# Audit: a nuclear type I counterexample to STW Problem XXII ## Verdict **Green as an internal proof audit; not a substitute for external refereeing.** The construction below gives a separable, unital, nuclear, type I C*-algebra `A` with compact Bauer trace simplex for which ```text T(A) proper_subset T(overline(A)^{T(A)}). ``` The proof has one standard topological input: the elementary Chern-class obstruction to a nowhere-zero section. Applied simultaneously to the two off-diagonal blocks of a finite packet, it gives late-coordinate vector states that are multiplicative on every prescribed pair. Weak-star compactness then produces a character of the product corona. The rank/dimension scaling makes its detecting element tend to zero in the *uniform tracial 2-norm*. A literature check through 2026-08-31 found the positive results recorded in the STW problem list, Evington's Z-stable theorem, the Bice--Farah asymptotic commutator construction, and Toms' quadratic-dimension topological obstructions, but not this exact counterexample to Problem XXII. Novelty is therefore a search result, not a priority claim. ## 1. Homogeneous blocks For `s>=1`, set ```text N_s = s^2, X_s = CP^(N_s), E_s = 1 + L_s^{+s}, D_s = Gamma(End(E_s)), ``` where `L_s -> X_s` is the tautological complex line bundle. Every fibre of `D_s` is `M_(s+1)`. Let `p_s` be the projection onto the trivial line and `q_s` the projection onto the first tautological summand. Put ```text h_s = p_s-q_s. ``` The two projections are orthogonal and rank one in every fibre, so every tracial state on `D_s` vanishes on `h_s`, while ```text ||h_s|| = 1, h_s^2 = p_s+q_s, ||h_s||_(2,T(D_s))^2 = 2/(s+1). (A1) ``` Thus the operator norm stays one but the uniform fibre 2-norm tends to zero. ## 2. Simultaneous Chern zeros give finite multiplicativity packets ### Lemma If `2 ell<=s`, then for arbitrary `z_1,...,z_ell in D_s` there is `x in X_s` such that the trivial line reduces every `z_j(x)`. Consequently its fibre vector state satisfies ```text omega_(s,x)(h_s)=1, omega_(s,x)(z_j z_k) =omega_(s,x)(z_j) omega_(s,x)(z_k) for all j,k. (A2) ``` ### Proof Relative to `E_s=1+L_s^{+s}`, write ```text z_j = [ a_j b_j ] [ c_j d_j ]. ``` The lower-left blocks of `z_j` and `z_j^*` are respectively `c_j` and `b_j^*`, each a section of `L_s^{+s}`. Their simultaneous join is a section of `L_s^{+(2s ell)}`. If `u` generates `H^2(CP^(s^2);Z)`, then ```text c_(2s ell)(L_s^{+(2s ell)}) = (+/-u)^(2s ell) != 0 ``` because `2s ell<=s^2`. A nowhere-zero section would split off a trivial line and force the top Chern class to vanish. Hence there is `x in X_s` with ```text b_1(x)=c_1(x)=...=b_ell(x)=c_ell(x)=0. (A3) ``` At this point each `z_j(x)` is block diagonal relative to the trivial line, so that line is reducing. Its vector state is therefore multiplicative on every pair from the packet, and it takes value one on `h_s`. This proves (A2). The quantifiers are load-bearing: for each *fixed* finite packet, (A2) holds for every tail index `s>=2 ell`. No bound uniform in packet size is asserted. ## 3. The separable nuclear algebra and its trace simplex Let ```text B = direct_sum_(s=1)^infinity D_s ``` be the c0-direct sum and let ```text A = unitization(B). ``` Each `D_s` is separable, nuclear and type I; these properties pass through a countable c0-sum and unitization. Hence `A` is separable, unital, nuclear and type I. Its center is the unitization of `direct_sum_s C(X_s)`, namely `C(K)` for ```text K = (disjoint_union_s X_s) union {infinity}, (A5) ``` the one-point compactification. Every tracial state on a homogeneous block is integration of normalized fibre trace against a probability measure, and the scalar quotient gives the point at infinity. Consequently ```text T(A) = Prob(K), partial_e T(A) = K. (A6) ``` Thus `T(A)` is Bauer and `K` is compact metrizable. It has infinite covering dimension because every `CP^(s^2)` is clopen in `K`. ## 4. Exact uniform tracial completion For `x in D_s`, write ```text ||x||_(2,s)=sup_(y in X_s) tr_(s+1)(x(y)^*x(y))^(1/2). ``` Define ```text J = {(x_s) in product_s D_s : sup_s ||x_s|| < infinity and ||x_s||_(2,s) -> 0}. (A7) ``` Then ```text M := overline(A)^{T(A)} = C1 + J. (A8) ``` For the forward inclusion, truncate a tail: if `x=lambda1+(x_s)` with `(x_s) in J`, then ```text x^(n)=lambda1+(x_1,...,x_n,0,0,...) ``` lies in `A` and ```text ||x-x^(n)||_(2,T(A)) = sup_(s>n)||x_s||_(2,s) -> 0. (A9) ``` Conversely, a uniformly operator-norm bounded uniform-2 Cauchy sequence from `A` has a scalar limit at infinity. At each fixed coordinate `s`, finite matrix rank gives ```text ||a|| <= sqrt(s+1) ||a||_(2,s), ``` so that coordinate converges in operator norm to an element of `D_s`. Uniform boundedness passes to the coordinatewise limit. Approximation by one member of `A`, whose non-scalar part is norm-null and hence 2-null in the tail, shows that the limiting non-scalar coordinates satisfy (A7). This proves (A8). The set `J` is a norm-closed two-sided ideal in `product_s D_s`, using ```text ||xy||_(2,s) <= ||x|| ||y||_(2,s) ``` and its right-handed analogue. ## 5. State compactness produces a corona character Regard `B=direct_sum_s D_s` as an ideal of the bounded product ```text P = product_s D_s. (A10) ``` By (A1), `h=(h_s)` belongs to `J`. Given a finite packet `z_1,...,z_ell in P` and a lower bound `S`, choose `s>=max(S,2 ell)` and apply (A2) to their `s`-coordinates. Pulling the fibre vector state back through coordinate evaluation gives a state `omega` on `P` such that ```text omega(h)=1, omega(z_j z_k)=omega(z_j)omega(z_k) for all j,k. (A11) ``` By taking `s` farther out, the same state is arbitrarily small on any prescribed finite family from `B`. For finite sets of pairs `(x,y)`, elements `a in B`, and integers `r>=1`, impose on the state space of `P` the conditions ```text rho(h)=1, rho(xy)=rho(x)rho(y), |rho(a)|<=1/r. (A12) ``` Each is weak-star closed. Include every left and right factor in the Chern packet and choose one sufficiently late coordinate; (A11) proves the finite intersection property. Compactness of the state space gives a state `rho` satisfying all pair constraints and all annihilation constraints. Therefore `rho` is multiplicative on `P`, `rho|_B=0`, and `rho(h)=1`. It descends to a character of `P/B`. Restriction to `M=C1+J` gives a character, hence a tracial state, `sigma_tilde` satisfying ```text sigma_tilde(h)=1, sigma_tilde(lambda 1+b)=lambda (b in B). (A13) ``` Thus `sigma_tilde|_A` is exactly the extreme trace `tau_infinity` from the scalar quotient, whereas the uniform-2-continuous extension of `tau_infinity` vanishes on `J`. ## 6. Explicit discontinuity The discontinuity can be seen without invoking uniqueness. Let ```text h^(n)=(0,...,0,h_n,h_(n+1),...). ``` Then `h-h^(n) in B`, and `sigma_tilde|_B=0`, so ```text sigma_tilde(h^(n)) = 1 (A14) ``` for every `n`. On the other hand ```text ||h^(n)||_(2,T(A)) = sup_(s>=n) sqrt(2/(s+1)) = sqrt(2/(n+1)) -> 0. (A15) ``` Therefore `sigma_tilde` is not uniform-2-norm continuous and ```text T(A) proper_subset T(M). (A16) ``` This is exactly a negative answer to STW Problem XXII. Since the uniform tracial completion with its designated compact trace face is factorial in the tracially-complete sense (and here the extreme designated fibres are the matrix factors `M_(s+1)` and the scalar fibre at infinity), the example also refutes the equivalent factorial-tracially-complete formulation. ## 7. Compatibility with the positive theory There is no conflict with the finite-dimensional theorem in this repository: `K` has unbounded covering dimension. The construction is designed so that topological complexity available to the Chern class produces simultaneous zeros for every fixed finite multiplicativity packet while the normalized rank fraction tends to zero. There is no conflict with Evington's Z-stable theorem: `A` is type I with finite-dimensional irreducible fibres and is not Z-stable. There is no conflict with the one-dimensional Farah--Vaccaro theorem or the existing finite-dimensional factorial-bundle results for the same dimension reason. The earlier Cairn node `stw22-fixed-replication-blocks-infinite-dimensional-selection` was therefore pointing at a genuine obstruction, but did not itself prove the trace problem false: failure of one Michael-selection route is not a trace. The present argument uses the topology to construct finite restrictions of a character; state-space compactness then assembles them into an actual character of the product corona. ## 8. Prior-art firewall The closest checked topological mechanism is Bice--Farah, *Traces, Ultrapowers and the Pedersen-Petersen C*-Algebras*, Houston J. Math. 41 (2015), 1175--1190, arXiv:1307.0111. They use projective-space homogeneous blocks to create additional traces in an asymptotic setting. That result does not supply the character constructed here, and it does not by itself address the uniform-tracial-2 completion in Problem XXII. The point that must not be blurred is the metric required by Problem XXII. The present blocks simultaneously satisfy ```text operator norm = 1, uniform tracial 2-norm = sqrt(2/(s+1)) -> 0, simultaneous multiplicativity packets of size <= s/2. ``` That simultaneous scaling is what lets compactness produce a character which fails uniform-2 continuity. ## 9. Hostile checks performed 1. **Compact trace space:** (A6) gives a concrete Bauer simplex. 2. **Supremum over all traces:** for positive `a^*a`, the supremum over `Prob(K)` is attained on point masses, so the uniform 2-norm is exactly the fibre supremum used above. 3. **Completion:** both inclusions in (A8) are proved; the reverse direction uses fixed-coordinate finite-rank norm/2-norm equivalence. 4. **Chern degree:** `rank(L_s^{+(2s ell)})=2s ell` and `2s ell<=s^2`, so its top Chern class survives in the cohomology ring of `CP^(s^2)`. 5. **Reducing, not merely invariant:** simultaneous vanishing of both `c_j(x)` and `b_j(x)` makes the trivial line reducing for every packet element, hence gives multiplicativity on every prescribed pair. 6. **Finite intersection property:** every finite family of product constraints is realized after putting all its left and right factors into one Chern packet; every finite family from `B` is small at the same late coordinate. 7. **Character:** the constraints range over all pairs in `P`, so their compactness limit is multiplicative on all of `P`; the constraints for all `r` annihilate `B` exactly. 8. **Discontinuity:** (A14)--(A15) is an explicit 2-null sequence on which the rogue character is constantly one. 9. **Nuclearity/type I:** inherited from homogeneous blocks through countable c0-sum and unitization. 10. **Scope:** the counterexample refutes the unrestricted problem while leaving every positive finite-dimensional theorem intact. ## References checked - T. Bice and I. Farah, *Traces, Ultrapowers and the Pedersen-Petersen C*-Algebras*, Houston J. Math. 41 (2015), 1175--1190; arXiv:1307.0111. - S. Evington, *Traces on the uniform tracial completion of Z-stable C*-algebras*, J. Lond. Math. Soc. (2025), DOI 10.1112/jlms.70207. - C. Schafhauser, A. Tikuisis, S. White, *Nuclear C*-algebras: 99 problems*, arXiv:2506.10902v2, Problem XXII. - A. S. Toms, *Schubert Calculus and uniform property Gamma*, arXiv:2606.12188 (2026).