1 , ∞ ) . Let ε > be suchthat a ∈ Commut( L , ε ) . Then we can write: (4.1) a ≈ ε a + a + a + a , where each a ii ′ is of the form θ ( f k ) k ∈ K ( gag ′ ) (see (2.1))for some g , g ′ ∈ C b ( X ) and someequicontinuous positive family ( f k ) k ∈ K in C b ( X ) with disjoint supports, such that each supp( f k ) is contained in some set in N L − + ( Y ) . In particular:(i) each a ii ′ is block diagonal with respect to N L − + ( Y ) ,(ii) if a ∈ Commut( L ′ , ε ′ ) for some L ′ , ε ′ > , then each a ii ′ is in Commut( L ′ , ε ′ ) aswell, and(iii) if B ⊆ B ( L p ( X ; E )) is a Banach subalgebra such that C b ( X ) BC b ( X ) = B and B isclosed under block cutdowns, and if a is in B, then each a ii ′ is in B as well.Proof of Theorem 3.3, “(i) ⇒ (iv)”. Although the proof is exactly the same as the onegiven in [34], we decide to include it here for the completeness and show thereader how straight finite decomposition complexity is used in the proof.Take b ∈ B such that it commutes with all f ∈ VL ∞ ( X ). Given ε >
0, we aim toconstruct a finite propagation operator in B , which is ε -close to b . It follows fromLemma 3.4 that for every ε n : = ε/ (2 · n ) , there exists some L n > b ∈ Commut ( L n , ε n ). Set R n : = L − n + + L − n − + + · · · + L − + . Since X has straight finite decomposition complexity, there exist metric families X = { X } , X , . . . , X m such that X n − R n −−→ X n for n ∈ { , . . . , m } and X m is uniformlybounded. An elementary observation shows that(4.2) N ( L − n − + + ··· + ( L − + ( X n − ) L − n + −−−−−−→ N ( L − n − + + ··· + ( L − + ( X n ) . QUASI-LOCAL CHARACTERISATION OF L p -ROE ALGEBRAS 19 Thus, we can apply Lemma 4.2 inductively with L n , ε n , the operators obtained inthe previous iteration, and metric families in (4.2). After m steps, we approximate b by an operator b ′ which is a sum of 4 m operators in B , each of which is blockdiagonal with respect to the bounded family N ( L − m + + ··· + ( L − + ( X m ). Hence, oper-ators which are block diagonal with respect to it clearly have finite propagation.Consequently, b ′ ∈ Roe( X , B ). Finally, the distance between b and b ′ is at most8 ε + ε + ε + . . . ))) = ε ( 12 + + + . . . ) = ε by Lemma 4.2. So we finish the proof. (cid:3) Approximation via block cutdowns.
Finally, we complete the proof of Propo-sition 4.1 as promised before.The main di ffi culty is the lack of reflexivity of the L p -Bochner space L p ( X ; E )for general p and general Banach space E (see e.g. [2, 5, 6]), which impedesus from applying the original proof in [34] directly. Instead, we establish somesubstituting results in functional analysis and state them in the context of generalBanach spaces, which conceivably would be of independent interests. In the rest of this subsection, suppose X is a Banach space and ˆ X is a closed subspace ofthe dual space X ∗ , which separates points in X (i.e., for any nonzero ξ ∈ X , there existssome η ∈ ˆ X such that η ( ξ ) , i : ˆ X ֒ → X ∗ induces a surjectiveadjoint map i ∗ : X ∗∗ → ˆ X ∗ . Composing it with the canonical map from X into itsdouble dual X ∗∗ , we obtain the following map(4.3) τ : X → ˆ X ∗ . It is clear that τ is injective, as ˆ X separates points in X .For any θ ∈ ˆ X ∗ and η ∈ ˆ X , we use the notation h θ, η i for θ ( η ). Consider theBanach space B ( X , ˆ X ∗ ) of all bounded operators from X to ˆ X ∗ , equipped with the weak* operator topology (W*OT) defined as follows: a net { T α } converges to T in B ( X , ˆ X ∗ ) if and only if for any ξ ∈ X and any η ∈ ˆ X , we have h T α ( ξ ) , η i → h T ( ξ ) , η i . The strong* topology with respect to ˆ X on B ( X ) is defined as follows: a net { T α } converges to T in B ( X ) if and only if for any ξ ∈ X and any η ∈ ˆ X , we have k T α ( ξ ) − T ( ξ ) k → k T ∗ α ( η ) − T ∗ ( η ) k → . We say that ˆ X is a ∗ -invariant for a ∈ B ( X ) if a ∗ ( ˆ X ) ⊆ ˆ X . In this case, the restriction a ∗ | ˆ X belongs to B ( ˆ X ). Hence, its adjoint ( a ∗ | ˆ X ) ∗ belongs to B ( ˆ X ∗ ) as well. In orderto simplify notations, we write a ( ∗∗ ) instead of ( a ∗ | ˆ X ) ∗ . Clearly, for any ζ ∈ ˆ X ∗ and η ∈ ˆ X we have: h a ( ∗∗ ) ζ, η i = h ζ, a ∗ η i . In [34] ˇSpakula and Tikuisis considered the weak operator topology (WOT) instead. However,(WOT) and (W*OT) agree when X ∗ (cid:27) X and taking ˆ X : = X ∗ . , ZHIJIE WANG , AND JIAWEN ZHANG Moreover, it is easy to check that if ˆ X is a ∗ -invariant for some a ∈ B ( X ), then(4.4) a ( ∗∗ ) τ = τ a . In other words, the following diagram commutes: X (cid:31) (cid:127) τ / / a (cid:15) (cid:15) ˆ X ∗ a ( ∗∗ ) (cid:15) (cid:15) X (cid:31) (cid:127) τ / / ˆ X ∗ . We say that ˆ X is A ∗ -invariant for a subset A ⊆ B ( X ) if ˆ X is a ∗ -invariant for all a ∈ A . If G is a subgroup of invertible isometries in B ( X ) and ˆ X is G ∗ -invariant,then u ∗ ( ˆ X ) = ˆ X for all u ∈ G . It is clear that if u is an invertible isometry, then soare u ∗ and u ( ∗∗ ) . Moreover, ( u ∗ ) − = ( u − ) ∗ and ( u ( ∗∗ ) ) − = ( u − ) ( ∗∗ ) , which are denotedby u −∗ and u − ( ∗∗ ) , respectively. Now suppose ( u α ) and u are invertible isometriesin G and u α → u in the strong* topology with respect to ˆ X , then k u −∗ α η − u −∗ η k → η ∈ ˆ X . Indeed, we have(4.5) k u −∗ α η − u −∗ η k = k η − u ∗ α u −∗ η k = k u ∗ ( u −∗ η ) − u ∗ α ( u −∗ η ) k → . We have the following technical lemma, which generalises [34, Lemma 4.1].
Lemma 4.3.
Suppose G is an abelian subgroup of the group of invertible isometriesin B ( X ) , which is compact in the strong* topology with respect to ˆ X . Suppose ˆ X isG ∗ -invariant, and define G ′ : = { a ∈ B ( X , ˆ X ∗ ) : au = u ( ∗∗ ) a , ∀ u ∈ G } .Then there exists a unique idempotent linear contraction E G : B ( X , ˆ X ∗ ) → G ′ with thefollowing properties:1) For any b , b ∈ G and a ∈ B ( X , ˆ X ∗ ) , E G ( b ( ∗∗ )1 ab ) = b ( ∗∗ )1 E G ( a ) b .2) The restriction of E G to the unit ball of B ( X , ˆ X ∗ ) is (W*OT)-continuous.In this case, for any a ∈ B ( X , ˆ X ∗ ) , we have that (4.6) kE G ( a ) − a k ≤ sup u ∈ G k au − u ( ∗∗ ) a k . Proof.
Since G is compact with respective to the strong* topology, we consider thenormalised Haar measure µ G on G . Fix a ∈ B ( X , ˆ X ∗ ), the map( G , strong* topology) → ( B ( X , ˆ X ∗ ) , W ∗ OT)defined by u u − ( ∗∗ ) au is clearly continuous. For each ξ ∈ X and each a ∈ B ( X , ˆ X ∗ ),we may consider the following functional on ˆ X : φ ξ, a : η Z G h u − ( ∗∗ ) au ξ, η i d µ G ( u ) , whose norm is bounded by k a k · k ξ k . Therefore, we obtain a linear contraction E G : B ( X , ˆ X ∗ ) → B ( X , ˆ X ∗ ) given by E G ( a )( ξ ) = φ ξ, a , where ξ ∈ X and a ∈ B ( X , ˆ X ∗ ).It remains to check that E G satisfies the required properties. First of all, we showthat E G has image in G ′ . More precisely, E G ( a ) v = v ( ∗∗ ) E G ( a ) for any a ∈ B ( X , ˆ X ∗ ) QUASI-LOCAL CHARACTERISATION OF L p -ROE ALGEBRAS 21 and any v ∈ G . Given ξ ∈ X and η ∈ ˆ X , it follows from the right-invariance of theHaar measure µ G that hE G ( a ) v ξ, η i = Z G h u − ( ∗∗ ) auv ξ, η i d µ G ( u ) = Z G h v ( ∗∗ ) u − ( ∗∗ ) au ξ, η i d µ G ( u ) = Z G h u − ( ∗∗ ) au ξ, v ∗ η i d µ G ( u ) = hE G ( a ) ξ, v ∗ η i = h v ( ∗∗ ) E G ( a ) ξ, η i . Hence, it follows that E G ( a ) v = v ( ∗∗ ) E G ( a ).Given a ∈ B ( X , ˆ X ∗ ), ξ ∈ X and η ∈ ˆ X , we have |h ( E G ( a ) − a ) ξ, η i| ≤ Z G k u − ( ∗∗ ) au − a k · k ξ k · k η k d µ G ( u ) = Z G k au − u ( ∗∗ ) a k · k ξ k · k η k d µ G ( u ) ≤ sup u ∈ G k au − u ( ∗∗ ) a k ! · k ξ k · k η k . Hence, (4.6) holds. In particular, E G ( a ) = a for any a ∈ G ′ , which implies that E G : B ( X , ˆ X ∗ ) → G ′ is an idempotent.Now let us check that E G ( b ( ∗∗ )1 ab ) = b ( ∗∗ )1 E G ( a ) b for any b , b ∈ G and a ∈ B ( X , ˆ X ∗ ).Since G is abelian, we have hE G ( b ( ∗∗ )1 ab ) ξ, η i = Z G h u − ( ∗∗ ) b ( ∗∗ )1 ab u ξ, η i d µ G ( u ) = Z G h b ( ∗∗ )1 u − ( ∗∗ ) aub ξ, η i d µ G ( u ) = Z G h u − ( ∗∗ ) aub ξ, b ∗ η i d µ G ( u ) = hE G ( a ) b ξ, b ∗ η i = h b ( ∗∗ )1 E G ( a ) b ξ, η i , for any ξ ∈ X and any η ∈ ˆ X . Hence, E G ( b ( ∗∗ )1 ab ) = b ( ∗∗ )1 E G ( a ) b .In order to prove the (W*OT)-continuity of the restriction of E G to the unitball of B ( X , ˆ X ∗ ), we have to approximate the integration by finite Riemann sums uniformly in the weak* operator topology:Indeed, fix ξ ∈ X , η ∈ ˆ X and u ∈ G and for any ε >
0, from (4.5) there exists anopen neighbourhood V u of u in the strong* topology such that for all v ∈ V u andall a ∈ B ( X , ˆ X ∗ ) , we have |h v − ( ∗∗ ) av ξ, η i − h u − ( ∗∗ ) au ξ, η i| < ε. , ZHIJIE WANG , AND JIAWEN ZHANG Since { V u : u ∈ G } forms an open cover of G and G is compact in the strong*topology, there exists a finite subcover { V u , . . . , V u n } of G . Let W = V u and weput W k = V u k \ S k − i = W i for 1 < k ≤ n . Without loss of generality, we may assumethat { W k } nk = forms a non-empty Borel partition of G . Take an arbitrary point w k ineach W k for k = , . . . , n . Then for any a ∈ B ( X , ˆ X ∗ ) and u ∈ W k , we have that |h u − ( ∗∗ ) au ξ, η i − h w − ( ∗∗ ) k aw k ξ, η i| < ε. In particular, we have that (cid:12)(cid:12)(cid:12) hE G ( a ) ξ, η i − n X k = h w − ( ∗∗ ) k aw k ξ, η i µ G ( W k ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) n X k = Z W k h u − ( ∗∗ ) au ξ, η i d µ G ( u ) − n X k = Z W k h w − ( ∗∗ ) k aw k ξ, η i d µ G ( u ) (cid:12)(cid:12)(cid:12) ≤ n X k = Z W k (cid:12)(cid:12)(cid:12) h u − ( ∗∗ ) au ξ, η i − h w − ( ∗∗ ) k aw k ξ, η i (cid:12)(cid:12)(cid:12) d µ G ( u ) ≤ n X k = Z W k ε d µ G ( u ) = ε, for all a ∈ B ( X , ˆ X ∗ ) . Since the map a P nk = µ G ( W k ) w − ( ∗∗ ) k aw k is continuous in theweak* operator topology, it is not hard to see that the restriction of E G to the unitball of B ( X , ˆ X ∗ ) is (W*OT)-continuous as well.Finally, we check the uniqueness of E G . If we have another E : B ( X , ˆ X ∗ ) → G ′ satisfying all the conditions in the lemma, then: E G ( a ) = E ( E G ( a )) ( E fixes G ′ ) = E (cid:16) W ∗ OT − Z G u − ( ∗∗ ) au d µ G ( u ) (cid:17) = W ∗ OT − Z G E ( u − ( ∗∗ ) au )d µ G ( u ) (W ∗ OT-continuity on the unit ball) = W ∗ OT − Z G u − ( ∗∗ ) E ( a ) u d µ G ( u ) (Property 1)) = E G ( E ( a )) = E ( a ) ( E G fixes G ′ )for all a ∈ B ( X , ˆ X ∗ ) . Thus, E G = E and we complete the proof. (cid:3) Now let us return to the setting of Proposition 4.1. Let ( X , d ) be a propermetric space equipped with a Radon measure µ whose support is X . Let q be theconjugate exponent to p when p ∈ (1 , + ∞ ), and q = p =
1. Suppose E isa Banach space and ( e j ) j ∈ J is an equicontinuous family of positive contractions in C b ( X ) with uniformly disjoint supports.In order to apply Lemma 4.3, we put X = L p ( X ; E ) and ˆ X = L q ( X ; E ∗ ). Clearly, ˆ X is a closed subspace of the dual space X ∗ , and separates points in X by Lemma 2.2. QUASI-LOCAL CHARACTERISATION OF L p -ROE ALGEBRAS 23 For each j ∈ J , set A j = supp( e j ) and B = X \ (cid:16) F j ∈ J A j (cid:17) . We consider p j and q c in B ( L p ( X ; E )) given by p j ( ξ ) = χ A j ξ and q c ( ξ ) = χ B ξ for ξ ∈ L p ( X ; E ). We define that(4.7) G = X j ∈ J ( − α j p j + ( − β q c : ( α j ) j ∈ J ⊆ ( Z / J , β ∈ Z / , where the sum converges in (SOT) and each element in G can be presented bya function of the form P j ∈ J ( − α j χ A j + ( − β χ B (in the pointwise convergence) viathe faithful multiplication representation ρ : L ∞ ( X ) → B ( L p ( X ; E )).Since g = id for all g ∈ G , G becomes a subgroup of the invertible isometrygroup in B ( L p ( X ; E )), and clearly G is abelian. Also notice that ˆ X is L ∞ ( X ) ∗ -invariant as for any f ∈ L ∞ ( X ) ⊆ B ( L p ( X ; E )) and η ∈ ˆ X , we have that f ∗ ( η ) = f · η by pointwise multiplications as functions on X . Consequently, ˆ X is G ∗ -invariantsince G ⊆ ρ ( L ∞ ( X )). Moreover, the strong* topology on G with respect to ˆ X iscompact, as it is homeomorphic to the product topology on ( Z / J ∪{ β } . The next lemma is a replacement of [34, Corollary 4.2], where ˇSpakula andTikuisis work within the setting of von Neumann algebras. Instead, we providea direct and concrete proof here as follows:
Lemma 4.4.
As above, the group G is defined as in (4.7) and q is the conjugate exponentto p when p ∈ (1 , ∞ ) , and q = when p = . Let X = L p ( X ; E ) and ˆ X = L q ( X ; E ∗ ) .If G ′ = { a ∈ B ( X , ˆ X ∗ ) : au = u ( ∗∗ ) a , ∀ u ∈ G } , then there exists a (W*OT)-continuousidempotent linear contraction E : B ( X , ˆ X ∗ ) → G ′ given by the formula E ( x ) = X j ∈ J p ( ∗∗ ) j xp j + q ( ∗∗ ) c xq c , where the sum converges in (SOT). Moreover, E ( b ( ∗∗ )1 ab ) = b ( ∗∗ )1 E ( a ) b for any b , b ∈ Gand a ∈ B ( X , ˆ X ∗ ) . Consequently, we have that kE ( a ) − a k ≤ sup u ∈ G k au − u ( ∗∗ ) a k , for any a ∈ B ( X , ˆ X ∗ ) .Proof. It is clear that E is a (W*OT)-continuous linear map on B ( X , ˆ X ∗ ) and the sumdefining E converges in (SOT), so we leave the details to the readers.Let us first verify that E is a contraction. When p =
1, we have kE ( x ) ξ k ≤ X j ∈ J k xp j ξ k + k xq c ξ k ≤ k x k · (cid:16) X j ∈ J k χ A j ξ k + k χ B ξ k (cid:17) = k x k · k ξ k , It is worth noting that C ( X , E ∗ ) is not L ∞ ( X ) ∗ -invariant and this is the reason why we use L ( X ; E ∗ ) instead of C ( X , E ∗ ) when p = However, it is false for L ∞ ( X ; E ∗ ) and this is the reason why we use L ( X ; E ∗ ) instead of L ∞ ( X ; E ∗ )when p = , ZHIJIE WANG , AND JIAWEN ZHANG for any ξ ∈ L ( X ; E ) by Lemma 2.2. It implies that E is a contraction in this case.When p >
1, it follows from H ¨older’s inequality that (cid:12)(cid:12)(cid:12)D X j ∈ J p ( ∗∗ ) j xp j ξ + q ( ∗∗ ) c xq c ξ, η E(cid:12)(cid:12)(cid:12) ≤ X j ∈ J |h xp j ξ, p ∗ j η i| + |h xq c ξ, q ∗ c η i|≤ k x k · (cid:16) X j ∈ J k p j ξ k p · k p ∗ j η k q + k q c ξ k p · k q ∗ c η k q (cid:17) ≤ k x k · (cid:16) X j ∈ J k p j ξ k pp + k q c ξ k pp (cid:17) p · (cid:16) X j ∈ J k p ∗ j η k qq + k q ∗ c η k qq (cid:17) q = k x k · k ξ k p · k η k q , for any ξ ∈ L p ( X ; E ) and η ∈ L q ( X ; E ∗ ). This implies that kE ( x ) ξ k = (cid:13)(cid:13)(cid:13) X j ∈ J p ( ∗∗ ) j xp j ξ + q ( ∗∗ ) c xq c ξ (cid:13)(cid:13)(cid:13) ≤ k x k · k ξ k p by Lemma 2.2. Hence, E is a contraction in this case as well.Now we show that the image of E sits inside G ′ . Indeed, given any x ∈ B ( X , ˆ X ∗ ),any u = P j ∈ J ( − α j p j + ( − β q c ∈ G , ξ ∈ X and η ∈ ˆ X , we have that hE ( x ) u ξ, η i = D(cid:16) X j ∈ J p ( ∗∗ ) j xp j + q ( ∗∗ ) c xq c (cid:17)(cid:16) X j ∈ J ( − α j p j + ( − β q c (cid:17) ξ, η E = X j ∈ J ( − α j h p ( ∗∗ ) j xp j ξ, η i + ( − β h q ( ∗∗ ) c xq c ξ, η i = X j ∈ J ( − α j h xp j ξ, p ∗ j η i + ( − β h xq c ξ, q ∗ c η i . On the other hand, h u ( ∗∗ ) E ( x ) ξ, η i = D(cid:16) X j ∈ J p ( ∗∗ ) j xp j + q ( ∗∗ ) c xq c (cid:17) ξ, u ∗ η E = X j ∈ J h xp j ξ, p ∗ j u ∗ η i + h xq c ξ, q ∗ c u ∗ η i = X j ∈ J ( − α j h xp j ξ, p ∗ j η i + ( − β h xq c ξ, q ∗ c η i . Hence, E ( x ) u = u ( ∗∗ ) E ( x ) for all u ∈ G .Next, we show that E ( x ) = x for all x ∈ G ′ . In other words, E is an idempotentonto G ′ . Fix an x ∈ G ′ and for any u = P j ∈ J ( − α j p j + ( − β q c in G , we have that p ( ∗∗ ) j xp i = ( − α i p ( ∗∗ ) j x ( up i ) = ( − α i ( p ( ∗∗ ) j u ( ∗∗ ) ) xp i = ( − α i + α j p ( ∗∗ ) j xp i . It follows that p ( ∗∗ ) j xp i = i , j . Similarly, p ( ∗∗ ) j xq c = q ( ∗∗ ) c xp j = j ∈ J .Therefore, we have that x = (cid:16) X i ∈ J p i + q c (cid:17) ( ∗∗ ) x (cid:16) X j ∈ J p j + q c (cid:17) = X j ∈ J p ( ∗∗ ) j xp j + q ( ∗∗ ) c xq c = E ( x ) for all x ∈ G ′ . QUASI-LOCAL CHARACTERISATION OF L p -ROE ALGEBRAS 25 Moreover, for any b , b ∈ G and any x ∈ B ( X , ˆ X ∗ ) we have that E ( b ( ∗∗ )1 xb ) = X j ∈ J p ( ∗∗ ) j b ( ∗∗ )1 xb p j + q ( ∗∗ ) c b ( ∗∗ )1 xb q c = X j ∈ J b ( ∗∗ )1 p ( ∗∗ ) j xp j b + b ( ∗∗ )1 q ( ∗∗ ) c xq c b = b ( ∗∗ )1 E ( x ) b , where we use the fact that b k p j = p j b k and b k q c = q c b k for any j ∈ J and k ∈ { , } .The final conclusion follows from the uniqueness of E in Lemma 4.3 and (4.6)therein. So we finish the proof. (cid:3) Proof of Proposition 4.1.
Let the group G be defined as in (4.7), and the map E : B ( L p ( X ; E ) , L q ( X ; E ∗ ) ∗ ) → G ′ be the idempotent defined in Lemma 4.4. Recall thatby Lemma 2.2, the map τ : L p ( X ; E ) → L q ( X ; E ∗ ) ∗ defined in (4.3) is an isometricembedding, hence it induces the following isometric embedding ι : B ( L p ( X ; E )) = B ( L p ( X ; E ) , L p ( X ; E )) ֒ → B ( L p ( X ; E ) , L q ( X ; E ∗ ) ∗ ) . In other words, ι ( a ) = τ ◦ a for any a ∈ B ( L p ( X ; E )).Now we define another map E ′ : B ( L p ( X ; E )) → B ( L p ( X ; E )) by the formula E ′ ( z ) = X j ∈ J p j zp j + q c zq c for z ∈ B ( L p ( X ; E )) and the sum converges in (SOT) by Lemma 2.4. It followseasily from Equation (4.4) that the following diagram commutes B ( L p ( X ; E )) (cid:31) (cid:127) ι / / B ( L p ( X ; E ) , L q ( X ; E ∗ ) ∗ ) B ( L p ( X ; E )) E ′ O O (cid:31) (cid:127) ι / / B ( L p ( X ; E ) , L q ( X ; E ∗ ) ∗ ) . E O O Furthermore, we have that kE ′ ( z ) − z k = k ι ( E ′ ( z )) − ι ( z ) k = kE ( ι ( z )) − ι ( z ) k ≤ sup u ∈ G {k ι ( z ) u − u ( ∗∗ ) ι ( z ) k} , for any z ∈ B ( L p ( X ; E )). While for u ∈ G , it follows from Equation (4.4) that ι ( z ) u − u ( ∗∗ ) ι ( z ) = τ zu − u ( ∗∗ ) τ z = τ zu − τ uz . Combining the above facts together, we obtain that kE ′ ( z ) − z k ≤ sup u ∈ G {k zu − uz k} . Let e : = P j ∈ J e j . Since p j e = e j = ep j and q c e = eq c =
0, we have that E ′ ( eae ) = X j ∈ J p j eaep j + q c eaeq c = X j ∈ J e j ae j . Also notice that for any u = P j ∈ J ( − α j p j + ( − β q c in G , we have that eu = ue = P j ∈ J ( − α j e j . Since { A j } j ∈ J are pairwise 2 / L -disjoint, there exists an L -Lipschitz , ZHIJIE WANG , AND JIAWEN ZHANG map f ∈ C b ( X ) such that f | A j ≡ ( − α j χ A j for all j . Hence, k [ a , f ] k ≤ ε since a ∈ Commut( L , ε ), and we clearly have e j f = f e j = ( − α j e j . Therefore, we obtainthat ueae = (cid:16) X j ∈ J ( − α j e j (cid:17) ae = e f ae ≈ ε ea f e = ea (cid:16) X j ∈ J ( − α j e j (cid:17) = eaeu . Finally, we complete the proof by the following computation: k eae − X j ∈ J e j ae j k = kE ′ ( eae ) − eae k ≤ sup u ∈ G k eaeu − ueae k ≤ ε, for any a ∈ Commut( L , ε ). (cid:3) Acknowledgments . The first-named author would like to thank Tomasz Kaniafor helpful discussions on Banach space valued L p -spaces.R eferences [1] Pere Ara, Kang Li, Fernando Lled ´o, and Jianchao Wu. Amenability and uniform Roe algebras. J. Math. Anal. Appl. , 459(2):686–716, 2018.[2] Bahattin Cengiz. On the duals of Lebesgue-Bochner L p spaces. Proc. Amer. Math. Soc. ,114(4):923–926, 1992.[3] Yeong Chyuan Chung and Kang Li. Rigidity of ℓ p R oe-type algebras. Bulletin of the LondonMathematical Society , 50(6):1056–1070, 2018.[4] Andreas Defant and Klaus Floret.
Tensor Norms and Operator Ideals . Number 176 in North-Holland Mathematics Studies. North-Holland Publishing Co., 1993.[5] J. Diestel and J. J. Uhl, Jr.
Vector measures . American Mathematical Society, Providence, R.I.,1977. With a foreword by B. J. Pettis, Mathematical Surveys, No. 15.[6] N. Dinculeanu.
Vector measures . International Series of Monographs in Pure and AppliedMathematics, Vol. 95. Pergamon Press, Oxford-New York-Toronto, Ont.; VEB DeutscherVerlag der Wissenschaften, Berlin, 1967.[7] Alexander Dranishnikov and Michael Zarichnyi. Asymptotic dimension, decompositioncomplexity, and Haver’s property C.
Topology Appl. , 169:99–107, 2014.[8] Alexander Engel. Index theory of uniform pseudodi ff erential operators. preprint , 2018.arXiv:1502.00494.[9] Alexander Engel. Rough index theory on spaces of polynomial growth and contractibility. preprint , 2018. arXiv:1505.03988.[10] Eske Ellen Ewert and Ralf Meyer. Coarse geometry and topological phases. preprint , 2018.arXiv:1802.05579.[11] Erik Guentner and Jerome Kaminker. Exactness and the Novikov conjecture. Topology ,41(2):411–418, 2002.[12] Erik Guentner, Romain Tessera, and Guoliang Yu. A notion of geometric complexity and itsapplication to topological rigidity.
Inventiones mathematicae , 189(2):315–357, 2012.[13] Erik Guentner, Romain Tessera, and Guoliang Yu. Discrete groups with finite decompositioncomplexity.
Groups Geom. Dyn. , 7(2):377–402, 2013.[14] Ra ff ael Hagger, Marko Lindner, and Markus Seidel. Essential pseudospectra and essentialnorms of band-dominated operators. J. Math. Anal. Appl. , 437(1):255–291, 2016.[15] Nigel Higson and John Roe. Amenable group actions and the Novikov conjecture.
J. ReineAngew. Math. , 519:143–153, 2000.[16] Julian Kellerhals, Nicolas Monod, and Mikael Rørdam. Non-supramenable groups acting onlocally compact spaces.
Doc. Math. , 18:1597–1626, 2013.[17] Yosuke Kubota. Controlled topological phases and bulk-edge correspondence.
Commun.Math. Phys. , 349(2):493–525, 2017.[18] B. V. Lange and V. S. Rabinovich. Noethericity of multidimensional discrete convolutionoperators.
Mat. Zametki , 37(3):407–421, 462, 1985.
QUASI-LOCAL CHARACTERISATION OF L p -ROE ALGEBRAS 27 [19] Kang Li and Hung-Chang Liao. Classification of uniform Roe algebras of locally finite groups. J. Operator Theory , 80(1):25–46, 2018.[20] Kang Li and Rufus Willett. Low-dimensional properties of uniform Roe algebras.
J. LondonMath. Soc. , (2) 97:98124, 2018.[21] Marko Lindner and Markus Seidel. An a ffi rmative answer to a core issue on limit operators. J. Funct. Anal. , 267(3):901–917, 2014.[22] Peter W Michor.
Functors and categories of Banach spaces: tensor products, operator ideals andfunctors on categories of Banach spaces , volume 651. Springer, 2006.[23] Narutaka Ozawa. Amenable actions and exactness for discrete groups.
C. R. Acad. Sci. ParisS´er. I Math. , 330(8):691–695, 2000.[24] N. Christopher Phillips. Analogs of Cuntz algebras on L p spaces. preprint, arXiv:1309.4196 ,2012.[25] N. Christopher Phillips. Crossed products of L p operator algebras and the K -theory of C untzalgebras on L p spaces. preprint, arXiv:1309.6406 , 2013.[26] John Roe. An index theorem on open manifolds. I, II. J. Di ff erential Geom. , 27(1):87–113, 115–136, 1988.[27] 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 MathematicalSciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996.[28] John Roe.
Lectures on coarse geometry , volume 31 of
University Lecture Series . American Math-ematical Society, Providence, RI, 2003.[29] Mikael Rørdam and Adam Sierakowski. Purely infinite C ∗ -algebras arising from crossedproducts. Ergodic Theory Dynam. Systems , 32(1):273–293, 2012.[30] Eduardo Scarparo. Characterizations of locally finite actions of groups on sets.
Glasg. Math.J. , 60(2):285–288, 2018.[31] Markus Seidel. Fredholm theory for band-dominated and related operators: a survey.
LinearAlgebra Appl. , 445:373–394, 2014.[32] Georges Skandalis, Jean-Louis Tu, and Guoliang Yu. The coarse Baum-Connes conjecture andgroupoids.
Topology , 41(4):807–834, 2002.[33] J´an ˇSpakula. Uniform K -homology theory. J. Funct. Anal. , 257(1):88–121, 2009.[34] J´an ˇSpakula and Aaron Tikuisis. Relative commutant pictures of R oe algebras. preprint,arXiv:1707.04552 , 2017.[35] J´an ˇSpakula and Rufus Willett. A metric approach to limit operators. Trans. Amer. Math. Soc. ,369(1):263–308, 2017.[36] J´an ˇSpakula and Jiawen Zhang. Quasi-locality and property a. arXiv preprint arXiv:1809.00532 ,2018.[37] ShuYun Wei. On the quasidiagonality of Roe algebras.
Sci. China Math. , 54(5):1011–1018, 2011.[38] Stuart White and Rufus Willett. Cartan subalgebras in uniform Roe algebras. preprint , 2017.arXiv:1808.04410.[39] Wilhelm Winter and Joachim Zacharias. The nuclear dimension of C ∗ -algebras. Adv. Math. ,224(2):461–498, 2010.[40] Guoliang Yu. Coarse Baum-Connes conjecture.
K-Theory , 9:199–221, 1995.[41] Guoliang Yu. Localization algebras and the coarse Baum-Connes conjecture.
K-Theory ,11(4):307–318, 1997.[42] Guoliang Yu. The Novikov conjecture for groups with finite asymptotic dimension.
Ann. ofMath. (2) , 147(2):325–355, 1998.[43] Guoliang Yu. The coarse Baum-Connes conjecture for spaces which admit a uniform embed-ding into Hilbert space.
Invent. Math. , 139(1):201–240, 2000.[44] Jiawen Zhang. Extreme cases of limit operator theory on metric spaces.
Integral Equations andOperator Theory , 90(6):73, 2018. , ZHIJIE WANG , AND JIAWEN ZHANG I nstitute of M athematics of the P olish A cademy of S ciences , ul . ´S niadeckich
8, 00-656W arsaw , P oland . E-mail address : [email protected] C ollege of M athematics P hysics and I nformation E ngineering , J iaxing U niversity , Y uexiu R oad (S outh ) 56, 314001 Z hejiang , C hina . 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 ::