Failure of rational injectivity for maximal coarse assembly

1 Introduction

Coarse assembly relates the topology visible at bounded scales in a metric space to the indices of operators controlled by its large-scale geometry. Its rational injectivity asks whether a non-torsion topological class can have zero coarse index. We prove that this can occur even when the analytic target is the maximal Roe algebra and the space is a coarse disjoint union of finite graphs of bounded degree.

Let Z be a uniformly discrete proper metric space of bounded geometry; thus every ball of a fixed radius contains a uniformly bounded number of points. Write P_r(Z) for the Rips complex whose simplices are the finite subsets of diameter at most r, and set

All K-theory and K-homology in this article are complex. On the ample module H_Z=\ell^2(Z)\otimes\ell^2(\N), let \C[Z] be the algebra of bounded, locally compact, finite-propagation operators. Local compactness means that aT and Ta are compact for every a\in C_0(Z), and finite propagation means that T_{xy}=0 whenever d(x,y) exceeds some fixed bound. In particular, the compact matrix entries of T need not satisfy a uniform rank bound. The universal C*-completion is the maximal Roe algebra C^*_{\max}(Z). The canonical maximal coarse assembly map is

We recall the needed norm estimates and the index construction in Section 4.

A coarse disjoint union of finite graphs is a metric on their disjoint union which restricts to the graph path metric on each component and for which only finitely many pairs of distinct components are within any fixed distance. We take such metrics to be uniformly discrete.

The theorem strengthens the reduced coarse-assembly result proved in the companion paper [13]. If C^*(X) denotes the operator-norm closure of \C[X], the canonical quotient C^*_{\max}(X)\to C^*(X) carries the maximal index to the reduced index. Our class \alpha therefore also lies in the reduced assembly kernel. The stronger conclusion requires a proof in the universal norm: vanishing of a reduced index alone does not imply vanishing of its maximal lift.

1.1 Context and prior work

Roe developed coarse cohomology and coarse index theory as tools for large-scale geometry and index-theoretic questions on noncompact manifolds [15]. Higson and Roe formulated and studied the coarse Baum–Connes conjecture in this framework [10]. The connection with Novikov-type questions is that injectivity retains the topological information carried by higher indices. The present theorem concerns the non-equivariant coarse assembly map displayed above.

Several fundamental positive results relate assembly to geometric control. Yu proved coarse assembly isomorphism for bounded-geometry spaces of finite asymptotic dimension [20, Theorem 7.1] and established the coarse Baum–Connes conjecture for bounded-geometry spaces admitting a uniform embedding into Hilbert space [21]. Skandalis, Tu, and Yu developed the coarse-groupoid approach and its relation to these embedding methods [17]. These results explain the importance of geometric hypotheses when seeking injectivity or isomorphism theorems.

Earlier failures of injectivity also show why bounded geometry and the domain of assembly matter. Yu constructed a coarse-assembly kernel class for a space without bounded geometry [20, Section 8, Proposition 8.1]. Dranishnikov, Ferry, and Weinberger constructed integral index-kernel examples with locally finite K-homology as domain; their kernel classes disappear under coarsening [6, Section 6, Remark 6.2]. Here the class belongs to the actual Rips colimit for a bounded-geometry graph union and remains nonzero after rationalization.

Finite quotients and expanding graphs also exposed limitations of general assembly assertions. Higson, Lafforgue, and Skandalis constructed counterexamples in group and groupoid settings and proved reduced coarse nonsurjectivity for certain unions of congruence Cayley graphs [9, Section 6]. Nonsurjectivity is a different obstruction from the kernel in Theorem 1.1. Their separate Hausdorff-groupoid kernel construction combines an odd Bott class with a Kazhdan projection: a nonzero full-assembly image becomes zero after reduction [9, Section 5]. This is a methodological predecessor, but its behavior also makes it essential to distinguish the two analytic targets.

The maximal Roe algebra and its assembly map were developed in work of Gong, Wang, and Yu [7] and Oyono-Oyono and Yu [14]; the latter analyzed maximal assembly for specific families of expanders arising from finite group quotients. Willett and Yu further studied higher indices for expanding and large-girth graph families [19]. Špakula and Willett proved that the maximal-to-reduced comparison induces a K-theory isomorphism for spaces admitting a coarse embedding into Hilbert space [18]. Chen, Wang, and Yu proved the maximal coarse Baum–Connes conjecture under the more general hypothesis of a fibred coarse embedding into Hilbert space [4]. More recent injectivity results concern finite-group extensions whose kernel and quotient families admit such fibred embeddings [5]; maximal isomorphism results also hold for coarse fibrations with uniformly property-A fibers, a fibred-coarsely-embeddable base, and uniform coarse equivalences between fibers and preimages of fixed-radius balls [8]. Finite fibers alone do not supply these uniform hypotheses. Theorem 1.1 shows that rational injectivity cannot hold for all bounded-geometry graph unions without further assumptions.

Our analytic tools belong to this established maximal-index framework. We give the norm and functoriality arguments needed here, with particular attention to the points where the universal and operator norms must be distinguished; see also [14, 16, 18]. Kazhdan’s property (T) [2, 12] supplies averaging in the full group C*-algebra. The topological part uses integral line-bundle classification, the K-theory of projective spaces, and the standard K-homology pairing [1, 3, 11].

1.2 The construction and the proof

The base graphs are three-dimensional discrete tori B_j whose side lengths tend to infinity. Over each B_j we place a finite quotient of \SL_3(\Z), obtaining a graph Y_j with Cayley edges inside the fibers. Transport around the first two base circles is given by commuting elements of that quotient. These elements cease to commute at a higher congruence level. The graph X_j joins two copies of B_j through Y_j: its middle has a long product collar, and toward either end the fiber quotients are reduced one level at a time until they become trivial. This keeps all vertex degrees bounded. Section 2 gives the construction.

Write B=\bigsqcup_jB_j, Y=\bigsqcup_jY_j, and let

be the middle projection and the two end inclusions. A torus K-homology class on each component defines a single class \beta\in\KX_1(B). The class of interest is

Its persistence and its zero index come from different features of the middle projection.

For persistence, take a line bundle \mathcal L on each continuous base torus, with first Chern class the product of the first two circle classes. Its reduced K-theory class, multiplied by the third circle’s odd class, pairs to one with the chosen K-homology class. At every fixed Rips scale, the lifted commutator calculation shows that the pullback of \mathcal L to the middle admits a 2^j-th root for all sufficiently large j. A uniform dimension bound for the Rips complexes converts these roots into divisibility by a growing power of two in integral K-theory. Mayer–Vietoris then prevents any fixed nonzero multiple of the endpoint difference from vanishing. Section 3 makes this argument before taking either the scale colimit or rational coefficients.

For the index, projection from the middle induces a surjection

To construct a splitting, we transport each base matrix entry horizontally between the fibers and then apply the Kazhdan projection onto the constant vectors in each fiber. The transport lift need not preserve products; averaging makes it multiplicative, and projecting the resulting homomorphism back to the base induces the identity on K-theory. Section 5 proves these identities in the maximal completion using two controls from Section 4: agreement of maximal and operator norms on finite vertex supports, and uniform norm estimates at fixed propagation.

Finally, Section 6 shows that the two composites (i_+)_*f_* and (i_-)_*f_* agree on analytic K_1. The end maps are not uniformly close: their separation grows with j. Instead, we place orthogonal copies of an operator at every height and rotate these copies simultaneously by one vertical step. After cancelling the common interior copies, the endpoint maps agree modulo compact operators, and injectivity of K_1(C^*_{\max}(X))\to K_1(C^*_{\max}(X)/\K) removes that quotient. Surjectivity of f_* and naturality of assembly then give the zero index.

2 Congruence fibers over discrete tori

We construct a graph with two torus ends and a long central region of constant fiber size. The fiber quotient changes one congruence level at a time outside this region. These gradual changes keep the degrees bounded, while the long central region will isolate the topological obstruction at any fixed Rips scale.

2.1 The finite quotients and their twists

Fix a finite symmetric generating set S of \Gamma=\SL_3(\Z). We use Kazhdan’s theorem that \Gamma has property (T) [2, 12]. For k\geq 1, let Q_k be the image of \Gamma under entrywise reduction modulo 2^k, and put Q_0=\{1\}. The natural reduction maps will always be denoted by their source and target.

For each j\geq1, set

where E_{ab} denotes a matrix unit. A direct multiplication gives

Thus U_j and V_j commute in Q_{2j} and in every lower quotient, although their commutator remains visible in Q_{3j}. This difference between the two levels is the source of the divisibility argument in Section 3.

Let B_j=(\Z/l_j\Z)^3 with its axis-neighbor graph. Represent each coordinate by an integer in \{0,\ldots,l_j-1\}. An oriented edge has signed seam numbers (a,b,c)\in\Z^3: a positive step from l_j-1 to 0 contributes 1 in that coordinate, a reverse step contributes -1, and all other contributions vanish. Equivalently, the unit displacement along the edge is the difference of the chosen representatives plus l_j(a,b,c). For 0\leq k\leq2j, define

with the evident interpretation in Q_0. Only the first two circle coordinates will twist the fibers.

The graph Y_{j,k} has vertex set B_j\times Q_k. Its edges are of two types:

• •

  over a fixed u\in B_j, connect (u,q) to (u,sq) for s\in S;

• •

  for an oriented base edge u\to v with seam numbers (a,b,c), connect (u,q) to (v,qg_{j,k}(a,b)).

The first action is on the left and the second on the right, so they commute. Reversing a base edge reverses its matching, because U_j and V_j commute in Q_k. We discard loops and repeated edges. Each fiber Cayley graph is connected, and the horizontal matchings connect the fibers over the connected graph B_j. Hence Y_{j,k} is connected. Write

The metric on Y_j is its intrinsic graph metric.

2.2 The two ends and the central collar

For each integer t with -3j\leq t\leq3j, put

At height t, place a copy of Y_{j,k_j(t)}. Connect successive slices over the same base vertex by the reduction map from the larger quotient to the smaller one, or by the identity when the levels agree. The resulting finite graph is X_j.

At height \pm3j the quotient is trivial, so each end is a copy of B_j. The induced graph on the heights -j,\ldots,j is the product of Y_j with the interval graph: its horizontal edges are those of Y_j, and its vertical edges preserve both u and q. We call this subgraph the central collar. Paths in X_j may take shortcuts outside the collar; whenever we use its product structure, we will first show that the paths under consideration cannot leave it.

The gradual reductions in Figure 1 have bounded fibers by Lemma 2.1. They therefore preserve a uniform degree bound even though the middle quotient grows rapidly with j.

2.3 Coarse disjoint unions

We now fix metrics on

In B_j choose the zero vertex, in Y_j the vertex (0,1), and in X_j the vertex (0,1) at height zero. In each union, join the chosen vertices of components j and j+1 by an auxiliary edge of length 100+j^2, and use the resulting path distances on the original vertex set. These edges define the metric; they add no vertices to the space.

A path which leaves and returns to the same component must pass through its chosen vertex both times. Consequently its use of the auxiliary edges cannot shorten an intrinsic component distance. At each fixed distance bound, cross-component pairs occur in only finitely many finite components. The resulting metrics are uniformly discrete and proper. They have bounded geometry because the weighted graphs have uniformly bounded degrees and all edge lengths are at least one: a ball of radius R contains no more vertices than a ball of graph radius \lfloor R\rfloor in a graph of the same degree bound.

Projection and end inclusion define maps

On each component they are 1-Lipschitz. For a fixed input distance bound, the remaining pairs lie in a finite set, so their image distances also have a finite bound. Thus the maps are bornologous. Preimages of bounded sets are finite, so the maps are proper. At scale 3 none of the Rips complexes has a cross-component simplex. In particular, the end maps induce proper simplicial inclusions

These are the maps used to form the K-homology class. The next section shows why its two endpoint contributions remain distinct at every fixed scale.

3 An integral class that persists at every scale

We construct a class \beta\in\KX_1(B) whose images at the two ends of X have an infinite-order difference. The obstruction to identifying these images will be integral: at any fixed Rips scale, an appropriate K-cohomology class becomes divisible by larger and larger powers of two over the middle graphs Y_j.

All K-groups in this section are complex and integral. For a finite complex M, we write K^*(M) for topological K-theory and K_*(M)=\KK_*(C(M),\C) for analytic K-homology. The pairing

is the Kasparov index pairing. In particular, for a continuous map h:M\to N between finite complexes,

We use the usual Mayer–Vietoris sequence for a finite complex covered by two subcomplexes; see [11] for these K-homology constructions.

3.1 The classes on the two ends

Let

Give T_j its unit cubical subdivision and triangulate the cubes periodically. The vertices are B_j, and the vertices of every cube have graph distance at most 3. The resulting triangulated torus is therefore a subcomplex of P_3(B_j); write \kappa_j:T_j\hookrightarrow P_3(B_j) for its inclusion.

Let u_1,u_2,u_3\in H^1(T_j;\Z) be the positively oriented coordinate classes. Choose a line bundle \mathcal L_j pulled back from the first two circles with

and let e_3\in K^1(T_j) be the pullback of the positive generator of K^1(S^1) under projection to the third circle. Define

This class is primitive. Indeed, the exterior-algebra description of K^*((S^1)^3) identifies [\mathcal L_j]-1 with a generator of the reduced K^0 of the first two circles, and \theta_j with the product of the three circle generators. Here the first assertion also follows from the first Chern class, which identifies \widetilde K^0((S^1)^2) with H^2((S^1)^2;\Z). The Künneth and universal coefficient theorems therefore give a class

More explicitly, K^*(T_j) is free abelian, so the universal coefficient theorem identifies K_1(T_j) with \operatorname{Hom}(K^1(T_j),\Z), and a primitive element admits an integer-valued functional taking value 1 on it; see [3, Theorems 23.1.1 and 23.1.3].

The component separation gives P_3(B)=\bigsqcup_jP_3(B_j). Countable additivity in the first variable of \KK gives

This is a product, so the entire sequence ((\kappa_j)_*\tau_j)_j determines a class \beta_3 in the group on the left; see [3, Theorem 19.7.1]. Write \beta for its image in \KX_1(B) and set

Thus \alpha=(i_+)_*\beta-(i_-)_*\beta. We will prove that every nonzero integer multiple of \alpha_3 remains nonzero at every later Rips scale.

3.2 Retractions to the tori

Fix an integer r\geq3. From now on, thresholds on j may depend on r, but not on a simplex or a vertex of the corresponding component.

The maps in Lemma 3.1 allow us to compare the same torus class at the ends and in the middle. At either end its pairing with \tau_j is 1. In the middle, a congruence calculation forces divisibility.

3.3 The congruence calculation

3.4 From line-bundle roots to divisibility in K^1

For fixed r, bounded geometry supplies an integer D_r\geq1 such that

Indeed, the vertices of a Rips simplex all belong to the radius-r ball about any one of them, and these balls have uniformly bounded cardinality. The dimension bound converts the line-bundle roots into integral divisibility in K-theory.

We have now obtained the growing divisor over the middle. It remains to show that an equality between the classes at the two ends would force their common pairing to have this divisor.

3.5 Mayer–Vietoris detection and the direct limit

For r\geq3, let

denote the inclusion of the triangulated torus at height \pm3j.

4 Maximal Roe algebras and the coarse index

The infinite-order class is now constructed. It remains to show that its two end contributions have the same maximal index. The subsequent averaging and sliding arguments will use operator formulas on the standard modules. We therefore establish the norm estimates that allow those formulas to be used in the maximal completion, and then verify the required naturality of the index. These are standard features of maximal Roe theory; see [14, Section 2] and [16, Section 2]. We give the details needed here because the distinction between operator norm and maximal norm is essential to the proof.

4.1 Norm control and multipliers

Let Z be a uniformly discrete proper metric space of bounded geometry, and write

The algebra \C[Z] consists of the bounded finite-propagation operators on H_Z that are locally compact for the multiplication representation of C_0(Z). We denote their concrete operator norm by \|\cdot\|_{\mathrm{op}} and their maximal norm by \|\cdot\|_{\max}. Write T_{xy}:H_0\to H_0 for the matrix entries of an operator T.

For a bounded finite-propagation operator, local compactness is equivalent to compactness of every entry T_{xy}. In one direction, compress by the characteristic functions of single vertices. Conversely, every row and column of a finite-propagation matrix has only finitely many entries. Multiplication on either side by a finitely supported function therefore gives a finite matrix of compact operators. Approximation by finitely supported functions proves the assertion for every function in C_0(Z). In particular, no uniform bound on the ranks, or on finite-rank approximations to the entries, is part of this definition.

This matching argument is the discrete-module form of the maximal-norm estimate in [14, Lemma 2.1]; compare [18, Lemma 1.10]. We shall also use the following stronger statement when the vertex support, rather than merely the propagation, is fixed.

We will need finite-propagation operators whose entries are bounded but need not be compact, such as the unitaries implementing the group action. Here and below M(A_Z) denotes the multiplier algebra of A_Z.

In particular, an operator identity involving bounded finite-propagation operators remains an identity between their multipliers. This statement does not assert that the concrete representation of the entire maximal Roe algebra is faithful. The finite-propagation control in the preceding lemmas is what will justify the specific limits below.

4.2 Maps induced by coarse maps

A map h:Z\to W between uniformly discrete proper bounded-geometry spaces is coarse if inverse images of bounded sets are bounded and, for every R, there is S such that d_Z(x,y)\leq R implies d_W(h(x),h(y))\leq S. Choose an isometry V_h:H_Z\to H_W that sends the summand over z into the summand over h(z), using orthogonal ranges when several vertices have the same image. Such a choice is possible because the vertex fibers are finite and H_0 is infinite dimensional. We call V_h an exact covering isometry for h.

This is the exact-covering version of [14, Lemma 2.15]. For the maps in our construction we henceforth choose such homomorphisms and write

By Proposition 4.4, all statements about their K-theory maps are independent of these choices.

4.3 The canonical odd index and naturality

The remaining index-theoretic input is compatibility with the two end inclusions.

To verify this identity in the maximal completion, we first identify the Roe algebra of a Rips complex with the vertex algebra and then recall the exponential formula for its canonical odd index.

For r>0, regard a point of P_r(Z) as a finitely supported probability measure on Z whose support has diameter at most r. Equip the complex with the 1-Wasserstein metric induced by d_Z. Every point is within r of every vertex of its support. Moreover, a bounded set in this metric uses only vertices from one bounded subset of Z: if v is a support vertex of p, then

That subset is finite. Consequently this metric is proper and induces the usual topology on the locally finite complex. The vertex inclusion and any selection of a support vertex are coarse inverses.

Choose a countable dense subset E\subset P_r(Z) containing the vertices, and use the ample module \ell^2(E)\otimes H_0 with pointwise multiplication by C_0(P_r(Z)). Assign each point of E to one vertex of its support, assigning a vertex to itself. This partitions the module into infinite-dimensional separable blocks H_v, one for each v\in Z; choose identifications H_v\cong H_0.

Under this regrouping, the algebraic Roe algebra of P_r(Z) is exactly \C[Z]. Propagation bounds in the two descriptions differ by at most 2r. For local compactness, the points in H_v lie in the compact closed star of v, so a compactly supported continuous cutoff equal to one on that star shows that a locally compact operator has compact vertex-block entries. Conversely, a compact subset of P_r(Z) meets only finitely many blocks. Compact vertex-block entries and finite propagation therefore give local compactness after multiplication by a compactly supported function, and then by any function in C_0(P_r(Z)). The algebraic identification extends to the maximal completions by universality. It is the coarse identification induced by the chosen support-vertex map. Pseudolocality below is always taken relative to C_0(P_r(Z)); the regrouping is used only for the Roe algebra and for Lemma 4.3.

We recall the cycle description of the odd index. On an ample module for P_r(Z), an odd K-homology class is represented, after stabilization and addition of degenerate cycles, by a bounded self-adjoint operator F such that

The first condition is pseudolocality. The use of a fixed ample module is the usual Paschke realization of locally finite K-homology; see [11].

Such a representative may be taken to have finite propagation. Indeed, let \varphi_v be the square roots of the barycentric coordinate functions. Their supports are compact, have diameter at most 2r, and form a locally finite family with \sum_v\varphi_v^2=1. Set

The sum converges strongly and has norm at most \|F\|: it is the compression of the direct sum of copies of F by the isometry \xi\mapsto(\varphi_v\xi)_v. It has propagation at most 2r. Furthermore,

is locally compact, since multiplication on the left by a compactly supported function leaves only finitely many compact summands; use self-adjointness for the right-hand assertion. Thus the straight-line path from F to F' consists of cycles and does not change the K-homology class. Replace F by F'.

After regrouping into vertex blocks, Lemma 4.3 makes F a self-adjoint multiplier of A_Z, and finite propagation gives F^2-1\in\C[Z]. Hence

The odd maximal index is

To relate this formula to the canonical index, let D^*_{\max}(P_r(Z)) be the universal completion of the algebra of finite-propagation pseudolocal operators on the Rips module. It contains A_Z as an ideal, and Paschke duality identifies

These assertions, including existence of the universal completion, are proved in [16, Proposition 2.22, Propositions 2.27–2.28, and Corollary 2.29]. The index is the boundary map of this extension. The multiplier action of Lemma 4.3 agrees with the inclusion on A_Z, so its exponential boundary formula is exactly (4.2). Thus this formula computes the canonical index, with its usual boundary-map sign convention. It agrees with the cycle construction in [14, Corollary 2.18]. Passing to the Rips direct limit gives \mu_Z^{\max}.

5 A transfer in the maximal Roe algebra

The projection f:Y\to B forgets a finite Cayley graph over each base vertex. We will lift operators on B to operators on Y by transporting their matrix entries between these fibers and then projecting onto the constant vectors. The resulting map supplies the analytic class that we will move from one end of X to the other.

Throughout this section, u,v,w denote vertices of B. Write F_u=f^{-1}(u) and let

These vectors define an isometry

The diameters of the fibers are not uniformly bounded, so V itself need not be a covering isometry for a coarse map. We will obtain an element of the maximal Roe algebra from this averaging operation by using property (T).

5.1 The projection onto fiber constants

For \gamma\in\Gamma, let U_\gamma act by left multiplication by \gamma on every fiber, with the identity on H_0. This is a unitary operator of propagation at most the word length of \gamma. Lemma 4.3 therefore gives a unitary representation

and hence a unital *-homomorphism from C^*_{\max}(\Gamma) into M(A_Y).

We recall explicitly the consequence of property (T) that will be used; see [2]. In C^*_{\max}(\Gamma) put

For every unitary representation \pi and every vector \eta,

In particular, 0\le\Delta\le2, and the kernel of \pi(\Delta) is the invariant subspace. Property (T), applied to the fixed finite generating set S, gives \kappa>0 such that every unit vector orthogonal to that subspace is moved by at least \kappa by some s\in S. Consequently, with \varepsilon=\kappa^2/(2|S|),

The continuous functional calculus thus gives a projection onto the invariant vectors in every representation, and

converges to this projection in the maximal group norm. Denote its image in M(A_Y) by P, and denote the images of the displayed approximants by P_n. We have

Each P_n is a finite-propagation multiplier preserving every fiber.

In the concrete representation on H_Y, the invariant vectors in \ell^2(F_u) are precisely the constant vectors: the map \Gamma\to Q_{2j} is onto and its left action on F_u is transitive. Thus the concrete image of P is VV^*. The transfer itself will use P in the maximal multiplier algebra M(A_Y).

5.2 Transporting matrix entries

For each pair u,v in the same component B_j, choose a shortest edge path from v to u, choosing the reverse path for the reversed pair. Horizontal transport along this path is a permutation

Take O_{uu}=1. When u,v lie in different components, instead put

In both cases,

Moreover, O_{uv} intertwines the left \Gamma-actions. For horizontal transport this follows because the matchings use right multiplication; for the rank-one operators it follows because both vectors are invariant.

Given T\in\C[B], use the decomposition H_Y=\bigoplus_{u\in B}\ell^2(F_u)\otimes H_0 to define

This is a linear, adjoint-preserving lift, but it need not be multiplicative. The next lemma isolates both its control and the precise failure of multiplicativity.

5.3 The splitting

We can now define the transfer and compute its composition with f_A. The first task used finite vertex support. The second will use a different control: averaging can increase propagation inside a fiber, but it does not increase propagation after the fiber is projected to its base vertex.

6 Moving the analytic class to the two ends

The two endpoint maps i_-f,i_+f:Y\to X are connected componentwise by the height slices of X_j. The distance between the endpoints grows with j, so a single bounded-distance homotopy does not compare their Roe-algebra maps. Instead, we place a copy of an operator at every height and compare this stack with its shift by one height. The common interior copies will cancel on K-theory.

For -3j\le t\le3j, let

where \overline q is reduction from Q_{2j} to Q_{k_j(t)}. The map is 1-Lipschitz: it carries both Cayley edges and horizontal matchings to edges or to vertices. Consecutive slices satisfy

Choose exact covering isometries

over these vertex maps. They exist because every vertex summand is a copy of the infinite-dimensional Hilbert space H_0. At the endpoints choose them to be the restrictions of the covering isometries defining i_{-,A}f_A and i_{+,A}f_A, respectively. For fixed j, the ranges of the D_{j,t} for distinct t are orthogonal, since they lie at distinct heights.

6.1 Stacks modulo the compact operators

Let

be the quotient map. If T\in\C[Y], write T_j=\chi_{Y_j}T\chi_{Y_j}. The difference T-\bigoplus_jT_j has finite vertex support: at a fixed propagation bound there are only finitely many cross-component entries. In particular it is compact. Define

The number of copies is unbounded, but their ranges are orthogonal. Thus \|S_a(T)\|_{\mathrm{op}}\le\|T\|_{\mathrm{op}}, for a=0,1. If \prop(T)\le R, the 1-Lipschitz property gives \prop(S_a(T))\le R. Every vertex entry is compact: it is a finite sum of isometric images of compact entries of T_j, since each Y_j is finite. Hence S_a(T)\in\C[X]. No bound on the ranks of these compact entries is required in the algebra used here.

The maps S_a need not be multiplicative before taking the quotient. For T,T'\in\C[Y], however,

is zero except for finitely many j. Each nonzero term is compact on H_{Y_j}, since it is a matrix of compact operators over a finite vertex set. Copying it into the finitely many heights of the corresponding X_j remains compact. Orthogonality of the height ranges and (6.4) therefore show that

is a *-homomorphism. By maximal universality, it extends to a *-homomorphism with domain A_Y.

6.2 A continuous rotation between the stacks

We next prove that \overline S_0 and \overline S_1 induce the same map on K-theory. The rotation takes place in a doubled target module; this ensures that the ranges stay orthogonal at every intermediate angle.

For 0\le\lambda\le\pi/2, define

Each W_{\lambda,j,t} is an isometry. For distinct pairs (j,t), their ranges are orthogonal: they lie in different components or, within one component, at different heights in each target copy. In the first copy the heights are t, and in the second they are t+1. Thus they assemble to an isometry

On \C[Y] define

Its diagonal matrix blocks are scalar multiples of the stacks in (6.2)–(6.3). The off-diagonal blocks are sums of D_{j,t}T_jD_{j,t+1}^* and their counterparts. If \prop(T)\le R, then (6.1) gives propagation at most R+1 for these blocks. Their vertex entries are again finite sums of compact operators. We conclude that

The same calculation as (6.4), using W_\lambda^*W_\lambda=1, shows that the images of F_\lambda in M_2(A_X/\K_X) are *-homomorphisms on \C[Y]. Each extends to A_Y by maximal universality.

For completeness, continuity is uniform in the number of height slices. The orthogonality just established gives

Consequently, in concrete operator norm,

The difference has propagation at most R+1, independently of the angles. Applying Lemma 4.1 to its four matrix blocks promotes (6.6) to an estimate

for a constant independent of j,\lambda,\mu. This proves point-norm continuity on the dense algebra \C[Y]. The extended homomorphisms are contractive for the maximal norm, so density proves point-norm continuity on all of A_Y.

At \lambda=0 the homomorphism is \overline S_0 in the first matrix corner, and at \lambda=\pi/2 it is \overline S_1 in the second. The corner identifications on K-theory agree, so

6.3 Canceling the interior copies