# Baum–Connes: the attempted general closure

**Date:** 20 September 2026. **Target:** reduced Baum–Connes with trivial
complex coefficients for every countable discrete group.
**Status: no full solution is established.** No counterexample group with a
verified assembly defect is constructed, and no inverse to general assembly
is proved.

This is the repository integration of the further attempt supplied in the
user's message. Sections 3–6 contain the supplied mathematical deductions,
with explicit proof details and the nonunital representation qualification
in Corollary 3.2. The earlier source audit is represented locally by the
[previous integration ledger](bc-nonhyperlinearity-integration-2026-09-20.md).
The historical uploaded archive and sandbox report are provenance, not new
local dependencies or claims of files inspected during this integration.

## 1. The actual target

For a countable discrete group G and j=0,1, the map in question is

\[
\mu_{G,j}:K_j^G(\underline EG)\longrightarrow K_j(C_r^*(G)).
\]

A positive solution must prove both injectivity and surjectivity for every
G. A negative solution must exhibit a nonzero assembly-kernel class or a
target class with no preimage for some G. Neither is supplied here.
Coefficient assembly, maximal completion, and groupoid statements do not
replace this target. In particular, [Z, Theorem 1.2](https://arxiv.org/html/2601.09615v1#S1)
retains a transformed coefficient algebra.

The Cairn endpoint is
[the open ordinary counterexample root](../baum-connes-counterexample-group-exists.md).
The results below constrain particular routes, rather than exhaust all
possible approaches to the conjecture.

## 2. Where the attempted general proof stops

### 2.1. Reduced descent is not formal

For a torsion-free host, the stabilized cylinder comparison for the
punctured finite-alphabet full shift is a weak equivalence on compact
subgroup tests. Going-down computes the topological domain, but does not
prove the reduced crossed-product comparison is onto. The
[host-and-tail splitting](../bernoulli-assembly-defects-split-into-host-and-tail.md)
leaves the rank-invisible subgroup of degree-zero K-theory and the entire
degree-one K-theory of the punctured-shift crossed product as the analytic
remainders. Locality retracts onto cylinder classes; it does not kill the
complement. [Product-trace invisibility](../bernoulli-product-traces-kill-rank-invisible-k0.md)
also does not kill a virtual class.

Even vanishing of both remainders for every torsion-free host would leave
the host assembly map as a direct summand. Such a wreath-transfer result
would not be general Baum–Connes.

### 2.2. The module route needs a K-class

Let V be a countable abelian G-module and W an invariant subgroup. Set

\[
X=\widehat V,\quad Z=W^\perp\cong\widehat{V/W},\quad U=X\setminus Z,
\qquad\Gamma=V\rtimes G,\quad Q=(V/W)\rtimes G,
\]
\[
C=C(X)\rtimes_rG\cong C_r^*(\Gamma),\quad
B=C(Z)\rtimes_rG\cong C_r^*(Q),\quad J=C_0(U)\rtimes_rG.
\]

The reduced inclusion embeds J as an ideal in C. Restriction gives a
surjection q:C→B, with J⊆ker q; equality is not assumed. The missing object
is an actual y∈K_j(C) such that

\[
q_*(y)=0,\qquad y\notin\operatorname{im}(K_j(J)\to K_j(C)). \tag{2.1}
\]

The [existing diagram proof](../k-inexact-module-triple-refutes-trivial-coefficient-bc-proof.md)
then forces ordinary assembly to fail: surjectivity for Γ or injectivity
for Q in degree j. Indeed, if the coefficient assembly for C(X) were onto,
lift y to a topological class x. If the coefficient assembly for C(Z) were
injective, naturality would kill the restriction of x. Topological
half-exactness lifts x from C_0(U); naturality then puts y in the forbidden
ideal image. Amenable-kernel partial assembly identifies these two
coefficient maps with the ordinary maps for Γ and Q.

[Existence of such a triple](../some-nonexact-group-has-a-k-inexact-module-triple.md)
remains open. A deficient norm or a nonzero algebra kernel does not
construct (2.1).

## 3. Tracial factorization for amenable-kernel group quotients

No Baum–Connes, exactness, nonhyperlinearity, finite-presentation or
property-(T) hypothesis is needed. The external input is the
[existing BKKO trace-support import](../bkko-reduced-traces-concentrate-on-amenable-radical.md).
No historical-priority claim is made for this deduction.

### Theorem 3.1. Tracial factorization

Let π:Γ→Q be a surjective homomorphism of discrete groups with amenable
kernel N. Let J_N be the closed two-sided ideal of C_r^*(Γ) generated by
λ_Γ(n)−1 for n∈N. Amenability of N gives the canonical surjection

\[
q:C_r^*(\Gamma)\longrightarrow C_r^*(Q),\qquad
q(\lambda_\Gamma(g))=\lambda_Q(\pi(g)).
\]

Every tracial state τ on C_r^*(Γ) annihilating J_N factors uniquely through
q. Equivalently, the induced surjection

\[
\bar q:D=C_r^*(\Gamma)/J_N\longrightarrow C_r^*(Q)
\]

induces an affine bijection on tracial state spaces.

**Proof.** Write R_H for the amenable radical of H.

1. N⊆R_Γ. The image π(R_Γ) is amenable and normal, hence lies in R_Q.
   Conversely π⁻¹(R_Q) is an amenable extension of R_Q by N and is normal
   in Γ. Thus
   \[
   R_\Gamma=\pi^{-1}(R_Q),\qquad R_\Gamma/N\cong R_Q. \tag{3.1}
   \]
2. [BKKO, Theorem 4.1](https://arxiv.org/html/1410.2518v3#S4)
   gives τ(λ_Γ(g))=0 for g∉R_Γ. For the canonical subgroup expectation
   E_(R_Γ), density of the group ring gives
   \[
   \tau=\tau|_{C_r^*(R_\Gamma)}\circ E_{R_\Gamma}. \tag{3.2}
   \]
3. R_Γ is amenable, so C_r^*(R_Γ)=C^*(R_Γ). The restriction of τ kills
   the ideal generated by n−1, n∈N. The full-group-algebra quotient
   therefore descends it to a tracial state σ_R on
   C^*(R_Γ/N)=C^*(R_Q)=C_r^*(R_Q). It is invariant under Q-conjugation:
   lift any conjugating element to Γ and use the trace identity for τ.
4. Define σ=σ_R∘E_(R_Q). This is a state. For a,b∈Q, normality gives
   ab∈R_Q iff ba∈R_Q. Outside R_Q both values vanish. Inside R_Q the
   elements ab and ba are Q-conjugate, so σ_R gives equal values.
   Linearity and continuity prove σ is tracial. The Q-invariance in step
   3 is essential; an arbitrary trace on a normal subgroup need not
   extend by the subgroup expectation to a trace of the whole group.
5. If g∉R_Γ, (3.1) makes both τ(λ_Γ(g)) and σ(q(λ_Γ(g))) zero.
   Inside R_Γ they agree by construction. Density proves τ=σ∘q.
   Surjectivity gives uniqueness. Conversely any trace on C_r^*(Q)
   pulls back to one annihilating J_N. ∎

### Corollary 3.2. The extra kernel is the tracial GNS radical

For E=ker q̄ and the GNS representations π_ρ of tracial states ρ on D,

\[
E=\bigcap_{\rho\in T(D)}\ker\pi_\rho. \tag{3.3}
\]

**Proof.** Factorization through q̄ puts E in every GNS kernel. The
canonical trace τ_Q on C_r^*(Q) is faithful. Its pullback has GNS kernel
exactly E: q̄(d)≠0 implies τ_Q(q̄(d)^*q̄(d))>0. This gives the reverse
inclusion. ∎

Every *-homomorphism φ:D→A into an algebra with a faithful tracial state
t kills E. For a unital φ, pull back t and apply (3.3). For a nonunital
φ, t∘φ is a bounded positive trace; if nonzero, normalize it, and if
zero, faithfulness gives φ=0. For e∈E in either case,
t(φ(e)^*φ(e))=0 forces φ(e)=0. In particular, every finite-dimensional
representation kills E. None of these assertions proves E=0.

### Corollary 3.3. Bounded positive traces miss the quotient K-defect

If z∈ker(q̄_*:K_0(D)→K_0(C_r^*(Q))), then

\[
\rho_*(z)=0\quad\text{for every }\rho\in T(D). \tag{3.4}
\]

**Proof.** Write ρ=σ∘q̄ by Theorem 3.1. Naturality gives
ρ_*(z)=σ_*(q̄_*(z))=0. Every nonzero bounded positive trace on the
unital algebra D is a positive scalar multiple of a tracial state; the
zero trace is immediate. Matrix pairings use the **unnormalized** matrix
extension of the trace. ∎

This assertion does not cover unbounded semifinite traces, index pairings,
cyclic cocycles or every other possible K-theoretic detector.

## 4. Application to the exact module obstruction

The kernel of Γ→Q in Section 2 is W, which is abelian and hence amenable.

### Lemma 4.1. The evident ideal is the relation ideal

J=C_0(X\Z)⋊_rG equals the closed ideal generated by λ_Γ(w)−1, w∈W.

**Proof.** Fourier duality identifies λ_Γ(w)−1 with
f_w(χ)=χ(w)−1 in C(X). Their common zero set is exactly W^⊥=Z.
Here is the closed-ideal argument explicitly. For h∈C_c(U), choose finitely
many w_i such that F=Σ_i|f_(w_i)|² is strictly positive on supp h. Then
h/F, extended by zero near Z, is continuous after a cutoff equal to one
on supp h. Consequently h=Σ_i(h/F)overline(f_(w_i))f_(w_i) belongs to
the generated ideal. Density of C_c(U) in C_0(U) gives that ideal as C_0(U).

Inside the crossed product, the closed span of finite sums Σ_g f_g u_g,
f_g∈C_0(U), is the embedded reduced crossed product J and is an ideal.
It contains the coefficient ideal, and any ideal containing the
coefficient ideal contains all these sums. Thus it is precisely the
generated ideal. ∎

Let p:C→D=C/J and q̄:D→B, with q=q̄p.

### Proposition 4.2. The missing class is liftable from C

For j=0,1, restriction of p_* induces an isomorphism

\[
\mathcal D_j=
\frac{\ker(q_*:K_j(C)\to K_j(B))}
{\operatorname{im}(K_j(J)\to K_j(C))}
\ \cong\ \operatorname{im}p_*\cap\ker\bar q_*\subseteq K_j(D). \tag{4.1}
\]

**Proof.** The genuine short exact sequence 0→J→C→D→0 gives
ker p_*=im(K_j(J)→K_j(C)) by its six-term exact sequence. Also
q_*=q̄_*p_*. Hence p_* on ker q_* has exactly the denominator as
kernel and has exactly the intersection on the right as image. ∎

Thus (2.1) is equivalent to a **nonzero** z in that intersection. A
nonzero element of ker q̄_* alone is not enough: it must lift from K_j(C).
Equivalently its boundary ∂z∈K_(j−1)(J) for 0→J→C→D→0 must vanish.
An algebra element of E=ker q̄, or even a nonzero class of K_j(E), alone
does not prove that its image in K_j(D) is nonzero and liftable.

Theorem 3.1, Corollaries 3.2–3.3 and Lemma 4.1 apply to this D→B.
In particular, every bounded positive trace on D kills every candidate
in 𝒟_0. This covers all tracial states of the quotient, not only ones
initially obtained from invariant measures on X.

### Corollary 4.3. The torsion-module version on C

If V is torsion and τ is a tracial state on C with τ_* zero on the image
of K_0(J), then τ factors through q:C→B. Thus τ_*(y)=0 for every y
satisfying (2.1) in degree zero.

**Proof.** A torsion abelian V has zero-dimensional compact dual X.
There is an approximate unit (e_i) of compact-open coefficient projections
for C_0(U). It is also an approximate unit for J: on a finite crossed-product
sum, left multiplication approximates each coefficient and right
multiplication approximates it using the corresponding translated
approximate unit; finite sums are dense. Each [e_i] is in K_0(J), so
τ(e_i)=0. For a≥0 in J,
0≤e_i a e_i≤||a||e_i, hence τ(e_i a e_i)=0 and norm convergence gives
τ(a)=0. Linear decomposition gives τ(J)=0. It descends to D, then to
B by Theorem 3.1. ∎

This extra zero-dimensional hypothesis is only needed to pass from
annihilation of the K_0(J) image to annihilation of J by a trace on C.
The quotient statements hold for every countable abelian module V.

## 5. Why trace invisibility does not settle K-theory

Construct a line bundle L over T² by gluing the ends of [0,1]×T with
transition y↦exp(2πiy). It is nontrivial: a nowhere-zero section would
give nonvanishing loops at x=0 and x=1 homotopic through the cylinder,
so they have equal winding number. The gluing condition makes the winding
numbers differ by one, a contradiction.

The virtual class ξ=[L]−[1] in K^0(T²)=K_0(C(T²)) is nonzero. If it
were zero, group completion would give a vector bundle F with
L⊕F≅1⊕F. Taking determinants and tensoring with (det F)^* would trivialize
L, contrary to the preceding argument.

Every tracial state on C(T²) is integration against a probability measure.
With the unnormalized matrix trace the pairing is the integral of the
pointwise rank difference, which is identically 1−1=0. Thus

\[
\bigl(\forall\rho\in T(C(\mathbb T^2))\bigr)\ \rho_*(\xi)=0
\quad\text{but}\quad\xi\ne0.
\]

Since C(T²)≅C_r^*(Z²), this occurs even in a reduced group algebra with
a faithful canonical trace. It is not an assembly counterexample and
does not construct a module quotient defect. Faithfulness kills an
individual positive projection with zero trace, but not an arbitrary
virtual difference of projections with equal traces.

## 6. A restriction on coefficient embeddings

### Proposition 6.1. Faithful trace forces full-support invariant measure

Let A be a unital C*-algebra with a faithful tracial state t. Suppose
ι:C(Y)→A is an injective unital *-homomorphism, Y is compact Hausdorff,
and a group acts on A preserving t with ι equivariant. Then Y has an
invariant probability measure of full support.

**Proof.** The invariant state t∘ι corresponds to a probability measure
m. If a nonempty open O had measure zero, compact Hausdorff regularity
provides a nonzero continuous f≥0 supported in O. Injectivity makes ι(f)
nonzero positive; faithfulness gives t(ι(f))>0, contradicting ∫f dm=0. ∎

Infinitely many pairwise disjoint translates of a nonempty open subset
are therefore impossible: each has the same strictly positive measure.
An infinite orbit of an isolated point is a special case. This obstructs
the specified unital equivariant coefficient realization in a reduced
group algebra with conjugation action, since conjugation preserves its
faithful canonical trace. It generalizes the geometric obstruction for
[module dual actions](../module-dual-actions-have-no-wandering-open-sets.md).

The same proof applies to an invariant nonzero finite-trace corner, with
its trace normalized and a unital embedding into that corner. It does not
automatically apply to a multiplier algebra carrying only a semifinite
trace. Such a strategy still needs a separate placement of the relevant
class in ordinary group-algebra K-theory. This is not an exclusion of all
possible coefficient-removal mechanisms.

## 7. Extension and completion source checks

[Z, Theorem 1.2](https://arxiv.org/html/2601.09615v1#S1)
assumes BCC with A for each preimage of a finite quotient subgroup and
the appropriate injective, surjective or bijective assembly assertion for
the quotient with C_0(Γ/N,A)⋊_rΓ. Taking A=ℂ leaves this transformed
coefficient; it does not establish assembly for it. Theorem 1.5 also
retains isometric-action, localized-coefficient and partial-assembly
hypotheses. These statements supply no general inverse here.

[Meyer's extension note](https://arxiv.org/html/2508.05726v2)
disproves a weakening to normal-subgroup-only assumptions using
coefficients. It is not an ordinary reduced Baum–Connes counterexample.

[BEW, Appendix A](https://arxiv.org/html/1804.02725v3#A1)
withdraws Proposition 4.4 and its listed consequences. Scalar agreement
for the smallest exact Morita-compatible functor is not established by
that argument. Surviving Lemma A.1 concerns a literal scalar quotient.
Even independent completion agreement would need a positive assembly
theorem to settle the target. See the
[existing completion boundary](../bgw-trivial-coefficient-agreement-is-open.md).

These were bounded primary-source statement checks on 20 September 2026,
not a complete literature survey or verification of every source proof.

## 8. Outcome and Cairn wiring

No class satisfying (2.1), general assembly inverse, vanishing theorem
for the Bernoulli remainders, or reduced projection with verified forbidden
trace has been constructed here. The unresolved step is an actual
K-theoretic construction or a general assembly inverse, not a missing
bounded-positive-trace estimate.

| Deduction | Canonical claim |
| --- | --- |
| Theorem 3.1 | [Amenable-kernel trace factorization](../amenable-kernel-quotient-traces-factor-reduced.md) |
| Corollary 3.2 | [Tracial GNS radical](../amenable-kernel-extra-kernel-is-tracial-gns-radical.md) |
| Lemma 4.1 | [Module relation ideal](../module-evident-ideal-is-kernel-relation-ideal.md) |
| Proposition 4.2 | [Liftable quotient description](../module-k-defect-is-liftable-quotient-kernel.md) |
| Corollaries 3.3 and 4.3 | [All bounded positive traces miss module defects](../module-quotient-k0-defects-are-trace-invisible.md) |
| Section 5 | [Nonzero trace-invisible torus class](../torus-has-nonzero-k0-class-invisible-to-every-trace.md) |
| Proposition 6.1 | [Faithful trace embedding restriction](../equivariant-faithful-trace-embeddings-force-measure.md) |
| Missing object | [Nonzero liftable quotient class](../some-module-quotient-has-a-nonzero-liftable-kernel-class.md) |

The last node is equivalent to the existing missing module triple, with
proof routes in both directions; it remains OPEN. The preceding claims
have written proofs and named external inputs where needed. Cairn's
ESTABLISHED/COMPLETE labels describe its dependency graph, not Lean or
independent referee certification.

## References and reproducibility

- **[BKKO]** E. Breuillard, M. Kalantar, M. Kennedy, N. Ozawa,
  *C*-simplicity and the unique trace property for discrete groups*,
  Publ. Math. IHÉS 126 (2017), 35–71, Theorem 4.1;
  [arXiv:1410.2518v3](https://arxiv.org/html/1410.2518v3),
  [DOI](https://doi.org/10.1007/s10240-017-0091-2).
- **[Z]** J. Zhang, *The Baum–Connes and the Mishchenko–Kasparov assembly
  maps for group extensions*, 14 January 2026,
  [arXiv:2601.09615v1](https://arxiv.org/html/2601.09615v1).
- **[M]** R. Meyer, *The Baum–Connes conjecture for extensions*, 2025,
  [arXiv:2508.05726v2](https://arxiv.org/html/2508.05726v2).
- **[BEW]** A. Buss, S. Echterhoff, R. Willett, *The minimal exact crossed
  product*, [arXiv:1804.02725v3](https://arxiv.org/html/1804.02725v3), Appendix A.

The supplied attempt referred to `group-approximation-main (1).zip`,
`Cairn_Baum_Connes_Proofs_and_Progress.md`, and a topic-filtered extraction
of 173 files. Those are the supplied attempt's historical provenance;
that extraction or full-report reread is not claimed for this integration.
This integration edits Cairn's live research nodes and adds this local
proof artifact. Validation is recorded in the
[integration receipt](bc-general-closure-validation-2026-09-20.json).