The Kadison–Kaplansky conjecture and scalar compression

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

The projection form of the Kadison–Kaplansky conjecture asserts that the reduced group \(C^*\)-algebra of a torsion-free group contains no projections other than \(0\) and \(1\). OpenAI constructed in 2026 a finitely generated torsion-free group and a projection in its reduced group \(C^*\)-algebra whose canonical trace lies strictly between \(0\) and \(\frac12\) [OpenAI-KK, Theorem 1.1]. This course proves that theorem. The present lesson proves the last step of the argument, which is independent of the rest: a projection matrix over \(C_r^*(\Lambda)\) whose unnormalized trace is less than \(\frac12\) becomes a scalar projection of the same trace after a free product with a free group [OpenAI-KK, Proposition 2.2]. It then states the matrix construction that the remaining lessons carry out.

We use the continuous functional calculus (Theorem 5.1) and spectral permanence (Theorem 3.2) of C*-algebras, continuous functional calculus, automatic continuity and positive cones, and the trace of Traces of projections and the Baum–Connes conjecture, Section 1.

1. The conjecture and the theorem

For a countable group \(\Lambda\) let \(\lambda_\Lambda(g)\delta_h=\delta_{gh}\) be the left regular representation on \(\ell^2(\Lambda)\), and \(C_r^*(\Lambda)\) the norm closure of \(\lambda_\Lambda(\mathbb C\Lambda)\). Its canonical trace is \(\tau_\Lambda(x)=\langle x\delta_1,\delta_1\rangle\). On \(N\times N\) matrices we always use the unnormalized trace \[ \tau_{\Lambda,N}(x)=\sum_{j=1}^N\tau_\Lambda(x_{jj}), \] so that \(\tau_{\Lambda,N}(I_N)=N\). It is tracial, \(\tau_{\Lambda,N}(xy)=\tau_{\Lambda,N}(yx)\), also for rectangular matrices \(x\in M_{N,M}\) and \(y\in M_{M,N}\), and faithful: a nonzero positive matrix has positive trace (Lemma 1.1 of the lesson on traces of projections, whose proof applies verbatim to rectangular products).

Conjecture (Kadison–Kaplansky, projection form). If \(\Lambda\) is torsion-free, the only projections in \(C_r^*(\Lambda)\) are \(0\) and \(1\).

Pimsner and Voiculescu proved it for free groups. For torsion-free groups that satisfy the Baum–Connes conjecture, the trace of every projection, over every matrix size, is an integer: by Lück's theorem the traces of classes in the image of the assembly map lie in the ring generated by the inverses of the orders of the finite subgroups, which is \(\mathbb Z\) here [OpenAI-KK, Section 1]. Then a projection \(e\in C_r^*(\Lambda)\) has \(\tau_\Lambda(e)\in\{0,1\}\), and \(\tau_\Lambda(e)=0\) or \(\tau_\Lambda(1-e)=0\) forces \(e=0\) or \(e=1\) by faithfulness. This covers amenable groups and torsion-free subgroups of hyperbolic groups [OpenAI-KK, Section 1]; see Traces of projections and the Baum–Connes conjecture, Section 3, for the assembly map and Lück's theorem.

Theorem 1.1 (OpenAI [OpenAI-KK, Theorem 1.1]). There are a finitely generated torsion-free group \(G_{\mathrm{proj}}\) and a projection \(e\in C_r^*(G_{\mathrm{proj}})\) with \[ 0<\tau_{G_{\mathrm{proj}}}(e)<\tfrac12 . \] In particular \(e\ne0\) and \(e\ne1\), and the conjecture is false.

For this group the complex group ring itself has no idempotents other than \(0\) and \(1\): this is Kaplansky's idempotent conjecture in characteristic zero, proved for all torsion-free groups in Idempotents and integral traces, Theorem 2.1. The projection \(e\) lives in the completion only.

Kaplansky's other problems of this circle are treated in the programme as well: zero divisors in group algebras of torsion-free groups in Group algebras and zero divisors, where OpenAI's counterexample is over \(\mathbb F_2\) and S. V. Ivanov extended it to every odd characteristic [Ivanov]; direct finiteness of group algebras in Directly finite group algebras; and quasitraces on \(C^*\)-algebras in Quasitraces and a uniform additivity defect.

2. Reduced subgroup inclusion

Lemma 2.1 (reduced subgroup inclusion; [OpenAI-KK, Lemma 2.1]). Let \(\Lambda\) be a subgroup of a countable group \(K\). The map \(\lambda_\Lambda(x)\mapsto\lambda_K(x)\), \(x\in\mathbb C\Lambda\), extends to an isometric unital \(*\)-homomorphism \(C_r^*(\Lambda)\to C_r^*(K)\) that preserves the canonical traces, and its matrix amplifications preserve the unnormalized matrix traces.

Proof. Choose a representative \(v\) in every right coset \(\Lambda v\subseteq K\). The unitary \(\ell^2(\Lambda)\to\ell^2(\Lambda v)\), \(\delta_\gamma\mapsto\delta_{\gamma v}\), intertwines \(\lambda_\Lambda(g)\) with the restriction of \(\lambda_K(g)\), \(g\in\Lambda\), because \(g\gamma v\) corresponds to \(g\gamma\). Since \(\ell^2(K)\) is the orthogonal sum of the invariant subspaces \(\ell^2(\Lambda v)\), the operator \(\lambda_K(x)\), \(x\in\mathbb C\Lambda\), is unitarily equivalent to a direct sum of copies of \(\lambda_\Lambda(x)\), and has the same norm. So the map is isometric on \(\lambda_\Lambda(\mathbb C\Lambda)\) and extends to the closures. Choosing the representative \(v=1\) for the coset \(\Lambda\) itself, \(\langle\lambda_K(x)\delta_1,\delta_1\rangle=\langle\lambda_\Lambda(x)\delta_1,\delta_1\rangle\); this passes to limits and to diagonal entries of matrices. \(\square\)

Through this inclusion we regard matrices over \(C_r^*(\Lambda)\) as matrices over \(C_r^*(K)\), acting on \(\ell^2(K)^N\); every subspace \(\ell^2(\Lambda v)^N\) is invariant, and on it the matrix acts as on \(\ell^2(\Lambda)^N\).

3. Free products

Let \(\Lambda_0,\Lambda_1,\ldots,\Lambda_N\) be groups. A reduced word is a finite sequence \((s_1,\ldots,s_n)\), \(n\ge0\), of nontrivial elements of the groups \(\Lambda_k\), called syllables, in which consecutive syllables belong to different groups. The free product \(\Lambda_0*\Lambda_1*\cdots*\Lambda_N\) is the group generated by copies of the \(\Lambda_k\), subject only to the relations of each \(\Lambda_k\). Every element is a product \(s_1\cdots s_n\) of the syllables of a reduced word: multiply adjacent elements of the same group and delete identities until none remain.

Lemma 3.1 (normal form). Every element of a free product is the product of exactly one reduced word.

Proof. Let \(\mathcal W\) be the set of reduced words. For \(g\in\Lambda_k\) define a map \(\theta_g:\mathcal W\to\mathcal W\) on \(w=(s_1,\ldots,s_n)\): if \(s_1\notin\Lambda_k\) or \(n=0\), put \(\theta_g(w)=(g,s_1,\ldots,s_n)\) for \(g\ne1\) and \(\theta_1(w)=w\); if \(s_1\in\Lambda_k\), put \(\theta_g(w)=(gs_1,s_2,\ldots,s_n)\) when \(gs_1\ne1\), and \((s_2,\ldots,s_n)\) when \(gs_1=1\). A check of the cases shows \(\theta_g\theta_{g'}=\theta_{gg'}\) for \(g,g'\in\Lambda_k\), so each \(\Lambda_k\) acts on \(\mathcal W\), and by the definition of the free product these actions extend to an action \(g\mapsto\theta_g\) of the free product. For a reduced word \(w\), the product \(\bar w=s_1\cdots s_n\) satisfies \(\theta_{\bar w}(\varnothing)=\theta_{s_1}\cdots\theta_{s_n}(\varnothing)=w\), by induction on \(n\). So \(w\) is recovered from \(\bar w\). \(\square\)

The length of an element is the length of its reduced word.

Lemma 3.2 (torsion in free products). If every \(\Lambda_k\) is torsion-free, so is the free product. If every \(\Lambda_k\) is finitely generated and \(N\) is finite, the free product is finitely generated.

Proof. Let \(g\ne1\) have reduced word \((s_1,\ldots,s_n)\). If \(n\ge2\) and \(s_1,s_n\) lie in the same factor, conjugation by \(s_1\) gives an element with reduced word \((s_2,\ldots,s_{n-1},s_ns_1)\), or \((s_2,\ldots,s_{n-1})\) if \(s_ns_1=1\), which is shorter. Repeating, \(g\) is conjugate to an element \(g'\ne1\) of length \(1\), or to one of length \(n'\ge2\) whose first and last syllables lie in different factors. In the first case \(g'\) is a nontrivial element of some \(\Lambda_k\), of infinite order. In the second case the concatenation of \(k\) copies of the reduced word of \(g'\) is reduced, so \(g'^k\) has length \(kn'\ge2\) and is not \(1\), for every \(k\ge1\). In both cases \(g\) has infinite order. The second statement holds because the free product is generated by the union of generating sets of the factors. \(\square\)

We use \(K=\Lambda*\mathbb F_N\), where \(\mathbb F_N\) is free on \(t_1,\ldots,t_N\); it is the free product of \(\Lambda\) and the \(N\) infinite cyclic groups \(\langle t_j\rangle\cong\mathbb Z\). A syllable of \(\langle t_j\rangle\) is a power \(t_j^k\), \(k\ne0\). We say that an element starts with \(t_j^{\pm}\) if its first syllable is \(t_j^k\) with \(\pm k>0\).

4. Scalar compression

Proposition 4.1 (scalar compression; [OpenAI-KK, Proposition 2.2]). Let \(\Lambda\) be a countable group and \(p\in M_N(C_r^*(\Lambda))\) a projection with \[ 0<a:=\tau_{\Lambda,N}(p)<\tfrac12 . \] Let \(K=\Lambda*\mathbb F_N\), regard \(p\) as a matrix over \(C_r^*(K)\) by Lemma 2.1, and put \[ \begin{aligned} U&=\bigl(\lambda_K(t_1),\ldots,\lambda_K(t_N)\bigr)\\ &\in M_{1,N}\bigl(C_r^*(K)\bigr),\\ A&=pU^*Up . \end{aligned} \] Then \(A\) is invertible in the corner \(pM_N(C_r^*(K))p\), whose unit is \(p\), and \[ e=(Up)A^{-1}(Up)^* \] is a projection in \(C_r^*(K)\) with \(\tau_K(e)=a\). If \(\Lambda\) is finitely generated and torsion-free, so is \(K\).

Proof. The matrix \(U\) maps \(\xi=(\xi_1,\ldots,\xi_N)\in\ell^2(K)^N\) to \(U\xi=\sum_j\lambda_K(t_j)\xi_j\in\ell^2(K)\). We show that \[ \begin{gathered} \|U\xi\|\ge c_a\|\xi\|\ \ \text{for }\xi=p\xi,\\ c_a=\sqrt{1-a}-\sqrt a>0 . \end{gathered}\tag{4.1} \]

Cosets. Let \(\mathcal R\subseteq K\) consist of \(1\) and the elements whose first syllable lies in \(\mathbb F_N\), that is, in one of the groups \(\langle t_j\rangle\). By Lemma 3.1, every element of \(K\) is uniquely \(\gamma v\) with \(\gamma\in\Lambda\) and \(v\in\mathcal R\): if its first syllable \(\gamma\) lies in \(\Lambda\), it is \(\gamma\) times the remaining word, and otherwise \(\gamma=1\). So \(\mathcal R\) is a set of representatives of the right cosets \(\Lambda v\). For \(\xi=p\xi\) put \(m_v=\|\xi|_{\Lambda v}\|^2\), so that \(\sum_{v\in\mathcal R}m_v=\|\xi\|^2\), and \(a_j=\tau_\Lambda(p_{jj})\ge0\), so that \(\sum_ja_j=a\).

A pointwise bound. Restriction to the invariant subspace \(\ell^2(\Lambda v)^N\) commutes with \(p\), so \(\xi|_{\Lambda v}=p(\xi|_{\Lambda v})\). Let \(\delta_{(v,j)}\) be the vector with entry \(\delta_v\) in coordinate \(j\). By the Cauchy–Schwarz inequality, \[ \begin{aligned} |\xi_j(v)|^2&=|\langle\xi|_{\Lambda v},p\delta_{(v,j)}\rangle|^2\\ &\le m_v\langle p\delta_{(v,j)},\delta_{(v,j)}\rangle\\ &=a_jm_v , \end{aligned}\tag{4.2} \] because the unitary \(\ell^2(\Lambda)\to\ell^2(\Lambda v)\) of Lemma 2.1 carries \(\delta_1\) to \(\delta_v\) and intertwines the actions of \(p_{jj}\), and \(\langle p_{jj}\delta_1,\delta_1\rangle=a_j\).

Good and bad inputs. Call \(g\in K\) bad for the coordinate \(j\) if \(g\) starts with \(t_j^-\), and good otherwise. A bad element lies in \(\mathcal R\), and it is bad for exactly one coordinate, read off from its first syllable. Split \(\xi=\xi_{\mathrm g}+\xi_{\mathrm b}\), where \(\xi_{\mathrm b}\) keeps the entries \(\xi_j(g)\) with \(g\) bad for \(j\). By (4.2), \[ \begin{aligned} \|\xi_{\mathrm b}\|^2&=\sum_{v\in\mathcal R}\ \sum_{j:\,v\text{ bad for }j}|\xi_j(v)|^2\\ &\le a\sum_{v\in\mathcal R}m_v=a\|\xi\|^2 . \end{aligned} \] If \(g\) is good for \(j\), then \(t_jg\) starts with \(t_j^+\): either \(g\) starts with \(t_j^k\), \(k>0\), and \(t_jg\) starts with \(t_j^{k+1}\), or the first syllable of \(g\) is not a power of \(t_j\) (or \(g=1\)), and \(t_jg\) starts with \(t_j\). The sets of elements starting with \(t_j^+\) are disjoint for different \(j\), and \(g\mapsto t_jg\) is injective. So the vectors \(\lambda_K(t_j)(\xi_{\mathrm g})_j\) have disjoint supports, and \[ \begin{aligned} \|U\xi_{\mathrm g}\|^2=\|\xi_{\mathrm g}\|^2&=\|\xi\|^2-\|\xi_{\mathrm b}\|^2\\ &\ge(1-a)\|\xi\|^2 . \end{aligned} \]

Bad outputs. For \(h\in K\), let \(J(h)\) be the set of \(j\) for which \(v_{h,j}=t_j^{-1}h\) is bad for \(j\). Then \((U\xi_{\mathrm b})(h)=\sum_{j\in J(h)}\xi_j(v_{h,j})\), and (4.2) with the Cauchy–Schwarz inequality gives \[ \begin{aligned} |(U\xi_{\mathrm b})(h)|^2&\le\Bigl(\sum_ja_j\Bigr)\sum_{j\in J(h)}m_{v_{h,j}}\\ &\le a\sum_{j\in J(h)}m_{v_{h,j}} . \end{aligned} \] The pairs \((h,j)\) with \(j\in J(h)\) correspond injectively to the bad elements \(v_{h,j}\in\mathcal R\): the first syllable of \(v_{h,j}\) determines \(j\), and then \(h=t_jv_{h,j}\). Summing over \(h\), \(\|U\xi_{\mathrm b}\|^2\le a\sum_{v\in\mathcal R}m_v=a\|\xi\|^2\). The triangle inequality now gives (4.1).

The corner. By (4.1), \(\langle A\xi,\xi\rangle=\|Up\xi\|^2\ge c_a^2\|p\xi\|^2\), so \(A\ge c_a^2p\) in the corner. Since \(UU^*=\sum_j\lambda_K(t_j)\lambda_K(t_j)^*=N\), the norm of \(U\) is \(\sqrt N\), and \(A\le Np\). Hence \(\|p-A/N\|\le1-c_a^2/N<1\) in the unital \(C^*\)-algebra \(pM_N(C_r^*(K))p\), and \[ A^{-1}=\frac1N\sum_{k\ge0}\Bigl(p-\frac AN\Bigr)^k \] converges there. The continuous functional calculus in the corner gives \(A^{-1/2}\), with spectrum in \([N^{-1/2},c_a^{-1}]\). Put \(V=UpA^{-1/2}\in M_{1,N}(C_r^*(K))\). Then \[ V^*V=A^{-1/2}\,pU^*Up\,A^{-1/2}=p , \] so \(V\) is a partial isometry, and \(e=VV^*=UpA^{-1}pU^*\) is a projection in \(C_r^*(K)\). Writing \(V=(V_1,\ldots,V_N)\), the trace property gives \[ \begin{aligned} \tau_K(e)&=\sum_j\tau_K(V_jV_j^*)=\sum_j\tau_K(V_j^*V_j)\\ &=\tau_{K,N}(p)=a , \end{aligned} \] using Lemma 2.1 for the last equality. The final statement is Lemma 3.2. \(\square\)

The hypothesis \(a<\frac12\) enters only through \(c_a>0\). For \(N=1\) the proposition is empty: then \(U=\lambda_K(t_1)\) is unitary and \(e\) is the conjugate of \(p\) (Exercise 7.2). The point is the passage from a matrix projection of small unnormalized trace to a scalar one.

5. The matrix target

A subset of the spectrum of a self-adjoint element \(T\) is separated if it has positive distance from the rest of the spectrum. Its indicator function is then continuous on the spectrum, and the continuous functional calculus places the corresponding spectral projection in every unital \(C^*\)-algebra that contains \(T\).

Theorem 5.1 (matrix construction; OpenAI [OpenAI-KK, Theorem 3.1]). There are a finitely generated torsion-free group \(\Gamma\), an integer \(N\ge1\), and a self-adjoint matrix \(T\in M_N(\mathbb C \Gamma)\) whose spectrum in the left regular representation has a nonempty separated subset with spectral projection \(p_0\) satisfying \[ p_0\in M_N\bigl(C_r^*(\Gamma)\bigr),\quad0<\tau_{\Gamma,N}(p_0)<\tfrac12 . \]

Proof of Theorem 1.1 from Theorem 5.1. Apply Proposition 4.1 to \(\Lambda=\Gamma\) and \(p=p_0\), and put \(G_{\mathrm{proj}}=\Gamma*\mathbb F_N\). \(\square\)

The same trace shows that \([p_0]\) is not in the image of the reduced Baum–Connes assembly map of \(\Gamma\): for a torsion-free group, Lück's theorem makes the traces of assembly classes integers, while \(0<\tau(p_0)<\frac12\) [OpenAI-KK, Section 1]. The same holds for \([e]\) and \(G_{\mathrm{proj}}\). Reduced assembly can also fail to be injective, even rationally, for a finitely generated torsion-free group [OpenAI-BCI, Theorem 1.1]; see The reduced assembly map and the torus class.

6. The plan of the course

The matrix \(T\) is a weighted adjacency operator on the Cayley graph of a group \(\Gamma\) defined by a graphical presentation, as in Graphical presentations and coned diagrams. The relator graphs are the components of a voltage cover of an infinite layered graph, the tower, with voltages in a torsion-free nilpotent group. After a Fourier transform in the center of that group, a model operator on the tower has a separated upper band, spanned by constant vectors on one or two adjacent layers tensored with a projection of small trace. The cover, the embedding into the Cayley graph and the removal of overlaps preserve the band, and the trace stays below \(\frac12\).

  1. This lesson: the conjecture and scalar compression.
  2. Separated bands and equivariant traces.
  3. Streams, slots and slot evaluations: the vertex sets of the tower.
  4. Gated permutations and the two towers.
  5. The mean incidence and its gap.
  6. Expansion of the realized incidence.
  7. Pruning and recovery of long paths.
  8. Voltage covers and embedded copies.
  9. Thin triangles, overlap balls and the residual forest.
  10. Torsion-freeness.
  11. A word detector in unitriangular matrices.
  12. An affine nilpotent group.
  13. Schwartz idempotents and lattice projections.
  14. The magnetic factor and the weighted trace.
  15. The model operator and finite tests.
  16. Masks, lists and one draw per edge.
  17. Girth and centered moments.
  18. From the cover band to the Cayley projection: the proof of Theorem 5.1.

The geometric lessons use the statements (VK) and (MAP) on van Kampen diagrams and plane maps, recorded in Section 4 of Voltage covers and embedded copies and proved in the course Van Kampen diagrams. Every step of Theorem 5.1 is proved in the course or in the lessons it names.

7. Exercises

7.1 (easy). Show that \(c_a=\sqrt{1-a}-\sqrt a\) is positive exactly when \(a<\frac12\), and compute \(c_a^2\) for \(a=\frac14\).

7.2 (easy). Let \(N=1\) in Proposition 4.1. Show that \(A=p\) and \(e=\lambda_K(t_1)\,p\,\lambda_K(t_1)^*\).

7.3 (medium). In \(\mathbb Z/2*\mathbb Z/2=\langle a\rangle*\langle b\rangle\), show that \(ab\) has infinite order, while \(a\) has order two. Which hypothesis of Lemma 3.2 fails?

7.4 (medium). Let \(N=2\), \(\Lambda\) any countable group, and \(p=\operatorname{diag}(q,0)\) with \(q\in C_r^*(\Lambda)\) a projection of trace \(a<\frac12\). Compute \(A\) and \(e\), and show that \(e=\lambda_K(t_1)\,q\,\lambda_K(t_1)^*\).

7.5 (medium). Let \(\Lambda=\mathbb Z\subseteq K=\mathbb Z^2\). For \(x=\sum_na_n\lambda(n)\in\mathbb C\mathbb Z\), show directly that \(\lambda_K(x)\) on \(\ell^2(\mathbb Z^2)\) is a direct sum of copies of \(\lambda_{\mathbb Z}(x)\), indexed by the second coordinate.

8. Solutions

7.1. \(\sqrt{1-a}>\sqrt a\) if and only if \(1-a>a\). For \(a=\frac14\), \(c_a=\frac{\sqrt3-1}2\) and \(c_a^2=1-\frac{\sqrt3}2\).

7.2. \(U=\lambda_K(t_1)\) is unitary, so \(A=pU^*Up=p\), the unit of the corner, and \(A^{-1}=p\). Then \(e=UpU^*\).

7.3. \(ab\) has reduced word \((a,b)\), whose first and last syllables lie in different factors, so \((ab)^k\) has reduced word \((a,b,\ldots,a,b)\) of length \(2k\), and is not \(1\). The factors \(\mathbb Z/2\) are not torsion-free; \(a\) is a nontrivial factor element of finite order.

7.4. \(U^*U\) has entries \(\lambda(t_i)^*\lambda(t_j)\), so \(pU^*Up=\operatorname{diag}(q\lambda(t_1)^*\lambda(t_1)q,0)\), which is \(\operatorname{diag}(q,0)=p\). Hence \(A^{-1}=p\) and \(e=UpU^*=\lambda(t_1)q\lambda(t_1)^*\). Here the second generator plays no role. In general each coordinate of \(p\) is moved by its own generator, and (4.1) says that the moved pieces stay almost orthogonal.

7.5. For fixed \(m\), the subspace spanned by \(\delta_{(n,m)}\), \(n\in\mathbb Z\), is invariant under \(\lambda_K(k,0)\), which maps \(\delta_{(n,m)}\) to \(\delta_{(n+k,m)}\). Identifying it with \(\ell^2(\mathbb Z)\) by \(\delta_{(n,m)}\mapsto\delta_n\), \(\lambda_K(x)\) acts as \(\lambda_{\mathbb Z}(x)\). These subspaces are the cosets \(\mathbb Z\times\{m\}\), and they span \(\ell^2(\mathbb Z^2)\) orthogonally.

References