Corners of free group factors

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

This lesson computes the corner of a free group factor cut down by a projection of trace \(\frac12\), and deduces the main theorem of [OAI].

Theorem 2.1 (Voiculescu). Let \(n\ge2\), \(Q=L(\mathbb F_n)\) with its trace \(\tau\), and \(p\in Q\) a projection with \(\tau(p)=\frac12\). Then \(pQp\), with the trace \(2\tau\), is isomorphic to \(L(\mathbb F_{4n-3})\) by a trace-preserving normal \(*\)-isomorphism.

Theorem 3.1 (OpenAI). \(L(\mathbb F_2)\) and \(L(\mathbb F_3)\) are isomorphic by a trace-preserving normal \(*\)-isomorphism. Consequently \(L(\mathbb F_m)\cong L(\mathbb F_n)\) for all integers \(m,n\ge2\).

The proof of Theorem 2.1 realizes \(L(\mathbb F_n)\) inside \(2\times2\) matrices over a semicircular algebra, as the von Neumann algebra generated by \(n-1\) semicircular matrices and one diagonal Haar unitary (Step 1). The diagonal unitary provides the projection \(p=E_{11}\otimes1\), and the polar part of the circular entry of the first matrix provides a partial isometry from \(1-p\) to \(p\) (Step 2). Moving the generators into the corner gives \(4n-3\) elements (Step 3), which are free by the polar decomposition theorem of the previous lesson and the lemma on conjugation by a free Haar unitary (Step 4). Theorem 3.1 then follows from Theorem 2.1 for \(n=2,3\) and the rank step of Absorbing a free generator. [OAI] proves Theorem 3.1 with Dykema's amplification formula for interpolated free group factors instead of the corner computation.

We use: Lemmas 1.1-1.3 of Semicircular elements on the full Fock space and its Lemma 3.1; Theorem 2.1, Corollary 2.2 and Proposition 3.1 of Matrices with semicircular and circular entries; Lemmas 2.1, 2.3 and Corollary 3.2 of Polar decomposition of a circular element; Corollary 3.2 of Absorbing a free generator; equivalence of projections of equal trace in a II₁ factor, Lemma 2.3 of Haar unitaries, freeness and tracial ultraproducts; the double commutant theorem, Theorem 4.4 of its lesson, and the polar decomposition inside a von Neumann algebra, Lemma 7.1 and Proposition 7.2 there.

1. Corners

Lemma 1.1 (generators of a corner). Let \(N\) be a von Neumann algebra generated by a self-adjoint set \(G\), \(p\in N\) a projection, and \(W\in N\) a partial isometry with \(W^*W=1-p\) and \(WW^*=p\). Then the von Neumann algebra \(pNp\), with unit \(p\), is generated by the elements \(ayb\) with \(y\in G\), \(a\in\{p,W\}\) and \(b\in\{p,W^*\}\).

Proof. Since \(W=WW^*W=pW\) and \(W^*=W^*p\), every \(ayb\) lies in \(pNp\). Let \(\mathcal A\) be the unital \(*\)-algebra generated by \(G\); it is \(\sigma\)-weakly dense in \(N\) (Theorem 4.4 of the double commutant lesson), and \(x\mapsto pxp\) is \(\sigma\)-weakly continuous, so \(p\mathcal Ap\) is \(\sigma\)-weakly dense in \(pNp\). The space \(p\mathcal Ap\) is spanned by \(p\) and the products \(py_1y_2\cdots y_kp\) with \(y_i\in G\). Insert \(1=p+W^*W\) between consecutive factors and expand: every term is a product of elements \(a_iy_ib_i\) with \(a_1=p\), \(b_k=p\), and \((b_i,a_{i+1})\) equal to \((p,p)\) or \((W^*,W)\). So \(p\mathcal Ap\) lies in the algebra generated by the \(ayb\), and the von Neumann algebra they generate is \(pNp\). \(\square\)

Lemma 1.2 (corners of trace one half). Let \(R\) be a II₁ factor with trace \(\tau\).

  1. If \(p,q\in R\) are projections with \(\tau(p)=\tau(q)\), then \(x\mapsto vxv^*\), for any \(v\in R\) with \(v^*v=p\) and \(vv^*=q\), is a trace-preserving \(*\)-isomorphism \(pRp\to qRq\).
  2. If \(\tau(p)=\frac12\), then \(R\cong M_2(\mathbb C)\otimes pRp\) by a \(*\)-isomorphism carrying \(\tau\) to \(\operatorname{tr}_2\otimes2\tau|_{pRp}\).

Proof. (1) Such \(v\) exists by Lemma 2.3 of the lesson on Haar unitaries. The inverse map is \(y\mapsto v^*yv\), and \(\tau(vxv^*)=\tau(xv^*v)=\tau(x)\). (2) Take \(v\) with \(v^*v=p\) and \(vv^*=1-p\), and put \(e_{11}=p\), \(e_{21}=v\), \(e_{12}=v^*\), \(e_{22}=1-p\), a system of matrix units with \(e_{11}+e_{22}=1\). The map \(\sum_{i,j}E_{ij}\otimes x_{ij}\mapsto\sum_{i,j}e_{i1}x_{ij}e_{1j}\) (\(x_{ij}\in pRp\)) is a \(*\)-isomorphism onto \(R\), with inverse \(x\mapsto\sum_{i,j}E_{ij}\otimes e_{1i}xe_{j1}\). It carries \(\tau\) to \(\sum_i\tau(x_{ii})=\operatorname{tr}_2\otimes2\tau\), because \(\tau(e_{i1}x_{ii}e_{1i})=\tau(x_{ii})\). \(\square\)

2. The corner theorem

Proof of Theorem 2.1. Step 1: a model. Take a Hilbert space with orthonormal basis \(e_d\) and \(e_{j}^l\) (\(1\le l\le n-1\), \(1\le j\le4\)), \(4n-3\) vectors in all, and let \(\mathcal S\) be its semicircular algebra with trace \(\varphi\), \(s_d=s(e_d)\), \(s^l_j=s(e^l_j)\), and \(J=\{d\}\). Let \(X_l\) be the semicircular matrix of the quadruple \((e^l_1,e^l_2,e^l_3,e^l_4)\), with diagonal entries \(\frac1{\sqrt2}s^l_1\), \(\frac1{\sqrt2}s^l_2\) and circular entry \(\frac1{\sqrt2}\gamma_l\), and let \(\gamma_l=v_l|\gamma_l|\) be the polar decomposition, with \(v_l\) unitary (Proposition 3.1 of the lesson on matrices). Let \(u=e^{2\pi iF_\sigma(s_d)}\in\mathcal S_J\), a Haar unitary (Lemma 1.2 of the lesson on the Fock space), \(\theta(e^{2\pi it})=t\) for \(t\in[0,1)\), \(a=\theta(u)\), and \[ \begin{gathered} h=\begin{pmatrix}e^{\pi ia}&0\\0&-e^{\pi ia}\end{pmatrix}\in M_2(\mathcal S_J),\\ N=W^*(X_1,\ldots,X_{n-1},h)\subseteq M_2(\mathcal S), \end{gathered} \] with the trace \(\psi=\operatorname{tr}_2\otimes\varphi\). The distribution of \(a\) is the uniform distribution on \([0,1]\), and \(h^k=\operatorname{diag}(e^{\pi ika},(-1)^ke^{\pi ika})\), so \[ \begin{gathered} \psi(h^k)=\tfrac12\big(1+(-1)^k\big)\int_0^1e^{\pi ikt}\,dt=0\\ (k\ne0): \end{gathered} \] \(h\) is a Haar unitary. By Corollary 2.2 of the lesson on matrices, \(W^*(X_1),\ldots,W^*(X_{n-1})\) and \(M_2(\mathcal S_J)\ni h\) form a free family. Each \(X_l\) is semicircular, so \(W^*(X_l)\) is generated by a Haar unitary (Lemma 1.2 of the lesson on the Fock space). By Lemma 1.3 there, \(N\cong L(\mathbb F_n)\), and \(N\) is a II₁ factor. By Lemma 1.2(1), it suffices to prove the theorem for \(N\) and one projection of trace \(\frac12\) in \(N\).

Step 2: the projection and the partial isometry. Let \(S=\{e^{it}:0<t<\pi\}\) and \(p=1_S(h)\in N\). By Lemma 1.1 of the lesson on the Fock space, applied to \(h=e^{\pi ia}\oplus(-e^{\pi ia})\) on \(\mathcal F\oplus\mathcal F\), \(p=1_{(0,1)}(a)\oplus0\), because \(e^{\pi it}\in S\) exactly for \(t\in(0,1)\) and \(-e^{\pi it}\notin S\) for \(t\in[0,1)\). Since \(a\) has no atoms, \(1_{(0,1)}(a)=1\), so \(p=E_{11}\otimes1\), of trace \(\frac12\).

The element \(pX_1(1-p)=\frac1{\sqrt2}E_{12}\otimes\gamma_1\) lies in \(N\). It equals \(\frac1{\sqrt2}(E_{12}\otimes v_1)(E_{22}\otimes|\gamma_1|)\), where \(E_{22}\otimes|\gamma_1|\ge0\) and \(E_{12}\otimes v_1\) is a partial isometry whose kernel, the range of \(E_{11}\otimes1\), is the kernel of \(E_{22}\otimes|\gamma_1|\) (the kernel projection of \(|\gamma_1|\) is \(0\)). By Lemma 7.1 of the double commutant lesson this is the polar decomposition, and by Proposition 7.2 there \(W=E_{12}\otimes v_1\in N\). Also \(W^*W=E_{22}\otimes1=1-p\) and \(WW^*=p\).

Step 3: generators of the corner. Apply Lemma 1.1 with \(G=\{X_1,\ldots,X_{n-1},h,h^*\}\). For \(T\in M_2(\mathcal S)\), \[ \begin{aligned} &pTp=E_{11}\otimes T_{11},\quad pTW^*\\ &\quad =E_{11}\otimes T_{12}v_1^*,\quad WTp\\ &\quad =E_{11}\otimes v_1T_{21},\quad WTW^*\\ &\quad =E_{11}\otimes v_1T_{22}v_1^*. \end{aligned} \] For \(T=X_l\) these are, up to the factor \(\frac1{\sqrt2}\), \(E_{11}\otimes s^l_1\), \(E_{11}\otimes\gamma_lv_1^*\), its adjoint, and \(E_{11}\otimes v_1s^l_2v_1^*\). For \(T=h\) they are \(E_{11}\otimes e^{\pi ia}\), \(0\), \(0\) and \(E_{11}\otimes v_1(-e^{\pi ia})v_1^*\), and similarly for \(h^*\). Hence \(pNp=E_{11}\otimes\mathcal C\), where \(\mathcal C\subseteq\mathcal S\) is the von Neumann algebra generated by \(s^l_1\), \(v_1s^l_2v_1^*\) and \(\gamma_lv_1^*\) for \(1\le l\le n-1\), together with \(e^{\pi ia}\) and \(v_1e^{\pi ia}v_1^*\), and \(c\mapsto E_{11}\otimes c\) carries \(\varphi\) on \(\mathcal C\) to \(2\psi\) on \(pNp\).

Simplify the generators. \(e^{\pi ia}\in W^*(u)\), and \((e^{\pi ia})^2=e^{2\pi ia}=u\) by Lemma 1.1 of the lesson on the Fock space, so \(W^*(e^{\pi ia})=W^*(u)\) and \(W^*(v_1e^{\pi ia}v_1^*)=v_1W^*(u)v_1^*\). Next \(\gamma_1v_1^*=v_1|\gamma_1|v_1^*\). For \(l\ge2\), \(\gamma_lv_1^*=(v_lv_1^*)(v_1|\gamma_l|v_1^*)\) is the product of a unitary and a positive element with zero kernel projection, hence its polar decomposition (Lemma 7.1 of the double commutant lesson); by Proposition 7.2 there, \(w_l=v_lv_1^*\) and \(v_1|\gamma_l|v_1^*\) lie in \(W^*(\gamma_lv_1^*)\), and they generate it. Therefore \(\mathcal C\) is generated by \(4n-3\) elements: the \(2n-2\) elements \(u\), \(s^l_1\) (\(l\le n-1\)) and \(w_l\) (\(2\le l\le n-1\)), and the \(2n-1\) elements \(v_1uv_1^*\), \(v_1s^l_2v_1^*\) (\(l\le n-1\)) and \(v_1|\gamma_l|v_1^*\) (\(l\le n-1\)).

Step 4: freeness. By Corollary 3.2 of the lesson on the polar decomposition, the von Neumann algebras \(\mathcal S_J\ni u\), \(W^*(s^l_1)\), \(W^*(s^l_2)\), \(W^*(v_l)\), \(W^*(|\gamma_l|)\) (\(l\le n-1\)) form a free family, and the \(v_l\) are Haar unitaries. By Lemma 2.3 there, \(v_1,w_2,\ldots,w_{n-1}\) are Haar unitaries with free von Neumann algebras generating \(W^*(v_1,\ldots,v_{n-1})\); by associativity (Lemma 3.1 of the lesson on the Fock space) the family consisting of \(\mathcal S_J\), the algebras \(W^*(s^l_1)\), \(W^*(s^l_2)\) and \(W^*(|\gamma_l|)\) for \(l\le n-1\), the algebras \(W^*(w_l)\) for \(2\le l\le n-1\), and \(W^*(v_1)\) is free. Let \(\mathcal A\) be the von Neumann algebra generated by all of them except \(W^*(v_1)\); then \(W^*(v_1)\) and \(\mathcal A\) are free (Lemma 1.2(2) of the first lesson). Put \[ \begin{gathered} P=W^*\big(u,\;s^l_1,\;w_l\big),\\ R=W^*\big(u,\;s^l_2,\;|\gamma_l|\big), \end{gathered} \] both contained in \(\mathcal A\). By Lemma 2.1 of the lesson on the polar decomposition, \(P\) and \(v_1Rv_1^*\) are free. Inside \(P\), the algebras \(W^*(u)\subseteq\mathcal S_J\), \(W^*(s^l_1)\), \(W^*(w_l)\) are free, being members of the free family above; inside \(R\), the algebras \(W^*(u)\), \(W^*(s^l_2)\), \(W^*(|\gamma_l|)\) are free, and the trace-preserving automorphism \(\operatorname{Ad}v_1\) of \(\mathcal S\) keeps them free after conjugation. By associativity, the von Neumann algebras of the \(4n-3\) generators of \(\mathcal C\) form a free family.

Step 5: Haar generators. The unitaries \(u\), \(w_l\) and \(v_1uv_1^*\) are Haar unitaries. The elements \(s^l_1\) and \(v_1s^l_2v_1^*\) are semicircular, and \(v_1|\gamma_l|v_1^*\) has the distribution of \(|\gamma_l|\), whose distribution function is continuous and strictly increasing on \([0,2]\) (Proposition 3.1 of the lesson on matrices). By Lemma 1.2 of the lesson on the Fock space, each of their von Neumann algebras is generated by a Haar unitary. By Lemma 1.3 there, \((\mathcal C,\varphi)\cong L(\mathbb F_{4n-3})\), and so \((pNp,2\psi)\cong L(\mathbb F_{4n-3})\). \(\square\)

3. The isomorphism of the free group factors

Proof of Theorem 3.1. Let \(p_2\in L(\mathbb F_2)\) and \(p_3\in L(\mathbb F_3)\) be projections of trace \(\frac12\). By Theorem 2.1, \(p_2L(\mathbb F_2)p_2\cong L(\mathbb F_5)\) and \(p_3L(\mathbb F_3)p_3\cong L(\mathbb F_9)\), with the normalized traces. By Corollary 3.2 of the lesson on absorbing a free generator, \(L(\mathbb F_5)\cong L(\mathbb F_9)\). So the two corners are isomorphic by a trace-preserving isomorphism \(\theta\), and by Lemma 1.2(2), \[ \begin{aligned} &L(\mathbb F_2)\cong M_2(\mathbb C)\otimes p_2L(\mathbb F_2)p_2\\ &\quad \xrightarrow{\ \mathrm{id}\otimes\theta\ }M_2(\mathbb C)\otimes p_3L(\mathbb F_3)p_3\cong L(\mathbb F_3), \end{aligned} \] all maps preserving the traces. For \(m,n\ge3\) the second statement is Corollary 3.2 of the lesson on absorbing a free generator, and the case \(m=2\) follows by composing. \(\square\)

Corollary 3.2. For a projection \(p\in L(\mathbb F_2)\) of trace \(\frac12\), \(pL(\mathbb F_2)p\cong L(\mathbb F_2)\) and \(M_2(\mathbb C)\otimes L(\mathbb F_2)\cong L(\mathbb F_2)\), by trace-preserving isomorphisms for the normalized traces. In terms of the fundamental group of Murray and von Neumann, \(\frac12\) and \(2\) belong to the fundamental group of \(L(\mathbb F_2)\).

Proof. \(pL(\mathbb F_2)p\cong L(\mathbb F_5)\cong L(\mathbb F_2)\) by Theorems 2.1 and 3.1. Then Lemma 1.2(2) gives \(L(\mathbb F_2)\cong M_2(\mathbb C)\otimes pL(\mathbb F_2)p\), which is isomorphic to \(M_2(\mathbb C)\otimes L(\mathbb F_2)\). \(\square\)

4. Further results

The next lesson proves Voiculescu's corner formula for every integer \(k\ge2\), \(pL(\mathbb F_n)p\cong L(\mathbb F_{1+k^2(n-1)})\) for a projection of trace \(\frac1k\), with \(k\times k\) semicircular matrices, and deduces with Theorem 3.1 that every positive rational number lies in the fundamental group of \(L(\mathbb F_2)\). The interpolated free group factors \(L(\mathbb F_r)\), \(r>1\) real, the factor \(L(\mathbb F_\infty)\), and the conclusion of [OAI, Corollary 6.1] that all of them are isomorphic with fundamental group \(\mathbb R_{>0}\), rest on the theory of Dykema [D] and Rădulescu, which this course does not develop. Nor does it treat the consequences for Voiculescu's free entropy dimensions in [OAI, Section 7].

Shlyakhtenko gave a self-contained proof of a C*-algebraic form of the theorem [Sh 2026]. His argument is organized around an automorphism approximation property of an inclusion \(N\subseteq M\) of tracial von Neumann algebras: trace-preserving automorphisms of \(M\) move any given element of \(M\) arbitrarily close to \(N\) in \(\|\cdot\|_2\) while moving a given finite subset of \(N\) arbitrarily little; a Baire category argument turns this into trace-preserving isomorphisms \(N\to M\) that are arbitrarily close to the identity on finite subsets of \(N\) [Sh 2026, Proposition 1.1]. For \(n\in\{2,3,\ldots,\infty\}\) let \(Q_n\) be the C*-algebra generated by a free semicircular family of \(n\) elements; for finite \(n\) its weak closure in the trace representation is \(L(\mathbb F_n)\), by Theorem 4.2(3) of the lesson on semicircular elements on the full Fock space. Shlyakhtenko proves \(Q_n\cong Q_2\) for every such \(n\), and notes that this implies \(L(\mathbb F_n)\cong L(\mathbb F_2)\). The same paper proves \(C^*_{\mathrm{red}}(\mathbb F_m)*_{\mathrm{red}}Q_n\cong C^*_{\mathrm{red}}(\mathbb F_m)\) for \(m\in\{2,3,\ldots,\infty\}\) and \(n\in\{1,2,\ldots,\infty\}\), and that the \(n\)-fold reduced free product of the Jiang–Su algebra with its trace is isomorphic to \(Q_2\) for every \(n\in\{2,3,\ldots,\infty\}\) [Sh 2026, Section 5]. The reduced group C*-algebras \(C^*_{\mathrm{red}}(\mathbb F_n)\) themselves are pairwise non-isomorphic for different ranks, by a theorem of Pimsner and Voiculescu recalled in [Sh 2026, Section 1]. Since the free entropy dimensions considered there all take the value \(n\) on a free semicircular family of \(n\) elements, none of them is an invariant of the generated C*-algebra [Sh 2026, Section 1].

5. Exercises

Exercise 5.1. Show that applying Theorem 2.1 twice gives \(pL(\mathbb F_n)p\cong L(\mathbb F_{16n-15})\) for a projection \(p\) of trace \(\frac14\), in agreement with the formula of Section 4 for \(k=4\).

Exercise 5.2. Write out the nine generators of the corner in Theorem 2.1 for \(n=3\).

Exercise 5.3. In Step 2, explain why the partial isometry \(W\) would not connect \(1-p\) to \(p\) if the circular entry \(\gamma_1\) had a nonzero kernel projection.

Exercise 5.4. Show that the projection \(p=E_{11}\otimes1\) does not belong to \(W^*(X_1,\ldots,X_{n-1})\). (Hint: compare the distribution of a semicircular matrix with that of an element commuting with \(p\).)

6. Solutions

5.1. Choose projections \(p\le q\) with \(\tau(q)=\frac12\) and \(\tau(p)=\frac14\). Then \(pQp=p(qQq)p\), where \(qQq\cong L(\mathbb F_{4n-3})\) and \(p\) has trace \(\frac12\) in \(qQq\) for the normalized trace. By Theorem 2.1 again, \(pQp\cong L(\mathbb F_{4(4n-3)-3})=L(\mathbb F_{16n-15})\), and \(1+16(n-1)=16n-15\).

5.2. \(u\), \(s^1_1\), \(s^2_1\), \(w_2=v_2v_1^*\); \(v_1uv_1^*\), \(v_1s^1_2v_1^*\), \(v_1s^2_2v_1^*\), \(v_1|\gamma_1|v_1^*\), \(v_1|\gamma_2|v_1^*\).

5.3. The polar part of \(E_{12}\otimes\gamma_1\) is \(E_{12}\otimes v_1\) with \(v_1^*v_1\) the complement of the kernel projection of \(\gamma_1\); then \(W^*W=E_{22}\otimes v_1^*v_1\) would be strictly smaller than \(1-p\), and the insertion \(1=p+W^*W\) in Lemma 1.1 would fail.

5.4. \(X_1\) does not commute with \(p\): \(pX_1(1-p)=\frac1{\sqrt2}E_{12}\otimes\gamma_1\ne0\). Suppose \(p\in W^*(X_1,\ldots,X_{n-1})\). The algebra \(W^*(X_1,\ldots,X_{n-1})\) is free from \(M_2(\mathcal S_J)\ni p\) by Corollary 2.2 of the lesson on matrices, so \(p\) would be free from itself; but \(\psi(p^\circ p^\circ)=\frac14\ne0\) for the centred \(p^\circ=p-\frac12\).

References