Quantum cellular automata are a coarse homology theory

Quan tum cellular automata are a coarse homology theory Matthias Ludewig Univ ersit¨ at Greifsw ald Marc h 27, 2026 Abstract W e show that quantum cellular automata naturally form the degree-zero part of a coarse homology theory . The recent result of Ji and Y ang that the space of QCA forms an Ω-sp ectrum in the sense of algebraic top ology is a direct consequence of the formal prop erties of coarse homology theories. 1 In tro duction Quan tum cellular automata (QCA) are certain lo cality preserving automorphisms of C ∗ - algebras that arise as tensor products o ver matrix algebras attached to p oin ts in space. In a very recent pap er [4], Ji–Y ang pro ve strong structural results on the space of QCA, whic h they call the QCA c onje ctur e : The space of QCA on Z n is homotop y equiv alen t to the lo op space of the space of QCA on Z n +1 . In this pap er, w e give a somewhat simplified pro of, as well as a conceptual reason for this equiv alence: QCA form a coarse homology theory . T o define QCA on non-compact spaces, one t ypically starts with a discrete metric space X and attaches matrix algebras A x to eac h p oint in space. Then, an automorphism α of the infinite tensor pro duct A X := O x ∈ X A x is a QCA if there exists R > 0 such that α sends each of the subalgebras A x to the tensor pro duct of A y o ver those y in an R -neighborho o d around x . One of the main p oints I would like to mak e in this pap er is that the “correct” type of space to consider for studying QCA is not a metric space but a b ornolo gic al c o arse sp ac e : The observ ation is that only the large scale structure of space matters for QCAs; how ev er, next to the large scale (coarse) structure, metric spaces also carry an unnecessary small scale structure (a top ology , even a uniform structure). 1 A b ornological coarse space is a set X together with the information on whic h sets are to b e considered b ounde d , as w ell as a collection of subsets of X × X whic h are called entour ages or c ontr ol le d subsets and which enco de uniform b ounds from ab o ve on distances (see § 2.1 for precise definitions). The collection B of b ounded subsets of a b ornological coarse space X is partially ordered by inclusion, and a net on X is just a functor from the p oset B to the category of finite-dimensional C ∗ -algebras and injective ∗ -homomorphisms, whic h satisfies a certain co con tin uity hypothesis. Particularly nice examples are the lo c al matrix nets or spin systems , which are given b y a tensor pro duct of matrix algebras indexed b y a lo cally finite subset of sites. The crucial observ ation is now that the notion of QCA can b e form ulated using only the coarse structure: An automorphism α of a local matrix net is a QCA if there exists an en tourage E ⊂ X × X suc h that α ( A x ) commutes with A y unless ( x, y ) ∈ E ; in other words, only pairs of p oin ts from E interact via α . The reason for w orking with the larger class of coarse spaces instead of the more rigid framew ork of metric spaces is not the desire to study QCA on strange coarse spaces. Instead, the reason is that—in con trast to the category of metric spaces—the category of b ornological coarse spaces is w ell-b eha ved enough to do homotop y theory with it [1]. This leads to the notion of a c o arse homolo gy the ory , whic h is a functor from b ornological coarse spaces to sp ectra that sends close maps to homotopic maps and satisfies a May er–Vietoris axiom for decomp ositions of spaces. Imp ortan t examples include coarse K -homology and coarse ordinary homology . Using the May er–Vietoris prop erty of coarse homology theories, the QCA hypothesis Q ( X ⊗ Z ) ∼ = Ω Q ( X ) from [4] is a direct consequence of the follo wing theorem, which is the main result of this pap er and identifies the QCA group as a generalized homology in v arian t of large-scale geometry . Theorem. Ther e is a c o arse homolo gy the ory Q such that the de gr e e zer o gr oup Q 0 ( X ) is the gr oup QCA( X ) of quantum c el lular automata on X . The category of lo cal matrix nets is to o rigid for the homotopical constructions required for the construction of Q . T o remedy this, inspired b y [4], we in tro duce in this pap er the new notion of an Azumaya net , whic h pla ys a role analogous to Azumay a algebras in algebra. These are nets A such that there exists another net A ′ on the same space such that we ha ve a (controlled) isomorphism A ⊗ A ′ ∼ = B of nets, where B is a lo cal matrix net. This is a straigh tforward generalization of the notion of an Azuma ya algebra ov er a comm utativ e ring R , whic h is an R -algebra A suc h that there exists an R -algebra A ′ with A ⊗ R A ′ ∼ = M n ( R ). Denoting by Az ( X ) the symmetric monoidal category of Azumay a nets ov er X , there are natural maps K ( Az ( X )) → Ω K ( Az ( X ⊗ R )), where K denotes the algebraic K -theory sp ectrum for symmetric monoidal categories. The spectrum Q ( X ) is then defined as Q ( X ) = colim n →∞ Ω n +1 K ( Az ( X ⊗ R n )) , 2 where the colimit is tak en with resp ect to the maps ab o ve. One further new result that follo ws directly from this construction is that we ha ve a canonical isomorphism QCA( Z n ) ∼ = K 0 ( Az ( Z n − 1 )) . In other w ords, the group of QCA on Z n is iden tified with the Grothendieck group of Azuma ya nets in one dimension lo w er. F or n = 1, this isomorphism is precisely the GNVW index [3], but in higher dimensions, this seems to b e new. Ac knowledgemen ts. I w ould lik e to thank Christoph Winges, Daniel Kaspro wski and Ulric h Bunk e for helpful discussions. I also gratefully ackno wledge supp ort from SFB 1085 “Higher in v arian ts” funded b y the German Research F oundation (DFG). Con ten ts 1 Introduction 1 2 Nets and QCA on coarse spaces 3 2.1 Preliminaries on coarse spaces . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Nets on coarse spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Nets and coarse maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 T ensor factors and homomorphisms . . . . . . . . . . . . . . . . . . . . . . 11 2.5 QCA and coarsely lo cal automorphisms . . . . . . . . . . . . . . . . . . . . 14 2.6 Quan tum circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3 A coarse homology theory 19 3.1 QCA and K -theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2 The May er-Vietoris axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.3 Definition of the coarse homology theory . . . . . . . . . . . . . . . . . . . 27 3.4 QCA and Azuma ya nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2 Nets and QCA on coarse spaces In this section, w e give a brief recollection of the basic notions of coarse geometry , b efore dev eloping the theory of nets on b ornological coarse spaces and giving the definition of the group of quantum cellular automata. 2.1 Preliminaries on coarse spaces Definition 2.1. Let X b e a set. (1) A b ornolo gy on a set X is a collection B of subsets of X (called b ounde d subsets) whic h is closed under taking finite unions and subsets and with the prop ert y that S B = X . A b ornolo gic al sp ac e is a set together with a bornology . 3 (2) A c o arse structur e on X is a collection C of subsets of X × X (called entour ages ), whic h con tains the diagonal ∆ = { ( x, x ) | x ∈ X } and which is closed under taking subsets, finite unions, comp ositions E ◦ F = { ( x, z ) ∈ X × X | ∃ y ∈ X : ( x, y ) ∈ F , ( y , z ) ∈ E } and inv erses E − 1 = { ( y , x ) ∈ X × X | ( x, y ) ∈ E } . A c o arse sp ac e is a set together with a coarse structure. F or a subset Y ⊆ X of a coarse space X and an entourage E , w e denote the E -fattening of Y b y Y E = { x ∈ X | ∃ y ∈ Y : ( x, y ) ∈ E } . Definition 2.2 (b ornological coarse spaces [1]). A b ornolo gic al c o arse sp ac e is a set X together with a b ornology B and a coarse structure C , which are compatible in the sense that for any B ⊆ X and any en tourage E ∈ C , the fattening B E is b ounded again. W e usually just write X instead of ( X , B , C ) for a b ornological coarse space, suppressing the b ornology and coarse structure in notation. Remark 2.3. If C is a coarse structure on a set X , then there is a canonical b ornology determined by the coarse structure, consisting of those subsets B ⊆ X suc h that B × B is an en tourage of C . Hence an y coarse space is canonically a b ornological coarse space. Ho w ever, b ornological coarse spaces allow for more flexibilit y with resp ect to the b ornology , which is sometimes useful. Example 2.4. If E is any collection of subsets of X × X with ∆ ⊆ S E , then one can consider the smallest coarse structure C = ⟨E ⟩ con taining E , the c o arse structur e gener ate d by E . Example 2.5. If d is a (pseudo-) metric on X , then the metric c o arse structur e is the coarse structure generated by the en tourages { ( x, y ) ∈ X × X | d ( x, y ) ≤ R } for R ≥ 0. The b ornology induced b y this coarse structure is the collection of subsets of X with finite d -diameter. Whenev er w e view metric spaces as coarse spaces (such as Z n or R n ), w e alw ays understand them to b e equipp ed with the metric coarse structure and the induced b ornology , unless otherwise sp ecified. Example 2.6. Given tw o b ornological coarse spaces X and X ′ , there is a b ornological coarse space X ⊗ X ′ whic h has underlying set X × X ′ , bornology B × B ′ and coarse structure C × C ′ , the elemen twise cartesian pro duct. As noted in [1, Example 2.32.], the b ornological coarse space X ⊗ X ′ is not the categorical pro duct of X and X ′ ; the latter 4 do es exist and has the same underlying set and coarse structure, but a differen t b ornology . The pro duct ⊗ yields an auxiliary symmetric monoidal structure on BornCoarse . In particular, we ha ve canonical identifications R n − 1 ⊗ R = R n , Z n − 1 ⊗ Z ∼ = Z n . Definition 2.7 (coarse maps). Let X and X ′ b e b ornological coarse spaces and let f : X → X ′ b e a map. (1) f is called pr op er if for ev ery B ′ ⊆ X ′ b ounded, f − 1 ( B ′ ) ⊆ X is b ounded. (2) f is c ontr ol le d if for ev ery entourage E of X , ( f × f )( E ) is an entourage of X ′ . (3) f is called c o arse if it is prop er and con trolled. W e denote b y BornCo arse the category of b ornological coarse spaces and coarse maps. What allo ws us to do homotop y theory with bornological coarse spaces is the follo wing notion of equiv alence for coarse maps. Definition 2.8 (closeness). Let X and X ′ b e coarse spaces. (1) f , g : X → X ′ are close if the set ( f × g )(∆) = { ( f ( x ) , g ( x )) | x ∈ X } is an en tourage of X ′ . (2) A coarse map f : X → X ′ is a c o arse e quivalenc e if there exists a coarse map g : X ′ → X suc h that the comp ositions f ◦ g and g ◦ f are each close to the identit y . Example 2.9. F or any dimension n , the inclusion maps Z n  → R n are coarse equiv a- lences. How ever, R n is not coarsely equiv alen t to R m for n  = m . T o formulate supp ort conditions in coarse geometry that are inv arian t under coarse fattenings, we use the notion of a big family , defined as follows. Definition 2.10. A big family on X is a collection Y of subsets of X that is closed under taking subsets, finite unions and fattenings. An y subset Y ⊆ X generates a big family of X that w e denote by { Y } , given b y { Y } = { Z ⊆ X | ∃ E ∈ C : Z ⊆ Y E } . If Y and Z are big families, then their element wise union and intersection Y ⋓ Z = { Y ∪ Z | Y ∈ Y , Z ∈ Z } Y ⋒ Z = { Y ∩ Z | Y ∈ Y , Z ∈ Z } are again big families. In the ordinary homotopy theory of top ological spaces, con tractible spaces are mo- tivic al ly trivial , in the sense that they ev aluate trivially on an y homology theory . In the theory of coarse spaces, this role is pla yed b y so-called flasque spaces: 5 Definition 2.11 (flasqueness). A b ornological coarse space X is called flasque if there exists a coarse map f : X → X with the following prop erties: (1) f is close to the identit y map of X ; (2) F or every b ounded set B ⊆ X , there exists n ∈ N such that f n ( X ) ∩ B = ∅ , where f n is the n -th iterate of f ; (3) F or every en tourage E , the set [ n ∈ N ( f n × f n )( E ) is an en tourage. Suc h a map f is called a map witnessing the flasqueness . 2.2 Nets on coarse spaces Let X b e a bornological coarse space. The b ornology B of X is a partially ordered set and hence may b e viewed as a category with exactly one morphism for eac h inclusion B ⊆ B ′ . Definition 2.12 (nets). A net on X is a functor A : B − → ( category of finite-dimensional C ∗ -algebras and injective ∗ -homomorphisms ) , B 7− → A B . with the follo wing prop erty: Whenever B ⊆ X is b ounded and B ′ ⊆ B is a directed subset suc h that S B ′ = B , then canonical map colim B ′ ∈B ′ A B ′ − → A B (2.1) is an isomorphism. The prop erty (2.1) of the functor A is there to a void pathological phenomena, see Remark 2.20 b elo w. Giv en a net A , w e define for an y subset Y ⊆ X A Y := colim B ∈B B ⊆ Y A B , where the colimit is taken in the category of ∗ -algebras. W e could also take the limit in the category of C ∗ -algebras (which w ould amount to taking the completion of A Y ), but for the curren t pap er it is more conv enient to work with uncompleted algebras. By (2.1), w e can as w ell tak e the colimit o v er an arbitrary directed system of b ounded subsets co vering B instead. Since the structure maps of A are required to b e injectiv e, all the algebras A Y for Y ⊆ X may b e viewed as subalgebras of A X , and making these iden tifications, A X is just the union of all A B . 6 Definition 2.13. If A and B are nets on X , their tensor pr o duct A ⊗ B is defined b y ( A ⊗ B ) B := A B ⊗ B B . Definition 2.14 (homomorphisms). Let A , B b e t w o nets. A homomorphism of nets ϕ : A → B is a ∗ -homomorphism ϕ : A X → B X whic h is controlled in the sense that there exists an entourage E of X such that for all b ounded sets B ⊆ X , w e ha ve ϕ ( A B ) ⊆ B B E . In this case, w e sa y more precisely , that ϕ is E -c ontr ol le d . W e call ϕ lo c al if it is ∆- con trolled. Let ϕ : A → B b e a homomorphism of nets suc h that its controlled ∗ -homomorphism A X → B X is an isomorphism. If the inv erse ∗ -homomorphism is again con trolled, then w e sa y that ϕ is an isomorphism of nets. W e denote b y Net ( X ) the group oid of nets on X whose morphisms are (con trolled) isomorphisms of nets. Definition 2.15 (types of nets). Let A b e a net on X . (1) A is semilo c al if whenev er Y , Y ′ ⊆ X satisfy Y ∩ Y ′ = ∅ , then A Y and A Y ′ comm ute inside A Y ⊔ Y ′ . (2) A is lo c al if it is semilo cal and whenever Y , Y ′ ⊆ X satisfy Y ∩ Y ′ = ∅ , then the inclusion maps induce an isomorphism A Y ⊗ A Y ′ ∼ = A Y ⊔ Y ′ . (3) A is called a matrix net if A B is a full matrix algebra for each B ⊆ X b ounded. (4) A is called Azumaya if there exists a net A ′ on X and a lo cal matrix net B on X suc h that A ⊗ A ′ ∼ = B . W e write Loc ( X ) ⊆ Az ( X ) ⊆ Net ( X ) for the full subgroup oids of lo cal matrix nets, resp ectively Azuma ya nets. Example 2.16. Whenever q : X → N is a function with lo cally finite supp ort (meaning that for eac h b ounded set B ⊆ X , w e ha ve q ( x ) = 1 for all but finitely man y x ∈ B ), we obtain a lo cal matrix net b y setting A B := O x ∈ B M q ( x ) ( C ) , where M q ( C ) denotes the algebra of complex q × q matrices. This is the class of nets considered in [4]. Any lo cal matrix net is isomorphic in Net ( X ) to one of this form. The more general class of lo cal matrix nets ma y b e viewed as a “co ordinate free v ersion” of these nets. 7 Remark 2.17. While the terminology “lo cal matrix net” seems more appropriate in our con text, w e remark that these are often called spin system in the mathematical ph ysics literature. Remark 2.18. It follows from Corollary 2.29 that any Azumay a net on X is isomorphic to a semilocal net A , b ecause subnets of local nets are necessarily semilocal. Throughout, w e ma y therefore replace Az ( X ) by the full sub category of semilo cal Azumay a nets. Definition 2.19 (supp ort). Let A b e a net on X . A subset Y ⊆ X is a supp ort for A if for ev ery B ∈ B , w e ha ve A B = A Y ∩ B . If Y is a big family in X , w e write Loc ( Y ) ⊆ Loc ( X ) , Az ( Y ) ⊆ Az ( X ) , Net ( Y ) ⊆ Net ( X ) for the full sub categories of those nets that are supp orted on some mem b er Y of Y . Remark 2.20. Any lo cal net has a lo cally finite support Y , meaning that Y ∩ B is finite for each B ⊆ X b ounded. Indeed, b y lo calit y , w e ha ve A F = O x ∈ F A x for each finite subset F ⊆ X . No w by (2.1), w e ha ve A B = colim F ⊆ B finite A F = O x ∈ B A x . Since A B is finite-dimensional, we ha ve A x  = C for only finitely many x ∈ B . Without (2.1) on the functor A , there are pathological examples of lo cal nets without lo cally finite supp ort using the axiom of choice. Namely , if U is a free ultrafilter on X (which alw ays exists as so on as X is infinite), we ma y define a function µ : B → { 0 , 1 } by µ ( B ) = 0 if B / ∈ U and µ ( B ) = 1 if B ∈ U . This function is additiv e b ecause filters cannot con tain t wo disjoin t sets. Therefore, setting A B := M 2 ( C ) ⊗ µ ( B ) defines a lo cal net on X . A do es not hav e lo cally finite supp ort b ecause U do es not con tain an y finite sets. If A is a net on Y ⊆ X , we obtain a net ˜ A on X giv en by ˜ A B := A B ∩ Y . This giv es iden tifications of nets on Y with nets on X . Using these iden tifications, we ha v e canonical isomorphisms Az ( Y ) ∼ = colim Y ∈Y Az ( Y ) , Loc ( Y ) ∼ = colim Y ∈Y Loc ( Y ) , Net ( Y ) ∼ = colim Y ∈Y Net ( Y ) . (2.2) 8 2.3 Nets and coarse maps If f : X → X ′ is a coarse map, then w e may form the pushforw ard net f ∗ A , given b y ( f ∗ A ) B ′ = A f − 1 ( B ′ ) for B ′ ⊆ X ′ b ounded, whic h is a net on X ′ . This is well defined b ecause f is prop er. W e ha ve ( f ∗ A ) X ′ = colim B ′ ∈B ′ ( f ∗ A ) B ′ = colim B ′ ∈B ′ A f − 1 ( B ′ ) = A X , since { f − 1 ( B ′ ) | B ′ ∈ B ′ } ⊆ B is cofinal. Hence if ϕ : A → B is a morphism of nets, then we obtain a ∗ -homomorphism f ∗ ϕ : ( f ∗ A ) X = A X φ X − − − − → B X = ( f ∗ B ) X . If ϕ is E -controlled, then for eac h B ′ ⊆ X ′ b ounded, we ha ve ϕ (( f ∗ A ) B ′ ) = ϕ ( A f − 1 ( B ′ ) ) ⊆ B f − 1 ( B ′ ) E ⊆ B f − 1 ( B ′ ( f × f )( E ) ) = ( f ∗ B ) B ′ ( f × f )( E ) , hence f ∗ ϕ is ( f × f )( E )-con trolled. The pushforw ard functor f ∗ resp ects tensor pro ducts, in the sense that there are canonical isomorphisms f ∗ ( A ⊗ B ) ∼ = f ∗ A ⊗ f ∗ B . Com bining the ab ov e observ ations, we obtain that for each coarse map f : X → X ′ , we get a symmetric monoidal functor f ∗ : Net ( X ) − → Net ( X ′ ) . One easily chec ks that the pushforw ard of a lo cal net is again lo cal. It follows that f ∗ also sends Azumay a nets to Azumay a nets, so f ∗ restricts to functors f ∗ : Az ( X ) − → Az ( X ′ ) , f ∗ : Loc ( X ) − → Loc ( X ′ ) . If f : X → X ′ and g : X ′ → X ′′ are t wo coarse maps, then we hav e ob vious natural isomorphisms of functors ( g ◦ f ) ∗ ∼ = g ∗ f ∗ . These structures assem ble to (pseudo-)functors Net , Az , Loc : BornCoarse − → CMon ( Ca t ) from b ornological coarse spaces to the bicategory of symmetric monoidal categories (i.e., comm utative monoid ob jects in Ca t ). Prop osition 2.21. Supp ose that f , g : X → X ′ ar e close. Then we have natur al isomor- phisms of functors f ∗ ∼ = g ∗ : Net ( X ) → Net ( X ′ ) . Sinc e they ar e ful l sub c ate gories, the same is true for the pushforwar d functors on Azumaya and lo c al nets. 9 Pr o of. F or any net A on X , w e hav e the identification ( f ∗ A ) X = A X = ( g ∗ A ) X of the global algebras. This is clearly natural in A , but w e need to sho w that this isomorphism is controlled. T o this end, let E = ( g × f )(∆), whic h is an en tourage of X ′ as f and g are close. Then g − 1 ( B ′ E ) = { z ∈ X | g ( z ) ∈ B ′ E } = { z ∈ X | ∃ x ′ ∈ B ′ : ( g ( z ) , x ′ ) ∈ E } = { x ∈ X | f ( x ) ∈ B ′ } = f − 1 ( B ′ ) , hence ( f ∗ A ) B ′ = A f − 1 ( B ′ ) = A g − 1 ( B ′ E ) = ( g ∗ A ) B ′ E . Hence the canonical ∗ -isomorphism ( f ∗ A ) X → ( g ∗ A ) X is con trolled b y E . Swapping the roles of f and g , we see that the inv erse is con trolled b y the inv erse entourage E − 1 . □ Corollary 2.22. F or any subset Y ⊆ X , the c anonic al functor Net ( Y ) − → Net ( { Y } ) is an e quivalenc e. The same is true for the sub c ate gories of Azumaya, r esp e ctively lo c al nets. Pr o of. This follows from Prop. 2.21, as for all en tourages E and F of X with E ⊆ F , the inclusion map Y E  → Y F is a coarse equiv alence. □ Prop osition 2.23. If X is flasque, then ther e exists a symmetric monoidal and faithful endofunctor S : Net ( X ) → Net ( X ) to gether with a natur al isomorphism S ∼ = id ⊗ S. This functor r estricts to a functor on Az ( X ) and Loc ( X ) (and so do es the natur al tr ans- formation b e c ause b oth ar e ful l sub c ate gories). Pr o of. Let f : X → X be the map witnessing flasqueness. F or an y net A on X , w e set S ( A ) := A ⊗ f ∗ A ⊗ f 2 ∗ A ⊗ f 3 ∗ A ⊗ · · · . This infinite tensor pro duct is to b e in terpreted as S ( A ) B = ∞ O n =0 f − n ( B ) ∩ B  = ∅ A f − n ( B ) , whic h is actually a finite tensor pro duct as f n ( X ) ∩ B = ∅ for all but finitely man y n . This defines a net on X . If A is Azuma y a or local matrix, then S ( A ) has these properties as well. 10 If ϕ : A → B is an E -controlled morphism, then we obtain a ∗ -homomorphism S ( A ) X → S ( B ) X b y applying ϕ to each tensor factor. This homomorphism is controlled b y ∞ [ n =0 ( f n × f n )( E ) , whic h is an en tourage b y the prop erties of f , hence it defines a homomorphism of nets. Finally , since f is close to the iden tit y , w e ha ve a natural isomorphism S ( A ) ∼ = f ∗ S ( A ) b y Prop. 2.21, which in turn is clearly naturally isomorphic to A ⊗ S ( A ), in fact by a lo cal homomorphism. □ 2.4 T ensor factors and homomorphisms Let B b e a net on X . A subnet of B is a net A on X such that A B ⊆ B B for eac h B ⊆ X b ounded. W e write A ⊆ B if A is a subnet of B . Definition 2.24. Let B b e a net and let A ⊆ B b e a subnet. The c ommutant of A in B is the net A ′ giv en b y A ′ B = ( A X ) ′ ∩ B B , where ( A X ) ′ = { b ∈ B X | ∀ a ∈ A X : ab = ba } is the comm utan t of A X in B X . A is a tensor factor of B if the canonical homomorphism A ⊗ A ′ → B is an isomorphism of nets. Notice that A ′ X = ( A X ) ′ and that multiplication induces a canonical algebra homo- morphism µ : A X ⊗ ( A X ) ′ → B X whic h restricts to ∗ -homomorphisms ( A ⊗ A ′ ) B = A B ⊗ ( A X ) ′ ∩ B B − → B B (2.3) for ev ery B ⊆ X . How ever, it is generally not clear that these maps are isomorphisms, even if µ is an isomorphism globally . W e ha ve the follo wing criterion to hav e an isomorphism of nets, compare [4, Lemma 63]. Lemma 2.25. A subnet A ⊆ B of a net B is a tensor factor if and only if the multipli- c ation map µ : A X ⊗ ( A X ) ′ → B X is inje ctive and ther e exists an entour age E such that B B ⊆ A B E · A ′ B E . (2.4) for al l b ounde d sets B ⊆ X . Pr o of. (= ⇒ ) Suppose that A is a tensor factor of B . Then m ultiplication yields a con- trolled ∗ -homomorphism µ : ( A ⊗ A ′ ) X ∼ = A X ⊗ ( A X ) ′ → B X whose inv erse is con trolled as well. If E is a con trol for the inv erse, then w e ha ve B B ⊆ µ  ( A ⊗ A ′ ) B E  = A B E · A ′ B E . 11 ( ⇐ =) Supp ose that µ is injectiv e and that (2.4) holds. By (2.4), we ha ve B X = [ B ∈B B B ⊆ [ B ∈B A B E · A ′ B E = A X ⊗ ( A X ) ′ . This shows that µ is also surjectiv e, hence an isomorphism. µ is alwa ys controlled (in fact, lo cal) and (2.4) states precisely that the inv erse is E -controlled. □ Definition 2.26. If A , B are nets and ϕ : A → B is a homomorphism, then the image net ϕ ∗ A is defined b y ( ϕ ∗ A ) B = ϕ ( A X ) ∩ B B . Remark 2.27. The functor ϕ ∗ A is again a net because the in tersection of tw o C ∗ - subalgebras is again a C ∗ -subalgebra. If ϕ is E -controlled, then ϕ ( A B ) ⊆ B B E , hence also ϕ ( A B ) ⊆ ( ϕ ∗ A ) B E and ( ϕ ∗ A ) B ⊆ ϕ ( A B E ). W e therefore obtain an E -con trolled isomorphism of nets A ∼ = ϕ ∗ A . The followin g structure results on tensor factors will b e used in § 3.2 to pro ve the Ma yer–Vietoris decomp osition result. Prop osition 2.28 (Images of tensor factors). L et B and ˜ B b e nets and let ϕ : B → ˜ B b e an isomorphism of nets. If A is a tensor factor of B with c ommutant A ′ , then ϕ ∗ A is a tensor factor of ˜ B with c ommutant ϕ ∗ A ′ . Pr o of. Because ϕ is an isomorphism, w e ha ve ϕ (( A X ) ′ ) = ϕ ( A X ) ′ . Hence for b ounded subsets B ⊆ X , we get ( ϕ ∗ A ′ ) B = ϕ (( A X ) ′ ) ∩ ˜ B B = ϕ ( A X ) ′ ∩ ˜ B B =  ( ϕ ∗ A ) X  ′ ∩ ˜ B B = ( ϕ ∗ A ) ′ B , so ϕ ∗ A ′ is indeed the comm utant of ϕ ∗ A . W e ma y therefore v erify the criterion of Lemma 2.25. Since A is a tensor factor of B , the multiplication map A X ⊗ ( A X ) ′ → B X is an isomorphism. Since ϕ is an isomorphism, we obtain that also the m ultiplication map ( ϕ ∗ A ) X ⊗ ( ϕ ∗ A ) ′ X = ϕ ( A X ) ⊗ ϕ ( A ′ X ) − → ϕ ( B X ) = ˜ B X is an isomorphism. Moreov er, by Lemma 2.25, there exists an en tourage E such that B B ⊆ A B E · A ′ B E for all B ⊆ X b ounded. If F is a control for b oth ϕ and ϕ − 1 , then ϕ ( A B ) ⊆ ϕ ( A X ) ∩ ˜ B B F = ( ϕ ∗ A ) B F and ˜ B B ⊆ ϕ ( B B F ) . W e therefore get ˜ B B ⊆ ϕ ( B B F ) ⊆ ϕ ( A B F ◦ E · A ′ B F ◦ E ) = ϕ ( A B F ◦ E ) · ϕ ( A ′ B F ◦ E ) ⊆ ( ϕ ∗ A ) B F ◦ E ◦ F · ( ϕ ∗ A ′ ) B F ◦ E ◦ F . □ 12 Corollary 2.29. Any Azumaya net on X is isomorphic to a tensor factor of a lo c al matrix net. Pr o of. If A is an Azuma ya net on X and A ′ is another net on X suc h that there exists an isomorphism ϕ : A ⊗ A ′ → B with a lo cal matrix net B , then by Prop. 2.28, ϕ ∗ A is a tensor factor of B and by Remark 2.27, w e ha ve ϕ ∗ A ∼ = A . □ Prop osition 2.30 (Nested tensor factors). L et A ⊆ B ⊆ C b e nets on X . Assume that b oth A and B ar e tensor factors of C . Then A is a tensor factor of B . Mor e pr e cisely, if A ′ denotes the c ommutant of A in C , then the c ommutant of A in B is the net A ′ ∩ B , given by ( A ′ ∩ B ) B = A ′ B ∩ B B . Pr o of. Since A is a tensor factor of C , the m ultiplication map A X ⊗ ( A X ) ′ → C X is an isomorphism, and it remains injectiv e when the second factor is restricted to the comm utant ( A X ) ′ ∩ B X of A X in B X . Since B is a tensor factor of C , Lemma 2.25 yields an entourage E such that C B ⊆ B B E · B ′ B E (2.5) for all b ounded B ⊆ X . W e therefore hav e the inclusion A ′ B = ( A X ) ′ ∩ C B ⊆ ( A X ) ′ ∩ ( B B E · B ′ B E ) =  ( A X ) ′ ∩ B B E  · B ′ B E (2.6) T o see the last equalit y , first notice that since B is a tensor factor of C , the multiplication maps B X ⊗ ( B X ) ′ → C X is injective. So after c ho osing a vector space basis x 1 , . . . , x n for B ′ B E , any elemen t c ∈ B B E · B ′ B E ∼ = B B E ⊗ B ′ B E can b e uniquely written as c = n X i =1 b i x i , b 1 , . . . , b n ∈ B B E . Supp ose that c is also con tained in ( A X ) ′ , hence comm utes with each a ∈ A X . Since A X ⊆ B X and B ′ B E ⊆ ( B X ) ′ ⊆ ( A X ) ′ , every x i comm utes with a . Hence 0 = ac − ca = n X i =1 ( ab i − b i a ) x i . By uniqueness of the representations, we get ab i − b i a = 0. Since a w as arbitrary , w e get that b i ∈ ( A X ) ′ ∩ B B E , as claimed. By Lemma 2.25, the inclusion (2.6) shows that A ′ ∼ = ( A ′ ∩ B ) ⊗ B ′ , hence C ∼ = A ⊗ A ′ ∼ = A ⊗ ( A ′ ∩ B ) ⊗ B ′ . In particular, w e get B ∼ = A ⊗ ( A ′ ∩ B ). □ 13 2.5 QCA and coarsely lo cal automorphisms Let X b e a bornological coarse space. A quantum c el lular automaton , or QCA for short, is just another name for a (con trolled) automorphism of a lo cal matrix net on X . QCA are usually considered mo dulo a sp ecial class of automorphisms, called finite depth quantum cir cuits . How ever, this notion is not coarsely in v arian t; instead, w e introduce the new notion of a c o arsely lo c al automorphism. In § 2.6 b elow, we show that the t wo notions agree on spaces of bounded geometry . W e need the follo wing notions: A collection ( B i ) i ∈ I of subsets of X is called uniformly b ounde d if S i ∈ I B i × B i is an entourage. In particular, eac h B i m ust b e b ounded. It is lo c al ly finite if eac h B ⊆ X bounded has non-trivial in tersection with at most finitely man y B i . Definition 2.31 (coarsely lo cal automorphisms). Let A b e a lo cal matrix net on X . An automorphism α of A is c o arsely lo c al if there exists a uniformly b ounded and lo cally finite collection ( B i ) i ∈ I of subsets of X , together with a decomp osition A ∼ = O i ∈ I A ( i ) in to pairwise comm uting tensor factors A ( i ) of A supp orted on B i , such that α restricts to an automorphism of A ( i ) for eac h i ∈ I . W e denote by LAut( A ) ⊆ Aut( A ) the subgroup consisting of finite products of coarsely local automorphisms. The infinite tensor pro duct in the ab ov e definition is to b e understo o d as A B ∼ = O i ∈ I B i ∩ B  = ∅ A ( i ) B , whic h is a finite tensor pro duct b ecause the family ( B i ) i ∈ I is lo cally finite. W e require that the isomorphism ab ov e is giv en b y m ultiplication, which mak es sense b ecause the subalgebras A ( i ) B ⊆ A B all pairwise commute. Note that α is con trolled by S i ∈ I B i × B i . An y lo cal automorphism is coarsely lo cal; here one ma y take the tensor decomp osition A ∼ = N x ∈ X A | x , which is sub ordinate to the decomposition of X in to singleton subsets. Lemma 2.32. LAut( A ) is a normal sub gr oup of Aut( A ) . Pr o of. Let α b e a coarsely lo cal automorphism sub ordinate to the tensor factorization A ∼ = N i ∈ I A ( i ) . Then if β ∈ Aut( A ) is arbitrary , A ∼ = N i ∈ I β ∗ A ( i ) is another tensor factorization, and β ◦ α ◦ β − 1 restricts to an automorphism of β ∗ A ( i ) for eac h i ∈ I . If A ( i ) is supp orted on B i and β is E -controlled, then β ∗ A ( i ) is supp orted on ( B i ) E . Clearly , the collection (( B i ) E ) i ∈ I is again uniformly b ounded. Since B ∩ ( B i ) E  = ∅ if and only if B E − 1 ∩ B i  = ∅ and ( B i ) i ∈ I is lo cally finite by assumption, (( B i ) E ) i ∈ I is also lo cally finite. Hence β ◦ α ◦ β − 1 is again coarsely lo cal. This implies the lemma. □ By the ab o ve lemma, we can make the following definition: 14 Definition 2.33 (unstable QCA group). F or a lo cal net A on X , the group of quan- tum c el lular automata on A is defined b y QCA( A ) := Aut( A ) / LAut( A ) . W e no w pro ceed to define the stable QCA group, where one is allo wed to tensor with additional degrees of freedom. Lemma 2.34. If ϕ : A → B is a lo cal isomorphism X , we obtain a gr oup isomorphism ϕ ∗ : QCA( A ) − → QCA( B ) , [ α ] 7− → [ ϕ ◦ α ◦ ϕ − 1 ] . (2.7) F or any two lo c al automorphisms ϕ, ψ : A → B , we have ϕ ∗ = ψ ∗ . Pr o of. The group homomorphism is well-defined because the conjugation of a coarsely lo- cal automorphism on A b y an isomorphism ϕ : A → B is a coarsely local automorphism on B , b y an argumen t similar to that of Lemma 2.32. Let Y ⊆ X b e a support for A , so that A B = N x ∈ B ∩ Y A x . Since ϕ and ψ are lo cal, they induce embeddings ϕ x , ψ x : A x  → B x . Since an y tw o embeddings of a complex matrix algebra in to another are conjugate, there exist unitaries u x ∈ B x suc h that ψ x = Ad u x ◦ ϕ x . These ( u x ) x ∈ Y define a lo cal automorphism Ad u := N x ∈ Y Ad u x , which is in particular coarsely lo cal. F or any automorphism α ∈ Aut( A ) and an y a ∈ A X , we then hav e ( ψ ∗ α )( ψ ( a )) = ψ ( α ( a )) = Ad u ( ϕ ( α ( a ))) = Ad u ( ϕ ∗ α ( ϕ ( a ))) = Ad u ( ϕ ∗ α (Ad − 1 u ( ψ ( a )))) . hence ψ ∗ α = Ad u ◦ ( ϕ ∗ α ) ◦ Ad − 1 u . Therefore ϕ ∗ α and ψ ∗ α agree in the quotien t QCA( B ), so ϕ ∗ = ψ ∗ . □ W e now define a preorder on the set 1 of lo cal matrix nets on X . W e set A ⪯ B if there exists another lo cal matrix net A ′ and a lo cal (i.e., ∆-con trolled) isomorphism A ⊗ A ′ ∼ = B . If A ⪯ B , there exists a canonical group homomorphism QCA( A ) − → QCA( A ⊗ A ′ ) ∼ = QCA( B ) , giv en by stabilization, [ α ] 7→ [ α ⊗ id A ′ ], follo wed by conjugation with an arbitrary lo cal isomorphism ϕ : A ⊗ A ′ → B . This do es not dep end on the c hoice of ϕ b y Lemma 2.34. 1 If the reader is worried ab out size issues here (the ob jects of Loc ( X ) migh t b e too large to form a set), one ma y instead only consider nets of the form Example 2.16 in this construction. 15 F or an y tw o lo cal matrix nets A and A ′ , we hav e A ⪯ A ⊗ A ′ and A ′ ⪯ A ⊗ A ′ , hence an y tw o elemen ts of our preorder hav e a common upp er b ound. Therefore the asso ciated category with precisely one morphism for eac h relation A ⪯ B is a filtered category and assigning QCA groups provides a functor from this category to the category of ab elian groups. W e can therefore make the following definition: Definition 2.35 (stable QCA group). Let X b e a b ornological coarse space. The group QCA( X ) of stable quantum c el lular automata on X is giv en as the filtered colimit QCA( X ) := colim A QCA( A ) , tak en o ver the preorder on the set of lo cal matrix nets on X , as defined ab ov e. Remark 2.36. Using the standard explicit represen tation for a filtered colimit of groups, the elemen ts of QCA( X ) are given by equiv alence classes of pairs ( A , α ) consisting of a lo cal matrix net A on X and an automorphism α of A , where the equiv alence relation is generated b y the following three relations: (1) ( A , α ) ∼ ( A ⊗ A ′ , α ⊗ id A ′ ) for eac h lo cal matrix net A ′ ; (2) ( A , α ) ∼ ( A ′ , ϕ ◦ α ◦ ϕ − 1 ) for eac h lo c al isomorphism ϕ : A → A ′ . (3) ( A , α ) ∼ ( A , id A ) for an y coarsely lo cal automorphism α . The pro duct of [ A , α ] and [ A , α ′ ] is giv en b y [ A , α ] · [ A , α ′ ] = [ A , α ◦ α ′ ] . Tw o general elemen ts [ A , α ] and [ A ′ , α ′ ] ma y first be represented b y automorphisms on a lo cal matrix net B with A , A ′ ⪯ B and then comp osed in Aut( B ). Lemma 2.37. The gr oup QCA( X ) is ab elian. Pr o of. There is another asso ciativ e monoid op eration on QCA( X ) giv en b y [ A , α ] ∗ [ B , β ] := [ A ⊗ B , α ⊗ β ] . W e ha ve  [ A , α ] · [ A , α ′ ]  ∗  [ B , β ] · [ B , β ′ ]  = [ A , α ◦ α ′ ] ∗ [ B , β ◦ β ′ ] = [ A ⊗ B , ( α ◦ α ′ ) ⊗ ( β ◦ β ′ )] = [ A ⊗ B , ( α ⊗ β ) ◦ ( α ′ ⊗ β ′ )] = [ A ⊗ B , α ⊗ β ] · [ A ⊗ B , α ′ ⊗ β ′ ] =  [ A , α ] ∗ [ B , β ]  ·  [ A , α ′ ] ∗ [ B , β ′ ]  , Hence this monoid op eration comm utes with the group structure. By the Ec kmann-Hilton argumen t, b oth op erations m ust agree and be comm utative. □ 16 Remark 2.38. If f : X → X ′ is a coarse map b et w een b ornological coarse spaces, then the image of a uniformly b ounded and lo cally finite collection of subsets is again uniformly b ounded and lo cally finite. Hence if α is a coarsely lo cal automorphism of a lo cal matrix net A on X , then f ∗ α is a coarsely lo cal automorphism of f ∗ A . This shows that X 7→ QCA( X ) is a functor from the category of b ornological coarse spaces to abelian groups. 2.6 Quan tum circuits Let X b e a b ornological coarse space. The follo wing is a straightforw ard adaptation to general coarse spaces of the usual notion of quantum circuits in QCA theory: Definition 2.39 (quantum circuits). Let A b e a local matrix net on X . An automor- phism α of A is a depth one quantum cir cuit if there exists a uniformly bounded, locally finite and pairwise disjoint collection ( B i ) i ∈ I of subsets suc h that α is controlled by the en tourage S i ∈ I B i × B i . W e say that ( B i ) i ∈ I is the supp ort of the quan tum circuit α . An automorphism α of A is a depth n quantum cir cuit if it is a comp osition of at most n depth one quan tum circuits and a finite depth quantum cir cuit if it is a quan tum circuit of some finite depth n . By Cir( A ) ⊆ Aut( A ), w e denote the subgroup of finite depth quan tum circuits. If α is a depth one quan tum circuit with supp ort ( B i ) i ∈ I , then α restricts to an automorphism of A B i for eac h i ∈ I and acts as the identit y on A X \ S i ∈ I B i . Since any automorphism of a matrix algebra is inner, there exist unitaries u i ∈ A B i suc h that α ( a ) = Y i ∈ I B i ∩ B  = ∅ u i ! · a · Y i ∈ I B i ∩ B  = ∅ u ∗ i ! for each a ∈ A B . Here eac h pro duct is finite since the collection ( B i ) i ∈ I is lo cally finite. Remark 2.40. Any depth one quan tum circuit is a coarsely lo cal automorphism. Remark 2.41. There are tw o interrelated problems with the notion of quantum circuits: (1) It is claimed in [4, Prop. 2.16] that Cir( A ) is a normal subgroup of Aut( A ) for general discrete metric spaces. Ho w ev er, the pro of con tains a gap that we do not kno w how to fix without additional hypothesis; in fact, it seems that Cir( A ) is not a normal subgroup in general. T o illustrate the problem, supp ose that α is a depth one quantum circuit with supp ort ( B i ) i ∈ I . Then if β ∈ Aut( A ) is E -con trolled, β ◦ α ◦ β − 1 has supp ort (( B i ) E ) i ∈ I . Now, since the fattened collection (( B i ) E ) i ∈ I is no longer pairwise disjoin t, one must try to regroup the sets ( B i ) E in to finitely many pairwise disjoin t sub collections (“la y ers”), sho wing that β ◦ α ◦ β − 1 is a finite depth quan tum circuit. Ho wev er, this “distribution into la yers” argumen t may fail in general. F or example, 17 consider a graph where every v ertex has finite degree, but which is not n -colorable for any n (for example, the disjoin t union S n ∈ N K n , where K n is the complete graph with n vertices). Equip the vertex set X with the pseudometric where t wo vertices ha ve distance k if one needs at least k edges to connect them with a path. Then the collection ( B x ) x ∈ X of singleton subsets of X is uniformly b ounded, locally finite and pairwise disjoint, but its 1-fattening may not b e regroup ed in to finitely many la yers with the same prop erties. (2) The notion of quantum circuits is not a coarsely inv ariant notion. The problem is that images of pairwise disjoint collections of subsets under coarse maps need not b e pairwise disjoint again, and it ma y not ev en b e p ossible to divide them into finitely man y pairwise disjoint families. There are conditions on the underlying space whic h ensure that the notion of a quan- tum circuit is nevertheless w ell-b ehav ed. Definition 2.42. Let X b e a coarse space. (1) A subset Y ⊆ X is called uniformly lo c al ly finite if for each entourage E , there exists a num b er n ∈ N suc h that for all B ⊆ X with B × B ⊆ E , w e ha ve #( B ∩ Y ) ≤ n. (2) X is said to ha ve b ounde d ge ometry if it admits a coarsely dense and uniformly locally finite subset Y ⊆ X . (Here Y is called c o arsely dense if there exists an entourage suc h that Y E = X .) Of course, basic examples for spaces with bounded geometry are Z n or R n . Ho wev er, R n is of course not uniformly lo cally finite and also contains lo cally finite subsets which are not uniformly lo cally finite. Prop osition 2.43. If A admits a uniformly lo c al ly finite supp ort, then we have LAut( A ) = Cir( A ) . In particular, it follo ws from Lemma 2.32 abov e that under the ab ov e support assump- tion on A , Cir( A ) is a normal subgroup of Aut( A ). Pr o of. W e alw ays ha v e the inclusion Cir( A ) ⊆ LAut( A ), so it remains to show the con verse inclusion. W e start with some preliminary observ ations. F or an en tourage E of X , write E [ x ] := { y ∈ X | ( y , x ) ∈ E } . Observ e that E [ x ] × E [ x ] ⊆ E − 1 ◦ E , hence the collection ( E [ x ]) x ∈ X is uniformly bounded. Morev er, if Y ⊆ X is a uniformly lo cally finite support for A , the collection ( E [ y ] ∩ Y ) y ∈ X is uniformly b ounded and locally finite. 18 Let n ∈ N b e such that for any B ⊆ X with B × B ⊆ E ◦ E − 1 , we ha ve #( B ∩ Y ) ≤ n . Then each x ∈ Y is con tained in E [ y ] for at most n p oin ts y ∈ Y : Indeed, w e ha ve x ∈ E [ y ] if and only if y ∈ E − 1 [ x ], and E − 1 [ x ] × E − 1 [ x ] ⊆ E ◦ E − 1 , hence #( E − 1 [ x ] ∩ Y ) ≤ n b y c hoice of n . Therefore, after w ell-ordering Y , w e obtain a decomp osition Y = Y 0 ⊔ · · · ⊔ Y n suc h that the collections ( E [ y ] ∩ Y ) y ∈ Y k , k = 0 , . . . , n are each pairwise disjoint. Let no w α be a coarsely lo cal automorphism of A and let A ∼ = N i ∈ I A ( i ) b e a tensor decomp osition as in the definition of coarse lo calit y , where A ( i ) is supp orted on the member B i of a uniformly b ounded and lo cally finite collection ( B i ) i ∈ I . W e assume that A ( i ) is non-trivial, so that B i ∩ Y  = ∅ for eac h i ∈ I . W e no w use the considerations ab o ve for the en tourage E := [ i ∈ I B i × B i . Here we ha ve E [ y ] = { x ∈ X | ∃ i ∈ I : x, y ∈ B i } = [ i ∈ I y ∈ B i B i . Hence eac h B i is con tained in some E [ y ]. W e may therefore c ho ose for each B i an elemen t y i ∈ Y such that B i ⊆ E [ y i ]. This gives a decomposition I = I 0 ⊔ · · · ⊔ I n , where I k = { i ∈ I | y i ∈ Y k } . Moreo ver, this results in a decomp osition α = α 0 ◦ · · · ◦ α n , where α k acts as α on N i ∈ I k A ( i ) and trivial on N i ∈ I \ I k A ( i ) . By construction, eac h automorphism α k is a depth one quan tum circuit with supp ort ( E [ y ] ∩ Y ) y ∈ Y k . □ Remark 2.44. If X has b ounded geometry , then an y lo cal matrix net A on X is isomor- phic to one allowing a uniformly locally b ounded supp ort. Indeed, let A b e an arbitrary lo cal matrix net on X and let Y ⊆ X b e a coarsely dense and uniformly lo cally finite subset. Let Z ⊆ X b e a lo cally finite supp ort for A , whic h exists by Remark 2.20. Let E b e an en tourage suc h that Y E = X . Then for eac h p oin t z ∈ Z , there exists a p oint y ∈ Y suc h that ( z , y ) ∈ E . Hence w e may choose a map f : Z → Y that is close to the iden tity (hence coarse). It is prop er by lo cal finiteness of Y and Z . By Prop. 2.21, the lo cal matrix net f ∗ A is therefore isomorphic to A and has supp ort Y by construction. T ogether with Prop. 2.43, this shows that in the case that X has b ounded geometry , the group QCA( X ) may equiv alently be presented b y pairs ( A , α ), where A is a lo cal matrix net admitting a uniformly locally finite supp ort and α is a (con trolled) automorphism of A . The relations are then the same as in Remark 2.36, except that one may declare that all quantum circuits are trivial instead of coarsely lo cal automorphisms. In particular, for X = Z n , QCA( Z n ) agrees with the usual QCA group. 3 A coarse homology theory In this section, w e construct a coarse homology theory whose degree zero groups are giv en b y quan tum cellular automata. Let us fix set-theoretic size issues for the constructions in this section: W e choose t wo Grothendiec k universes, whose elements are called smal l sets and lar ge sets . All 19 b ornological coarse spaces and nets on these are assumed to ha v e underlying sets coming from the first Grothendiec k universe. The set of all nets on a b ornological coarse space will then form a large set. 3.1 QCA and K -theory If C is a symmetric monoidal category , its K -theory group K 0 ( C ) is the ab elian group obtained by group completing the comm utative monoid of isomorphism classes of ob jects of C , with monoid operation giv en b y the tensor product of C , [ x ] + [ y ] := [ x ⊗ y ] . T o define the higher K -theory groups of C , one uses the classifying space functor B : Ca t → Sp a ces , the latter denoting the ∞ -category of (large) simplicial sets. This functor preserves commutativ e monoid ob jects: If C is a symmetric monoidal category , then B C is a comm utative monoid in Sp aces . Here one may form the ∞ -categorical group completion functor CMon ( Sp aces ) → CMon grp ( Sp aces ) to obtain a grouplike commutativ e monoid in spaces, which by May’s recognition principle is the same information as an infinite lo op space, i.e., a connectiv e sp ectrum. In total, w e get a K -theory functor K : CMon ( Ca t ) B − − − → CMon ( Sp aces ) group completion − − − − − − − − → CMon grp ( Sp aces ) ≃ Spectra ≥ 0 from symmetric monoidal categories to connectiv e sp ectra. The functor B commutes with colimits, and group completion is a left adjoin t, so K also preserv es colimits. Remark 3.1. The K -theory functor has the property that π 0 ( K ( C )) = K 0 ( C ) , the K -theory group describ ed ab ov e and one defines the higher K -groups of C as the higher homotop y groups of K ( C ). If C is a group oid, the first K -theory group K 1 ( C ) = π 1 ( K ( C )) can b e described as follows: Its elements are equiv alence classes of pairs ( x, α ), where x is an ob ject of C and α is an automorphism of x . The equiv alence relation is generated by (1) ( x, α ) ∼ ( x ⊗ y , α ⊗ id y ) for an y ob ject y of C ; (2) ( x, α ) ∼ ( x ′ , ϕ ◦ α ◦ ϕ − 1 ) for eac h isomorphism ϕ : x → x ′ . The group op eration is giv en b y tensor pro duct, [ x, α ] + [ y , β ] := [ x ⊗ y , α ⊗ β ] , or, alternatively , by representing tw o given elements of K 1 ( C ) b y automorphisms of the same ob ject x of C and then taking the pro duct of Aut( x ). 20 Applying the K -theory functor to the v arious categories of nets, w e obtain v arious functors from the category of b ornological coarse spaces to connective sp ectra. By defini- tion, the category Loc ( X ) is monoidal ly c ofinal in Az ( X ), meaning that eac h ob ject of Az ( X ) is a tensor factor of an ob ject of Loc ( X ). By the explicit description of K 1 from Remark 3.1, w e therefore get that inclusion induces an isomorphism K 1 ( Loc ( X )) ∼ = K 1 ( Az ( X )) . (3.1) By standard prop erties of the K -theory functor, we also hav e an isomorphism of the higher homotopy groups, see Lemma 3.6. The relation to quantum cellular automata is given b y the follo wing result, which w as essentially shown in [4], using prop erties of Quillen’s plus construction. W e give a differen t, more elementary pro of. Theorem 3.2. F or any b ornolo gic al c o arse sp ac e X , ther e is a natur al gr oup isomorphism QCA( X ) ∼ = K 1 ( Az ( X )) . Pr o of. By (3.1), we ha ve K 1 ( Loc ( X )) ∼ = K 1 ( Az ( X )), so we ma y equiv alen tly construct a homomorphism to K 1 ( Loc ( X )). By the generators and relations description of b oth groups from Remarks 3.1, resp ectively 2.36, we observe that b oth groups are given by pairs ( A , α ) of a local matrix net and an automorphism α of A . W e claim that the map QCA( X ) − → K 1 ( Loc ( X )) , [ A , α ] 7− → [ A , α ] is well-defined, for which w e hav e to chec k that the relations defining the QCA group also hold in the K 1 group. In b oth groups, we hav e the stabilization relations ( A , α ) ∼ ( A ⊗ A ′ , α ⊗ id A ′ ). The second relation in QCA( X ) is ( A , α ) ∼ ( A ′ , ϕ ◦ α ◦ ϕ − 1 ) for lo cal isomorphisms ϕ , which also holds in K 1 (ev en for al l isomorphisms). The last relation of QCA( X ) is that [ A , α ] = 0 when α is a coarsely lo cal automorphism. Let A ∼ = N i ∈ I A ( i ) b e a tensor decomp osition as in the definition of a coarsely lo cal automorphism, where A ( i ) is supp orted on the member B i of a uniformly b ounded and lo cally finite collection ( B i ) i ∈ I of subsets of X and α restricts to an automorphism α i of A ( i ) for eac h i ∈ I . Because we ha ve an isomorphism A X ∼ = O i ∈ I A ( i ) B i with the right hand side a tensor pro duct of matrix algebras, eac h A ( i ) B i m ust b e a matrix algebra, so the automorphism α i of A ( i ) B i is implemen ted b y a unitary , unique up to phase. As the pro jectiv e unitary group P U ( n ) is simple for n ≥ 2, it follo ws from [2] that eac h of its elemen ts ma y be written as a multiplicativ e comm utator. Hence for eac h i ∈ I , w e ha ve α i = Ad u i Ad v i Ad − 1 u i Ad − 1 v i for unitaries u i , v i ∈ A ( i ) B i and globally , α is the multiplicativ e comm utator of Ad u = O i ∈ I Ad u i and Ad v = O i ∈ I Ad v i , 21 whic h are con trolled since ( B i ) i ∈ I is uniformly b ounded. W e obtain that [ A , α ] is a comm utator, hence v anishes as K 1 ( Loc ( X )) is ab elian. This sho ws that the map is well defined. It is ob vious from the definition of the group structures on b oth sides that the map is a group homomorphism. It is also clear that it is natural in X . It is clear that the group homomorphism is surjective. T o see that it is injective, w e ha ve to show that the second relation of K 1 ( Loc ( X )) also holds in QCA( X ), which is ( A , α ) ∼ ( A ′ , ϕ ◦ α ◦ ϕ − 1 ) whenever ϕ : A → A ′ is an isomorphism b etw een lo cal matrix nets A and A ′ . T o this end, consider the automorphism Φ of A ⊗ A ′ giv en b y Φ( a ⊗ a ′ ) := ϕ − 1 ( a ′ ) ⊗ ϕ ( a ) . Since Φ − 1 = Φ, we ha v e  Φ ◦ ( α ⊗ id A ′ ) ◦ Φ − 1  ( a ⊗ a ′ ) =  Φ ◦ ( α ⊗ id A ′ )  ϕ − 1 ( a ′ ) ⊗ ϕ ( a )  = Φ  ( α ◦ ϕ − 1 )( a ′ ) ⊗ ϕ ( a )  = a ⊗ ( ϕ ◦ α ◦ ϕ − 1 )( a ′ ) , hence com bined with the first relation, the second relation of K 1 ( Loc ( X )) is equiv alent to the relation that ( A , α ) ∼ ( A , ϕ ◦ α ◦ ϕ − 1 ) whenever ϕ is an auto morphism of A . But by Lemma 2.37, QCA( X ) is ab elian, so the same relation holds here. This sho ws injectivit y . □ Theorem 3.3. If X is flasque, then K ( Az ( X )) ≃ 0 . Pr o of. Because X is flasque, by Prop. 2.23 there exists an endofunctor S : Az ( X ) → Az ( X ) suc h that S ∼ = id ⊗ S . Since tensor pro duct corresp onds to addition in K -theory , this yields a self-map of sp ectra S ∗ : K ( Az ( X )) → K ( Az ( X )) with S ∗ ≃ id + S ∗ . This implies id ≃ 0, hence K ( Az ( X )) ≃ 0. □ The following result is more difficult to prov e; the pro of will b e carried out in the next section. In the statement b elow, X ⊗ R is the b ornological coarse space with underlying set X × R and the pro duct coarse structure and b ornology , see Example 2.6. Theorem 3.4. F or e ach b ornolo gic al c o arse sp ac e X , we have a c anonic al map of sp e ctr a K ( Az ( X )) − → Ω K ( Az ( X ⊗ R )) , which induc es an isomorphism on homotopy gr oups in non-ne gative de gr e es. 22 3.2 The Ma y er-Vietoris axiom F ollowing [4, § 4], w e use the double mapping cylinder construction of Thomason, see [7, 8]. Supp ose w e are giv en a span of symmetric monoidal categories and symmetric monoidal functors C D 1 D 2 P ι 1 ι 2 (3.2) suc h that C is a group oid. The double mapping cylinder of this span is the symmetric monoidal category P defined as follows. (1) Ob jects are triples ( c, d 1 , d 2 ) with c an ob ject of C and d i ob jects of D i . (2) A morphism ( c, d 1 , d 2 ) → ( c ′ , d ′ 1 , d ′ 2 ) is an equiv alence class of quintuples ( f , f 1 , f 2 , c 1 , c 2 ), where c 1 , c 2 are ob jects of C , f : c → c 1 ⊗ c ′ ⊗ c 2 is an isomorphism in C and f i : ι i ( c i ) ⊗ d i → d ′ i are morphisms in D i . Tw o such tuples are equiv alent if they differ b y replacing the ob jects c 1 and c 2 with isomorphic ones. The category P receives symmetric monoidal functors from D 1 and D 2 in the obvious w ay , and there is a natural transformation witnessing the commutativit y of the square (3.2). The significance of this construction is that after applying the K -theory functor, one obtains a pushout diagram of sp ectra K ( C ) K ( D 1 ) K ( D 2 ) K ( P ) . (3.3) Let X b e a b ornological coarse space. Definition 3.5. F or a subset Y ⊆ X , the r estriction A | Y of a net A on X to Y is the net on X given b y ( A | Y ) B := A Y ∩ B . F or a big family W in a bornological coarse space X , denote b y Az W ( X ) ⊆ Az ( X ) the full subcategory of those Azuma ya nets that are a tensor pro duct of a lo cal net on X and an Azumay a net supported on W . Put differen tly , Az W ( X ) consists of those nets A on X for which there exists a mem b er W of W such that we hav e a tensor decomp osition A = A | W ⊗ A | X \ W , where A | W is an Azumay a net and A | X \ W is a lo cal matrix net. F or a net A with these prop erties, we say that A is lo c al away fr om W . If V is another big family with W ⊆ Y , w e denote b y Az W ( V ) the full sub category of Az W ( X ) consisting of those nets that are supp orted on some member of V . 23 Lemma 3.6. F or any p air of big families W ⊆ V , the inclusions induc e isomorphisms K n ( Loc ( V )) ∼ = K n ( Az W ( V )) ∼ = K n ( Az ( V )) in al l de gr e e n ≥ 1 . Pr o of. Both Az ( V ) and Az W ( V ) are monoidally cofinal in Loc ( V ), meaning that eac h ob ject of Az ( V ), resp ectively Az W ( V ), is a tensor factor of an ob ject of Loc ( V ). This implies that the inclusion induces an isomorphism on the higher K -theory groups. □ F or t wo big families Y , Z in X , we then get a commutativ e diagram Az ( Y ⋒ Z ) Az Y ⋒ Z ( Y ) Az Y ⋒ Z ( Z ) Az Y ⋒ Z ( Y ⋓ Z ) . (3.4) In the follo wing, let P b e the double mapping cylinder for the top left span in the abov e diagram. There is an ob vious symmetric monoidal functor T : P − → Az Y ⋒ Z ( Y ⋓ Z ) , ( A , B Y , B Z ) − → A ⊗ B Y ⊗ B Z , whic h is easily seen to b e faithful using the equiv alence relation on morphisms in P . W e then hav e the follo wing result: Prop osition 3.7. The induc e d map K ( T ) : K ( P ) → K ( Az Y ⋒ Z ( Y ⋓ Z )) is a homotopy e quivalenc e. Conse quently, the diagr am K ( Az ( Y ⋒ Z )) K ( Az Y ⋒ Z ( Y )) K ( Az Y ⋒ Z ( Z )) K ( Az Y ⋒ Z ( Y ⋓ Z )) . is a c artesian squar e of sp e ctr a. Since b y the prop erties of the double mapping cylinder construction, the square (3.3) in volving P is cartesian, w e see that for the pro of of Prop. 3.7, it suffices to show that K ( T ) is a homotopy equiv alence. Quillen’s theorem A [6] states that this is the case provided that w e chec k that the classifying spaces of the comma categories C ↓ T are weakl y con tractible for every ob ject C . Recall that the ob jects of C ↓ T are morphisms in Az Y ⋒ Z ( Y ⋓ Z ) of the form ϕ : C → T ( A , B Y , B Z ) (3.5) and the morphisms of the comma category are commutativ e diagrams T ( A , B Y , B Z ) C T ( ˜ A , ˜ B Y , ˜ B Z ) . T ( ψ,ψ Y ,ψ Z , A Y , A Z ) φ ˜ φ 24 Lemma 3.8. F or every obje ct C in Az Y ⋒ Z ( Y ⋓ Z ) , the classifying sp ac e of the c omma c ate gory C ↓ T is we akly c ontr actible. Pr o of. Fix an ob ject C of Az Y ⋒ Z ( Y ⋓ Z ). A standard w ay to establish this is that C ↓ T admits an initial ob ject. How ever, this do es not seem to b e the case, so we will instead sho w the w eak er statemen t that C ↓ T is a filtered union of sub categories, each of whic h do admit an initial ob ject. This suffices as then C ↓ T is a filtered union of contractible spaces, hence con tractible. As preparation, first notice that by the supp ort conditions on C , there exist members Y ∈ Y and Z ∈ Z suc h that C is supp orted on Y ∪ Z and lo cal a wa y from Y ∩ Z . Hence ( C | Y ∩ Z , C | Y \ Z , C | Z \ Y ) is an ob ject of P and C ∼ = C | Y ∩ Z ⊗ C | Y \ Z ⊗ C | Z \ Y = T ( C | Y ∩ Z , C | Y \ Z , C | Z \ Y ) , defines an ob ject split Y ,Z ( C ) of C ↓ T . F or ev ery en tourage E and an y t wo members Y ∈ Y , Z ∈ Z suc h that C is supp orted on Y ∪ Z and lo cal aw a y from Y ∩ Z , we consider the full sub category ( C ↓ T ) Y ,Z ,E of C ↓ T consisting of those ob jects of the form (3.5) suc h that • ϕ is E -controlled; • A is supp orted on Y ∩ Z ; • B Y is supp orted on Y ; • B Z is supp orted on Z . It is clear that C ↓ T is the filtered union of the subcategories ( C ↓ T ) Y ,Z ,E . W e claim that split Y E ,Z E ( C ) is an initial ob ject of ( C ↓ T ) Y ,Z ,E . Since the functor T is faithful and all net categories are group oids, w e see that there is at most one morphism b etw een an y t wo giv en ob jects, determined as the pre-image under T of ˜ ϕ ◦ ϕ − 1 . Therefore, it suffices to establish that any ob ject of ( C ↓ T ) Y ,Z ,E admits a morphism from split Y E ,Z E ( C ); uniqueness is then automatic. T o sho w the claim, w e tak e an arbitrary ob ject ϕ : C − → T ( A , B Y , B Z ) = A ⊗ B Y ⊗ B Z =: T , of ( C ↓ T ) Y ,Z ,E . Prop. 2.28 implies that its image ϕ ∗ ( C | Y E \ Z E ) is a tensor factor of T . Moreo ver, since E is a control for ϕ , the net ϕ ∗ ( C | Y E \ Z E ) is supp orted in X \ Z , which is disjoin t from the supp orts of A and B Z . Therefore ϕ ∗ ( C | Y E \ Z E ) ⊆ B Y ⊆ T , where the first tw o are tensor factors in T . F rom Prop. 2.30, we therefore get that ϕ ∗ ( C | Y E \ Z E ) is also a tensor factor of B Y , so B Y = ϕ ∗ ( C | Y E \ Z E ) ⊗ A Y 25 for some Azumay a net A Y supp orted on Y . Since A Y is the commutan t of ϕ ∗ ( C | Y E \ Z E ) in B Y , it must lie in the image of the commutan t C | Z E of C | Y E \ Z E in C , hence b ecause ϕ is E -con trolled, w e hav e that A Y is supp orted on Z E ◦ E . In total, A Y is therefore supp orted on Y ⋒ Z . Similarly , we obtain another Azuma ya net A Z supp orted on Y ⋒ Z suc h that B Z = ϕ ∗ ( C | Z E \ Y E ) ⊗ A Z . In total, w e get isomorphisms of nets ψ : C | Y E ∩ Z E − → A ⊗ A Y ⊗ A Z , and ψ Y : C | Y E \ Z E ⊗ A Y − → B Y , ψ Z : C | Z E \ Y E ⊗ A Z − → B Z , whic h define a morphism in P and further a map in ( C ↓ T ) Y ,Z ,E from split Y E ,Z E ( C ) to our given ϕ : C → T ( A , B Y , B Z ). This pro ves the claim and finishes the pro of. □ Thm. 3.3 generalizes as follo ws. Lemma 3.9. If Y is flasque, then K ( Az Y ⋒ Z ( Y )) is c ontr actible. Pr o of. After applying Lemma 3.6, it remains to show that K 0 ( Az Y ⋒ Z ( Y )) = 0. T o this end, consider the functor S from Prop. 2.23. If A is an ob ject of Az Y ⋒ Z ( Y ) supp orted on some flasque member Y of Y , w e ma y choose an ob ject A ′ in Az Y ⋒ Z ( Y ) such that A ⊗ A ′ is lo cal. Then if f : Y → Y is the map implemen ting flasqueness, then S ( A ⊗ A ′ ) = A ⊗ A ′ ⊗ f ∗ A ⊗ f ∗ A ′ ⊗ f 2 ∗ A ⊗ f 2 ∗ A ′ ⊗ · · · . is a lo cal matrix net on Y , hence a w ell-defined ob ject of Az Y ⋒ Z ( Y ). W e then hav e an isomorphism of nets S ( A ⊗ A ′ ) ∼ = S ( A ) ⊗ S ( A ′ ) ∼ = A ⊗ S ( A ) ⊗ S ( A ′ ) ∼ = A ⊗ S ( A ⊗ A ′ ) , giv en b y shifting the copies of A . In K 0 , this implies [ S ( A ⊗ A ′ )] = [ A ] + [ S ( A ⊗ A ′ )] = ⇒ [ A ] = 0 . □ Corollary 3.10. If Y and Z ar e flasque big families on a b ornolo gic al c o arse sp ac e, then we have a c anonic al homotopy e quivalenc e K ( Az ( Y ⋒ Z )) ≃ Ω K ( Az Y ⋒ Z ( Y ⋓ Z )) Pr o of. By Lemma 3.9, the flasqueness assumption on Y and Z imply that the pushout diagram from Prop. 3.7 becomes K ( Az ( Y ⋒ Z )) ∗ ∗ K ( Az Y ⋒ Z ( Y ⋓ Z )) , whic h establishes K ( Az ( Y ⋒ Z )) as the lo op space of K ( Az Y ⋒ Z ( Y ⋓ Z )) (as the ∞ -category of sp ectra is stable, pushout and pullback diagrams coincide). □ 26 Pr o of (of Thm. 3.4). Consider the big families Y := { X ⊗ R ≤ 0 } and Z := { X ⊗ R ≥ 0 } in X ⊗ R . Their elemen twise in tersection is the big family generated b y X ⊗ { 0 } , hence b y Corollary 2.22 and the fact that the K -theory functor commutes with filtered colimits, the canonical map K ( Az ( X )) ∼ = K ( Az ( X ⊗ { 0 } )) − → K ( Az ( Y ⋒ Z )) is a homotop y equiv alence. Since X ⊗ R ≤ 0 and X ⊗ R ≥ 0 are flasque, Corollary 3.10 yields a canonical map K ( Az ( X )) ≃ K ( Az ( Y ⋒ Z )) ≃ Ω K ( Az Y ⋒ Z ( Y ⋓ Z )) − → Ω K ( Az ( X ⊗ R )) . By Lemma 3.6, it induces an isomorphism on homotopy groups of nonnegativ e degree. □ 3.3 Definition of the coarse homology theory The notion of group-v alued coarse homology theories was first axiomatically in tro duced b y Mitchener [5]. The mo dern p oint of view is that a group-v alued homology theory should b e the “shado w” of a functor v alued in sp ectra (or some other stable ∞ -category), whic h yields a group-v alued functor b y taking homotopy groups. The follo wing definition is equiv alen t to that of Bunke–Engel [1, Definition 4.22]. Definition 3.11. A functor F : BornCo arse − → Spectra is a c o arse homolo gy the ory if it satisfies the follo wing axioms: (1) F is c o arsely invariant , i.e., whenev er f , g : X → X ′ are close, then F ( f ) ≃ F ( g ). (2) F vanishes on flasques , i.e., whenev er X is flasque, then F ( X ) ≃ 0. (3) F satisfies the Mayer-Vietoris axiom : Whenever Y and Z are t wo big families in a coarse space X , then w e ha ve a pushout square F ( Y ⋒ Z ) F ( Y ) F ( Z ) F ( Y ⋓ Z ) . Here for an y big family in X , w e define F ( Y ) := colim Y ∈Y F ( Y ). (4) F is u -c ontinuous . This means that if for an en tourage E on a b ornological coarse space X , X E denotes the space X with the same b ornology but equipp ed with the smallest coarse structure containing E , then F ( X ) = colim E ∈C F ( X E ) . 27 T aking F to b e one of the functors K ◦ Az of K ◦ Loc , w e hav e already c heck ed axioms (1) and (2) and axiom (4) is also easy to c hec k. Ho wev er, neither functor satisfies the Ma yer-Vietoris axiom. The problem is that all the K -theory sp ectra are connectiv e, whic h generates a problem at the degree zero level. F ortunately , non-connectiv e delo opings are pro vided b y Thm. 3.4. Definition 3.12. F or a b ornological coarse space X , w e define Q ( X ) := colim n →∞ Ω n +1 K  Az ( X ⊗ R n )  , where the colimit is taken in the ∞ -category of spectra along the maps pro vided b y Thm. 3.4. V arying X , this defines a functor Q : BornCoarse − → Spectra . Since all the sp ectrum maps from Thm. 3.4 induce homotop y equiv alences in p osi- tiv e degree, they particularly induce homotop y equiv alences on the corresp onding infinite lo op spaces, hence the sp ectrum Q ( X ) ma y equiv alently b e describ ed as the Ω-sp ectrum E 0 , E 1 , . . . , where the n -th space E n is given b y E n = Ω ∞ +1 K  Az ( X ⊗ R n )  . Its homotopy groups are giv en b y the form ula Q n ( X ) := π n ( Q ( X )) = K n + m +1  Az ( X ⊗ R m )  , for m ≥ − n − 1 . The definition is made so that for an y b ornological coarse space X , w e ha ve Q 0 ( X ) = K 1 ( Az ( X )) ∼ = QCA( X ) , using the isomorphism from Thm. 3.2. Our main result is now the following. Theorem 3.13. The functor Q is a c o arse homolo gy the ory. Pr o of. Coarse in v ariance follows from Prop. 2.21 and the properties of the K -theory functor. That Q v anishes on flasques follo ws from Thm. 3.3. Both these prop erties hold already b efore taking the colimit. Since Q ( X ) is represen ted by a Ω-sp ectrum with spaces Ω ∞ + n +1 K ( Az ( X ⊗ R n )), it suffices that at each level the corresp onding square of spaces is a homotop y pushout. Precisely , we must chec k that for any pair Y , Z of big families in a b ornological coarse space X , w e ha ve homotop y pushouts Ω ∞ +1 K  Az (( Y ⋒ Z ) ⊗ R n )  Ω ∞ +1 K  Az ( Y ⊗ R n )  Ω ∞ +1 K  Az ( Z ⊗ R n )  Ω ∞ +1 K  Az (( Y ⋓ Z ) ⊗ R n )  28 of spaces (here Y ⊗ R n etc. denotes the big family in X ⊗ R n generated by the sets Y × R n for Y ∈ Y ). But this follo ws from Prop. 3.7 together with Lemma 3.6. Finally , to see u -contin uit y , observ e that Az ( X ) is the colimit of the sub categories Az ⟨ E ⟩ ( X ) containing all ob jects and those morphisms that are controlled b y an entourage in the coarse structure ⟨ E ⟩ generated by E . The statemen t therefore follo ws b ecause the K -theory functor comm utes with colimits. □ Remark 3.14. Similar to what is done in [4], when defining nets, we may choose a com- m utative ring R and replace the category of finite-dimensional C ∗ -algebras and injective ∗ -homomorphisms by the category of R -algebras and injective homomorphisms. Matrix nets are then required to assign algebras isomorphic to M n ( R ) to b ounded subsets of a b ornological coarse space X and Azumay a nets may b e defined in a similar w ay , yielding a category Az R ( X ). The further constructions go through in a similar w ay , and we obtain a homology theory Q ( X , R ) with Q − 1 ( X , R ) = K 0 ( Az R ( X )). Ho wev er the iden tification with quan tum cellular automata is not quite as straigh t for- w ard, as we used sp ecific results on the pro jectiv e unitary group in the proof of Thm. 3.2, compare [4]. 3.4 QCA and Azuma ya nets As coarse spaces, Z and R are equiv alent, hence w e hav e canonical homotop y equiv alences Ω K ( Az ( X ⊗ Z )) ≃ Ω K ( Az ( X ⊗ R )) ≃ K ( Az ( X )) . In view of Thm. 3.2, taking π 0 , we obtain the follo wing result. Theorem 3.15. F or any b ornolo gic al c o arse sp ac e X , we have QCA( X ⊗ Z ) ∼ = K 1 ( Az ( X ⊗ Z )) ∼ = K 0 ( Az ( X )) . In p articular, for X = Z n − 1 , we obtain QCA( Z n ) ∼ = K 0 ( Az ( Z n − 1 )) . Thm. 3.15 shows that the classification of quantum cellular automata in an y dimension is equiv alent to the classification of Azuma y a nets one dimension lo w er. This classification problem may not b e m uch simpler, but at least in one case, it is easy to do: Example 3.16. In the case that X is a p oint, w e hav e Az ( X ) ∼ = Loc ( X ) and since any matrix algebra is classified, up to isomorphism, by its dimension and we ha ve M n ( C ) ⊗ M m ( C ) ∼ = M nm ( C ), the monoid of isomorphism classes of ob jects is ( N , · ), natural num b ers with multiplication. The group completion yields therefore K 0 ( Az ( ∗ )) ∼ = K 0 ( Loc ( ∗ )) ∼ = Q > 0 and the isomorphism ab ov e is giv en b y the GNVW index, see [3]. 29 Remark 3.17. The isomorphism from Thm. 3.15 is just the Ma yer–Vietoris b oundary K 1 ( Az ( X ⊗ Z )) − → K 0 ( Az ( X )) of the decomp osition of X ⊗ Z into the big families generated by X ⊗ Z ≥ 0 resp ectiv ely X ⊗ Z ≤ 0 . An in verse to this map map be given as follo ws. T ake an Azuma ya net A on X . By Corollary 2.29, we may realize A as a subnet of a lo cal matrix net B ∼ = A ⊗ A ′ . Let ˆ B b e the lo cal net on X ⊗ Z obtained b y placing a cop y of B on eac h p oin t of Z . Then the image of [ A ] ∈ K 0 ( Az ( X )) under the ab ov e isomorphism is the QCA on X ⊗ Z that acts as shift on the A factors of ˆ B in the Z direction and lea ves the A ′ factors inv ariant. A ⊗ A ′ A ⊗ A ′ A ⊗ A ′ A ⊗ A ′ A ⊗ A ′ . . . . . . One has the following general statement: Lemma 3.18. Inclusion of c ate gories induc es an inje ctive gr oup homomorphism K 0 ( Loc ( X )) − → K 0 ( Az ( X )) . Pr o of. Since the inclusion functor is fully faithful, the monoid π 0 ( Loc ( X )) of isomorphism classes of ob jects is a submonoid of π 0 ( Az ( X )). Generally , if M is a commutativ e monoid, tw o elements m 1 , m 2 ∈ M yield the same elemen t in the group completion K 0 ( M ) if there exists n ∈ M such that m 1 + n = m 2 + n . Hence if A 1 and A 2 are t wo local matrix nets that define the same elemen t of K 0 ( Az ( X )), then there exists an Azumay a net C suc h that A 1 ⊗ C ∼ = A 2 ⊗ C . Therefore also A 1 ⊗ C ⊗ C ′ ∼ = A 2 ⊗ C ⊗ C ′ if C ′ is another Azuma ya net such that C ⊗ C ′ is isomorphic to a lo cal matrix net. But this implies that A 1 and A 2 already define the same element in K 0 ( Loc ( X )). □ It was sho wn in [4, Thm. D] that one has a canonical isomorphism K 0 ( Loc ( X )) ∼ = H X 0 ( X , Q > 0 ) , where H X n ( X ) denotes the coarse homology groups v alued in the abelian group Q > 0 (see [1, § 6.3]) and Z P denotes the set of maps from the set P of all prime num b ers to Z . The group H X 0 ( X , Z P ) is just a quotient of the group of functions X → Q > 0 of lo cally finitely supp ort, and the isomorphism ab o v e is given b y sending a local matrix A net to its dimension function q ( x ) = dim( A x ) 1 / 2 , see Example 2.16. Since Q − 1 ( X ) = K 0 ( Az ( X )), w e get a canonical map H X 0 ( X , Q > 0 ) − → Q − 1 ( X ) . 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