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] 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]. Dranishnikov, Ferry, and Weinberger constructed integral index-kernel examples with locally finite K-homology as domain; their kernel classes disappear under coarsening [6]. 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]. Nonsurjectivity is a different obstruction from the kernel in Theorem 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]. 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 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, 18, 16]. Kazhdan’s property (T) [12, 2] 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].

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 \(\mathop{\mathrm{SL}}_3(\mathbb Z)\), 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 \[f:Y\longrightarrow B,\qquad i_\pm:B\longrightarrow X\] be the middle projection and the two end inclusions. A torus K-homology class on each component defines a single class \(\beta\in\mathop{\mathrm{KX}}_1(B)\). The class of interest is \[\alpha=(i_+)_*\beta-(i_-)_*\beta.\] 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 \[f_*:K_1(C^*_{\max}(Y))\longrightarrow K_1(C^*_{\max}(B)).\] 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 \(K_1(C^*_{\max}(X))\to K_1(C^*_{\max}(X)/\mathcal K)\) removes that quotient. Surjectivity of \(f_*\) and naturality of assembly then give the zero index.

The finite quotients and their twists

Fix a finite symmetric generating set \(S\) of \(\Gamma=\mathop{\mathrm{SL}}_3(\mathbb Z)\). We use Kazhdan’s theorem that \(\Gamma\) has property (T) [12, 2]. 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 \[ m_j=2^j,\qquad l_j=100(j+1),\qquad U_j=1+m_jE_{12},\qquad V_j=1+m_jE_{23}, \tag{1}\] where \(E_{ab}\) denotes a matrix unit. A direct multiplication gives \[ U_jV_jU_j^{-1}V_j^{-1}=1+m_j^2E_{13}. \tag{2}\] 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 \(B_j=(\mathbb Z/l_j\mathbb Z)^3\) with its axis-neighbor graph. Represent each coordinate by an integer in \(\{0,\ldots,l_j-1\}\). An oriented edge has signed seam numbers \((a,b,c)\in\mathbb Z^3\): 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 \[ g_{j,k}(a,b)=V_j^bU_j^a\pmod {2^k}\in Q_k, \tag{3}\] 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 \[Y_j=Y_{j,2j}.\] The metric on \(Y_j\) is its intrinsic graph metric.

The two ends and the central collar

For each integer \(t\) with \(-3j\leq t\leq3j\), put \[ k_j(t)= \begin{cases} 2j,& |t|\leq j,\\ 3j-|t|,& j\leq |t|\leq3j. \end{cases} \tag{4}\] 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. They therefore preserve a uniform degree bound even though the middle quotient grows rapidly with \(j\).

Coarse disjoint unions

We now fix metrics on \[B=\bigsqcup_{j\geq1}B_j,\qquad Y=\bigsqcup_{j\geq1}Y_j,\qquad X=\bigsqcup_{j\geq1}X_j.\] 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 \[ f:Y\longrightarrow B,\qquad i_\pm:B\longrightarrow X. \tag{5}\] 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 \[i_{\pm,3}:P_3(B)\longrightarrow P_3(X).\] 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.

The classes on the two ends

Let \[T_j=(\mathbb R/l_j\mathbb Z)^3.\] 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 \(u_1,u_2,u_3\in H^1(T_j;\mathbb Z)\) be the positively oriented coordinate classes. Choose a line bundle \(\mathcal L_j\) pulled back from the first two circles with \[c_1(\mathcal L_j)=u_1\smile u_2,\] 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 \[ \theta_j=([\mathcal L_j]-1)e_3\in K^1(T_j). \tag{7}\] 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 \(H^2((S^1)^2;\mathbb Z)\). The Künneth and universal coefficient theorems therefore give a class \[ \tau_j\in K_1(T_j),\qquad \langle\theta_j,\tau_j\rangle=1. \tag{8}\] More explicitly, \(K^*(T_j)\) is free abelian, so the universal coefficient theorem identifies \(K_1(T_j)\) with \(\operatorname{Hom}(K^1(T_j),\mathbb Z)\), and a primitive element admits an integer-valued functional taking value \(1\) on it; see [3].

The component separation gives \(P_3(B)=\bigsqcup_jP_3(B_j)\). Countable additivity in the first variable of \(\mathop{\mathrm{KK}}\) gives \[ \mathop{\mathrm{KK}}_1(C_0(P_3(B)),\mathbb C) \cong\prod_{j\geq1}K_1(P_3(B_j)). \tag{9}\] 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]. Write \(\beta\) for its image in \(\mathop{\mathrm{KX}}_1(B)\) and set \[ \alpha_3=(i_{+,3})_*\beta_3-(i_{-,3})_*\beta_3, \qquad \alpha=[\alpha_3]\in\mathop{\mathrm{KX}}_1(X). \tag{10}\] Thus \(\alpha=(i_+)_*\beta-(i_-)_*\beta\). We will prove that every nonzero integer multiple of \(\alpha_3\) remains nonzero at every later Rips scale.

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 4 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.

The congruence calculation

From line-bundle roots to divisibility in \(K^1\)

For fixed \(r\), bounded geometry supplies an integer \(D_r\geq1\) such that \[\dim P_r(Y_j)\leq D_r\qquad(j\geq1).\] 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.

Mayer–Vietoris detection and the direct limit

For \(r\geq3\), let \[t_{\pm,j,r}:T_j\longrightarrow P_r(X_j)\] denote the inclusion of the triangulated torus at height \(\pm3j\).

Norm control and multipliers

Let \(Z\) be a uniformly discrete proper metric space of bounded geometry, and write \[H_0=\ell^2(\mathbb N),\qquad H_Z=\ell^2(Z)\otimes H_0, \qquad A_Z=C^*_{\max}(Z).\] The algebra \(\mathbb C[Z]\) 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]; compare [18]. 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.

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]. For the maps in our construction we henceforth choose such homomorphisms and write \[f_A:A_Y\longrightarrow A_B,\qquad i_{\pm,A}:A_B\longrightarrow A_X.\] By Proposition 12, all statements about their \(K\)-theory maps are independent of these choices.

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 \[d_Z(v,z_0)\leq r+\int_Z d_Z(z,z_0)\,dp(z) =r+d_{W_1}(p,\delta_{z_0}).\] 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 \(\mathbb C[Z]\). 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 11.

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 \[[F,a]\text{ is compact},\qquad a(F^2-1)\text{ is compact} \quad(a\in C_0(P_r(Z))).\] 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 \[F'=\sum_v\varphi_vF\varphi_v.\] 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, \[F'-F=\sum_v\varphi_v[F,\varphi_v]\] 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 11 makes \(F\) a self-adjoint multiplier of \(A_Z\), and finite propagation gives \(F^2-1\in\mathbb C[Z]\). Hence \[p=\frac{1+F}{2}\in M(A_Z),\qquad p^2-p\in\mathbb C[Z],\qquad u_F=\exp(2\pi i p)\in 1+A_Z.\] The odd maximal index is \[ \operatorname{Ind}_{Z,\max}(F)=[u_F]\in K_1(A_Z). \tag{23}\] 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 \[\mathop{\mathrm{KK}}_1(C_0(P_r(Z)),\mathbb C) \cong K_0\bigl(D^*_{\max}(P_r(Z))/A_Z\bigr), \qquad [F]\longmapsto[p\bmod A_Z].\] These assertions, including existence of the universal completion, are proved in [16]. The index is the boundary map of this extension. The multiplier action of Lemma 11 agrees with the inclusion on \(A_Z\), so its exponential boundary formula is exactly (23). Thus this formula computes the canonical index, with its usual boundary-map sign convention. It agrees with the cycle construction in [14]. Passing to the Rips direct limit gives \(\mu_Z^{\max}\).

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 11 therefore gives a unitary representation \[\Gamma\longrightarrow M(A_Y),\qquad\gamma\longmapsto U_\gamma,\] 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 \[\Delta=1-\frac1{|S|}\sum_{s\in S}s.\] For every unitary representation \(\pi\) and every vector \(\eta\), \[\langle\pi(\Delta)\eta,\eta\rangle =\frac1{2|S|}\sum_{s\in S} \|\pi(s)\eta-\eta\|^2.\] 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 \(\varepsilon=\kappa^2/(2|S|)\), \[\sigma_{C^*_{\max}(\Gamma)}(\Delta) \subseteq\{0\}\cup[\varepsilon,2].\] The continuous functional calculus thus gives a projection onto the invariant vectors in every representation, and \[\left(1-\frac{\Delta}{2}\right)^n\] 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 \[ \|P_n-P\|_{M(A_Y)}\longrightarrow0. \tag{25}\] 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 \(M(A_Y)\).

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 \[O_{uv}:\ell^2(F_v)\longrightarrow\ell^2(F_u).\] Take \(O_{uu}=1\). When \(u,v\) lie in different components, instead put \[O_{uv}=|\xi_u\rangle\langle\xi_v|.\] In both cases, \[ O_{vu}=O_{uv}^*,\qquad O_{uv}\xi_v=\xi_u,\qquad O_{uv}^*\xi_u=\xi_v. \tag{26}\] 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 \(T\in\mathbb C[B]\), use the decomposition \(H_Y=\bigoplus_{u\in B}\ell^2(F_u)\otimes H_0\) to define \[ L(T)_{uv}=O_{uv}\otimes T_{uv}. \tag{27}\] 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.

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.

Stacks modulo the compact operators

Let \[q:A_X\longrightarrow A_X/\mathcal K_X\] be the quotient map. If \(T\in\mathbb C[Y]\), 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 \[\begin{align*} S_0(T)&=\bigoplus_j\sum_{t=-3j}^{3j-1} D_{j,t}T_jD_{j,t}^*,\tag{32}\\ S_1(T)&=\bigoplus_j\sum_{t=-3j}^{3j-1} D_{j,t+1}T_jD_{j,t+1}^*. \tag{33}\end{align*}\] 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 \(\mathop{\mathrm{prop}}(T)\le R\), the \(1\)-Lipschitz property gives \(\mathop{\mathrm{prop}}(S_a(T))\le R\). 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 \(S_a(T)\in\mathbb C[X]\). 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 \(T,T'\in\mathbb C[Y]\), however, \[ (TT')_j-T_jT'_j =\chi_{Y_j}T(1-\chi_{Y_j})T'\chi_{Y_j} \tag{34}\] 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 (34) therefore show that \[\overline S_a=qS_a:\mathbb C[Y]\longrightarrow A_X/\mathcal K_X\] is a \(*\)-homomorphism. By maximal universality, it extends to a \(*\)-homomorphism with domain \(A_Y\).

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 \[W_{\lambda,j,t}= \begin{pmatrix} \cos\lambda\,D_{j,t}\\ \sin\lambda\,D_{j,t+1} \end{pmatrix} :H_{Y_j}\longrightarrow H_{X_j}\oplus H_{X_j}, \qquad -3j\le t<3j.\] 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 \[W_\lambda:\bigoplus_{j}\bigoplus_{t=-3j}^{3j-1}H_{Y_j} \longrightarrow H_X\oplus H_X.\] On \(\mathbb C[Y]\) define \[ F_\lambda(T)=W_\lambda \left(\bigoplus_j\bigoplus_{t=-3j}^{3j-1}T_j\right) W_\lambda^*. \tag{35}\] Its diagonal matrix blocks are scalar multiples of the stacks in (32)–(33). The off-diagonal blocks are sums of \(D_{j,t}T_jD_{j,t+1}^*\) and their counterparts. If \(\mathop{\mathrm{prop}}(T)\le R\), then (31) gives propagation at most \(R+1\) for these blocks. Their vertex entries are again finite sums of compact operators. We conclude that \[F_\lambda(T)\in M_2(\mathbb C[X]),\qquad \mathop{\mathrm{prop}}(F_\lambda(T))\le R+1.\] The same calculation as (34), using \(W_\lambda^*W_\lambda=1\), shows that the images of \(F_\lambda\) in \(M_2(A_X/\mathcal K_X)\) are \(*\)-homomorphisms on \(\mathbb C[Y]\). Each extends to \(A_Y\) by maximal universality.

For completeness, continuity is uniform in the number of height slices. The orthogonality just established gives \[\|W_\lambda-W_\mu\| =\bigl((\cos\lambda-\cos\mu)^2 +(\sin\lambda-\sin\mu)^2\bigr)^{1/2} \le |\lambda-\mu|.\] Consequently, in concrete operator norm, \[ \|F_\lambda(T)-F_\mu(T)\|_{\mathrm{op}} \le2\|T\|_{\mathrm{op}}\,|\lambda-\mu|. \tag{36}\] The difference has propagation at most \(R+1\), independently of the angles. Applying Lemma 9 to its four matrix blocks promotes (36) to an estimate \[\|F_\lambda(T)-F_\mu(T)\|_{M_2(A_X)} \le C_R\|T\|_{\mathrm{op}}\,|\lambda-\mu|\] for a constant independent of \(j,\lambda,\mu\). This proves point-norm continuity on the dense algebra \(\mathbb C[Y]\). 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 \[ (\overline S_0)_*=(\overline S_1)_*. \tag{37}\]

Canceling the interior copies