Corners of rational trace

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

The previous lesson computed the corner of \(L(\mathbb F_n)\) cut down by a projection of trace \(\frac12\). This lesson treats every trace \(\frac1k\), \(k\ge2\).

Theorem 3.1 (Voiculescu). Let \(n\ge2\) and \(k\ge2\) be integers, and \(p\in L(\mathbb F_n)\) a projection with \(\tau(p)=\frac1k\). Then \(pL(\mathbb F_n)p\), with the trace \(k\tau\), is isomorphic to \(L(\mathbb F_{1+k^2(n-1)})\) by a trace-preserving normal \(*\)-isomorphism.

With the isomorphism theorem of the previous lesson, this gives Corollary 4.1: every corner of \(L(\mathbb F_2)\) by a projection of rational trace, and every matrix algebra \(M_m(\mathbb C)\otimes L(\mathbb F_2)\), is isomorphic to \(L(\mathbb F_2)\). So every positive rational number lies in the fundamental group of \(L(\mathbb F_2)\).

The proof follows the previous lesson with \(k\times k\) matrices. \(L(\mathbb F_n)\) is realized in \(M_k\) of a semicircular algebra, generated by \(n-1\) semicircular matrices of size \(k\) and one diagonal Haar unitary whose spectral projections are the diagonal matrix units. Section 1 proves that a semicircular matrix of size \(k\) is free from all \(k\times k\) matrices over the other variables. Section 2 provides corners with \(k\) matrix units and freeness under conjugation by several free Haar unitaries. Section 3 proves the theorem, and Section 4 draws the consequences.

We use: Lemmas 1.1-1.3, 3.1 and 4.1, Theorems 3.2 and 4.2, and the distribution of a Haar unitary (Section 1) from Semicircular elements on the full Fock space; Proposition 3.1 of Matrices with semicircular and circular entries; Lemma 2.1 and Theorem 3.1 of Polar decomposition of a circular element; Lemma 1.2(1) and Theorem 3.1 of Corners of free group factors; Lemmas 1.2 and 2.1 and Corollary 2.5 of Free independence and Haar tuples; equivalence of projections of equal trace in a II₁ factor and the existence of Haar unitaries in its corners, Lemma 2.3 and Corollary 2.4 of Haar unitaries, freeness and tracial ultraproducts; measures determined by continuous functions, Corollary 1.2 of the spectral theorem lesson; 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; and the Weierstrass approximation theorem, Corollary 14.3 of the Stone–Weierstrass lesson.

1. Semicircular matrices of size k

As before, \(e_1,\ldots,e_m\) is an orthonormal basis of a Hilbert space \(H\), \(s_j=s(e_j)\) on the full Fock space \(\mathcal F(H)\), \(\mathcal S=W^*(s_1,\ldots,s_m)\) with the vacuum trace \(\varphi\), and \(\mathcal S_J=W^*(s_j:j\in J)\). Fix \(k\ge2\), and let \(E_{pq}\) (\(1\le p,q\le k\)) be the matrix units of \(M_k(\mathbb C)\). The algebra \(M_k(\mathcal S)\) acts on \(\mathbb C^k\otimes\mathcal F(H)\), with the faithful normal tracial state \[ \psi(T)=\frac1k\sum_{p=1}^k\varphi(T_{pp}). \] It is a vector state: let \(T\) act as \(T\otimes1\) on \(K=\mathbb C^k\otimes\mathcal F(H)\otimes\mathbb C^k\), and put \(\xi=k^{-1/2}\sum_p\varepsilon_p\otimes\Omega\otimes\varepsilon_p\), where \((\varepsilon_p)\) is the standard basis of \(\mathbb C^k\). Then \(\langle(T\otimes1)\xi,\xi\rangle=\psi(T)\).

A block (of size \(k\)) is a family of \(k^2\) distinct indices, \(j_{pp}\) for \(1\le p\le k\) and \(j'_{pq},j''_{pq}\) for \(1\le p<q\le k\). Its circular entries are \(\gamma_{pq}=\frac1{\sqrt2}(s_{j'_{pq}}+is_{j''_{pq}})\) for \(p<q\), and its semicircular matrix is \[ \begin{aligned} &X=\frac1{\sqrt k}\big(x_{pq}\big)_{p,q=1}^k,\\ &x_{pp}=s_{j_{pp}},\quad x_{pq}=\gamma_{pq},\quad x_{qp}\\ &\quad =\gamma_{pq}^*\quad(p<q). \end{aligned} \] For \(k=2\) this is the semicircular matrix of a quadruple from the lesson on matrices.

Theorem 1.1. Let \(X\) be the semicircular matrix of a block and \(J\) a set of indices disjoint from the block. Then \(X\) is semicircular with respect to \(\psi\), and \(W^*(X)\) and \(M_k(\mathcal S_J)\) are free in \((M_k(\mathcal S),\psi)\).

Proof. Put \(g_{pp}=e_{j_{pp}}\), and for \(p<q\), \(g_{pq}=\frac1{\sqrt2}(e_{j'_{pq}}+ie_{j''_{pq}})\) and \(g_{qp}=\frac1{\sqrt2}(e_{j'_{pq}}-ie_{j''_{pq}})\). These \(k^2\) vectors are orthonormal and orthogonal to \(K_0=\operatorname{span}\{e_j:j\in J\}\). Since \(\ell\) is linear and \(\ell(g)^*\) conjugate-linear in \(g\), \[ \begin{gathered} s_{j_{pp}}=\ell(g_{pp})+\ell(g_{pp})^*,\\ \gamma_{pq}=\ell(g_{pq})+\ell(g_{qp})^*\quad(p<q), \end{gathered} \] and \(\gamma_{pq}^*=\ell(g_{qp})+\ell(g_{pq})^*\). Let \[ L=\frac1{\sqrt k}\sum_{p,q=1}^kE_{pq}\otimes\ell(g_{pq})\otimes1 . \] Its adjoint is \(k^{-1/2}\sum_{p,q}E_{qp}\otimes\ell(g_{pq})^*\otimes1\), so the \((p,q)\) entry of \(L+L^*\) is \(k^{-1/2}(\ell(g_{pq})+\ell(g_{qp})^*)\), and \(L+L^*=X\otimes1\). We check the conditions of Theorem 3.2 of the lesson on the Fock space for \(\mathcal D=M_k(\mathcal S_J)\otimes1\) and \(\xi\).

(a) Let \(d=\sum_{a,b}E_{ab}\otimes b_{ab}\) with \(b_{ab}\in\mathcal S_J\). By Lemma 4.1(2) of the lesson on the Fock space, \(\ell(g_{pq})^*b_{ab}\ell(g_{p'q'})=\varphi(b_{ab})\delta_{(p,q),(p',q')}1\). Hence \[ \begin{aligned} &L^*(d\otimes1)L\\ &\quad =\frac1k\sum_{p,q,a,b}\varphi(b_{ab})\,E_{qp}E_{ab}E_{pq}\otimes1\otimes1\\ &\quad =\frac1k\sum_{p,q}\varphi(b_{pp})\,E_{qq}\otimes1\otimes1=\psi(d)1 . \end{aligned} \] (b) \(L^*\xi=0\), since \(\ell(g)^*\Omega=0\).

(c) \((d\otimes1)\xi=k^{-1/2}\sum_{a,p}\varepsilon_a\otimes b_{ap}\Omega\otimes\varepsilon_p\), and every \(\ell(g_{pq})^*\) annihilates \(\mathcal F(K_0)\ni b_{ap}\Omega\) (Lemma 4.1(1) there). So \(L^*(d\otimes1)\xi=0\).

Theorem 3.2 there gives \(\psi(X^r)=\int t^r\,d\sigma\) for all \(r\), and freeness of the polynomials in \(X\) from \(M_k(\mathcal S_J)\). With \(d=1\), (a) says \(L^*L=1\), so \(\|X\|\le2\), and the distribution of \(X\) equals \(\sigma\), as both are measures on \([-2,2]\) with the same moments (Weierstrass and Corollary 1.2 of the spectral theorem lesson). Freeness passes to \(W^*(X)\) by Lemma 1.2(1) of the first lesson. \(\square\)

Corollary 1.2. Let \(X_1,\ldots,X_r\) be the semicircular matrices of pairwise disjoint blocks, \(\gamma^l_{pq}\) the circular entries of \(X_l\), and \(J\) a set of indices disjoint from the blocks.

  1. \(W^*(X_1),\ldots,W^*(X_r)\) and \(M_k(\mathcal S_J)\) form a free family in \((M_k(\mathcal S),\psi)\).
  2. Let \(\gamma^l_{pq}=v^l_{pq}|\gamma^l_{pq}|\) be the polar decompositions; the \(v^l_{pq}\) are unitaries (Proposition 3.1 of the lesson on matrices). The von Neumann algebras \[ \begin{gathered} \mathcal S_J,\qquad W^*(x^l_{pp}),\\ W^*(v^l_{pq}),\qquad W^*(|\gamma^l_{pq}|)\\ (1\le l\le r,\ 1\le p\le k,\ p<q\le k) \end{gathered} \] form a free family in \((\mathcal S,\varphi)\), and the \(v^l_{pq}\) are Haar unitaries.

Proof. (1) By induction on \(r\), as in Corollary 2.2 of the lesson on matrices: \(W^*(X_r)\) is free from \(M_k(\mathcal S_{J'})\), where \(J'\) is \(J\) together with the other blocks (Theorem 1.1); that algebra contains the other members, which form a free family by induction; apply associativity (Lemma 3.1 of the lesson on the Fock space).

(2) The algebras \(\mathcal S_J\), \(\mathcal S_{\{j^l_{pp}\}}=W^*(x^l_{pp})\) and \(\mathcal S_{\{j'^l_{pq},j''^l_{pq}\}}=W^*(\gamma^l_{pq})\) are attached to disjoint sets of indices and form a free family (Theorem 4.2 of the lesson on the Fock space and Lemma 1.2 of the first lesson). Inside each \(W^*(\gamma^l_{pq})\), Theorem 3.1 of the lesson on the polar decomposition says that \(v^l_{pq}\) is a Haar unitary and that \(W^*(v^l_{pq})\) and \(W^*(|\gamma^l_{pq}|)\) are free. Associativity gives the claim. \(\square\)

2. Matrix units and several free Haar unitaries

Lemma 2.1 (generators of a corner). Let \(N\) be a von Neumann algebra generated by a self-adjoint set \(G\), let \(P_1,\ldots,P_k\in N\) be mutually orthogonal projections with sum \(1\), and let \(W_1=P_1,W_2,\ldots,W_k\in N\) be partial isometries with \(W_q^*W_q=P_q\) and \(W_qW_q^*=P_1\). Then the von Neumann algebra \(P_1NP_1\), with unit \(P_1\), is generated by the elements \(W_pyW_q^*\) with \(y\in G\) and \(1\le p,q\le k\).

Proof. \(W_p=W_pW_p^*W_p=P_1W_p\) and \(W_q^*=W_q^*P_1\), so each \(W_pyW_q^*\) lies in \(P_1NP_1\). The unital \(*\)-algebra \(\mathcal A\) generated by \(G\) is \(\sigma\)-weakly dense in \(N\) (Theorem 4.4 of the double commutant lesson), so \(P_1\mathcal AP_1\) is \(\sigma\)-weakly dense in \(P_1NP_1\). It is spanned by \(P_1\) and the products \(P_1y_1y_2\cdots y_mP_1\) with \(y_i\in G\). Insert \(1=\sum_qW_q^*W_q\) between consecutive factors and expand: each term is a product \((W_{p_1}y_1W_{q_1}^*)\cdots(W_{p_m}y_mW_{q_m}^*)\) with \(p_1=q_m=1\) and \(q_i=p_{i+1}\). \(\square\)

Lemma 2.2 (matrix units). Let \(R\) be a II₁ factor with trace \(\tau\).

  1. If \(q_1,\ldots,q_m\in R\) are mutually orthogonal projections of equal trace with sum \(p\), then \(pRp\cong M_m(\mathbb C)\otimes q_1Rq_1\) by a \(*\)-isomorphism carrying \(\frac1{\tau(p)}\tau\) to \(\operatorname{tr}_m\otimes\frac1{\tau(q_1)}\tau\).
  2. Every nonzero projection \(p\in R\) is the sum of \(m\) mutually orthogonal projections of trace \(\tau(p)/m\), for every \(m\ge1\).

Proof. (1) By Lemma 2.3 of the lesson on Haar unitaries there are \(w_i\in R\) with \(w_i^*w_i=q_1\) and \(w_iw_i^*=q_i\); then \(w_i=q_iw_iq_1\in pRp\). Put \(e_{ij}=w_iw_j^*\): these are matrix units in \(pRp\) with \(\sum_ie_{ii}=p\) and \(e_{11}=q_1\). The map \(\sum_{i,j}E_{ij}\otimes x_{ij}\mapsto\sum_{i,j}e_{i1}x_{ij}e_{1j}\) (\(x_{ij}\in q_1Rq_1\)) is a \(*\)-homomorphism onto \(pRp\), with inverse \(x\mapsto\sum_{i,j}E_{ij}\otimes e_{1i}xe_{j1}\). Since \(\tau(e_{i1}x_{ii}e_{1i})=\tau(x_{ii})\), it carries \(\tau\) to \(\sum_i\tau(x_{ii})\), which is the claim after normalization.

(2) By Corollary 2.4 of the lesson on Haar unitaries, \(pRp\) has a Haar unitary \(h\) of the corner. Its distribution, for the state \(\tau(p)^{-1}\tau\) on \(pRp\), is the normalized arc length (Section 1 of the lesson on the Fock space). The spectral projections of \(h\) for the \(m\) arcs \(\{e^{2\pi it}:\frac{i-1}m\le t<\frac im\}\) are mutually orthogonal with sum \(p\) and have trace \(\tau(p)/m\). \(\square\)

Lemma 2.3 (conjugation by several free Haar unitaries). Let \(v_2,\ldots,v_k\in M\) be Haar unitaries and \(\mathcal A\subseteq M\) a von Neumann subalgebra such that \(\mathcal A,W^*(v_2),\ldots,W^*(v_k)\) form a free family. For von Neumann subalgebras \(B_1,R_2,\ldots,R_k\subseteq\mathcal A\), the algebras \(B_1,v_2R_2v_2^*,\ldots,v_kR_kv_k^*\) form a free family.

Proof. By induction on \(k\); for \(k=2\) this is Lemma 2.1 of the lesson on the polar decomposition. For \(k\ge3\), let \(\mathcal A_1\) be the von Neumann algebra generated by \(\mathcal A\) and \(v_2,\ldots,v_{k-1}\). By grouping (Lemma 1.2(2) of the first lesson), \(W^*(v_k)\) and \(\mathcal A_1\) are free. The algebra \(Q\) generated by \(B_1,v_2R_2v_2^*,\ldots,v_{k-1}R_{k-1}v_{k-1}^*\) and the algebra \(R_k\) lie in \(\mathcal A_1\), so \(Q\) and \(v_kR_kv_k^*\) are free by Lemma 2.1 there. By the induction hypothesis the generators of \(Q\) form a free family, and associativity gives the claim. \(\square\)

3. The corner theorem

Proof of Theorem 3.1. Step 1: a model. Take a Hilbert space with an orthonormal basis consisting of one vector \(e_d\) and \(n-1\) pairwise disjoint blocks of \(k^2\) vectors, \(1+k^2(n-1)\) vectors in all, and let \(\mathcal S\) be its semicircular algebra with trace \(\varphi\), \(J=\{d\}\). Let \(X_l\) be the semicircular matrix of the \(l\)-th block, with diagonal entries \(k^{-1/2}x^l_{pp}\) and circular entries \(k^{-1/2}\gamma^l_{pq}\) (\(p<q\)), and let \(\gamma^l_{pq}=v^l_{pq}|\gamma^l_{pq}|\) be the polar decompositions. 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 \(t\in[0,1)\), and \(a=\theta(u)\), which has the uniform distribution on \([0,1]\). With \(\omega=e^{2\pi i/k}\) put \[ \begin{aligned} &h=\operatorname{diag}\big(e^{2\pi ia/k},\,\omega e^{2\pi ia/k},\,\ldots,\\ &\qquad\qquad\omega^{k-1}e^{2\pi ia/k}\big)\in M_k(\mathcal S_J),\\ &N=W^*(X_1,\ldots,X_{n-1},h)\subseteq M_k(\mathcal S). \end{aligned} \] For \(m\ne0\), \(h^m=\operatorname{diag}(\omega^{(p-1)m}e^{2\pi ima/k})_p\), so \[ \begin{aligned} \psi(h^m)&=\Big(\frac1k\sum_{p=1}^k\omega^{(p-1)m}\Big)\int_0^1e^{2\pi imt/k}\,dt\\ &=0 : \end{aligned} \] the sum vanishes unless \(k\) divides \(m\), and then the integral vanishes. So \(h\) is a Haar unitary. By Corollary 1.2(1), \(W^*(X_1),\ldots,W^*(X_{n-1})\) and \(M_k(\mathcal S_J)\ni h\) 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, \(N\cong L(\mathbb F_n)\), a II₁ factor. By Lemma 1.2(1) of the previous lesson it suffices to prove the theorem for \(N\) and one projection of trace \(\frac1k\).

Step 2: matrix units in \(N\). For \(1\le p\le k\) let \(A_p=\{e^{2\pi it/k}:p-1<t<p\}\) and \(P_p=1_{A_p}(h)\in N\). By Lemma 1.1 of the lesson on the Fock space, applied to the direct sum \(h=\bigoplus_q\omega^{q-1}e^{2\pi ia/k}\), and since \(\omega^{q-1}e^{2\pi it/k}=e^{2\pi i(q-1+t)/k}\) lies in \(A_p\) exactly when \(q=p\) and \(0<t<1\) (for \(t\in[0,1]\)), we get \(P_p=E_{pp}\otimes1_{(0,1)}(a)=E_{pp}\otimes1\), because \(a\) has no atoms. So the \(P_p\) are mutually orthogonal projections in \(N\) with sum \(1\) and trace \(\frac1k\).

For \(2\le q\le k\), the element \(P_1X_1P_q=k^{-1/2}E_{1q}\otimes\gamma^1_{1q}\) lies in \(N\), and equals \(k^{-1/2}(E_{1q}\otimes v^1_{1q})(E_{qq}\otimes|\gamma^1_{1q}|)\). The second factor is positive, and its kernel, the range of \((1-E_{qq})\otimes1\), is the kernel of the partial isometry \(E_{1q}\otimes v^1_{1q}\), because the kernel projection of \(|\gamma^1_{1q}|\) is \(0\) (Proposition 3.1 of the lesson on matrices). By Lemma 7.1 of the double commutant lesson this is the polar decomposition, and by Proposition 7.2 there \(W_q=E_{1q}\otimes v^1_{1q}\in N\). Then \(W_q^*W_q=E_{qq}\otimes1=P_q\) and \(W_qW_q^*=P_1\). Put \(W_1=P_1\), and \(v_1=1\), \(v_q=v^1_{1q}\) for \(q\ge2\), so that \(W_q=E_{1q}\otimes v_q\) for all \(q\).

Step 3: generators of the corner. For \(T\in M_k(\mathcal S)\), \[ W_pTW_q^*=E_{11}\otimes v_pT_{pq}v_q^* . \] By Lemma 2.1 with \(G=\{X_1,\ldots,X_{n-1},h,h^*\}\), \(P_1NP_1=E_{11}\otimes\mathcal C\), where \(\mathcal C\subseteq\mathcal S\) is the von Neumann algebra generated by the elements \(v_pT_{pq}v_q^*\), \(T\in G\); and \(c\mapsto E_{11}\otimes c\) carries \(\varphi\) on \(\mathcal C\) to \(k\psi\) on \(P_1NP_1\).

We simplify the generators. For \(T=h\) the nonzero ones are \(\omega^{p-1}v_pe^{2\pi ia/k}v_p^*\) and their adjoints. Now \(e^{2\pi ia/k}\in W^*(u)\), and its \(k\)-th power is \(e^{2\pi ia}=u\) (Lemma 1.1 of the lesson on the Fock space), so \(W^*(e^{2\pi ia/k})=W^*(u)\). For \(T=X_l\), the diagonal positions give \(k^{-1/2}v_px^l_{pp}v_p^*\). For \(p<q\) the position \((p,q)\) gives \(k^{-1/2}v_p\gamma^l_{pq}v_q^*\), and the position \((q,p)\) its adjoint. If \((l,p)=(1,1)\), then \(v_1\gamma^1_{1q}v_q^*=v_q|\gamma^1_{1q}|v_q^*\) is positive. Otherwise \[ \begin{gathered} v_p\gamma^l_{pq}v_q^*=w^l_{pq}\,\big(v_q|\gamma^l_{pq}|v_q^*\big),\\ w^l_{pq}=v_pv^l_{pq}v_q^*, \end{gathered} \] the product of a unitary and a positive element with zero kernel projection; by Lemma 7.1 and Proposition 7.2 of the double commutant lesson both factors lie in \(W^*(v_p\gamma^l_{pq}v_q^*)\), and they generate it. Therefore \(\mathcal C\) is generated by the following elements:

Their number is \[ \begin{aligned} &k+k(n-1)+\frac{k(k-1)}2(n-1)\\ &\quad+\Big(\frac{k(k-1)}2(n-1)-(k-1)\Big)\\ &\quad=1+k^2(n-1). \end{aligned} \]

Step 4: freeness. By Corollary 1.2(2), the algebras \(\mathcal S_J\), \(W^*(x^l_{pp})\), \(W^*(|\gamma^l_{pq}|)\) and \(W^*(v^l_{pq})\) form a free family, and the \(v^l_{pq}\) are free Haar unitaries. As in the proof of Lemma 1.3 of the lesson on the Fock space, the tuple of all \(v^l_{pq}\) is a freely generating Haar tuple of \(\mathcal V=W^*(v^l_{pq}:\text{all }l,p<q)\). Index the free generators of the corresponding free group by the triples \((l,p,q)\), write \(y_1=e\) and \(y_q=x_{(1,1,q)}\) for \(q\ge2\), and let \(\vartheta\) fix the generators \(x_{(1,1,q)}\) and send \(x_{(l,p,q)}\) to \(y_px_{(l,p,q)}y_q^{-1}\) when \((l,p)\neq(1,1)\). It is an automorphism of the free group, with inverse \(x_{(l,p,q)}\mapsto y_p^{-1}x_{(l,p,q)}y_q\) (Lemma 2.1 of the first lesson). The new tuple consists of \(v_2,\ldots,v_k\) and the \(w^l_{pq}\) with \((l,p)\neq(1,1)\). By Corollary 2.5(2)-(3) of the first lesson, these are Haar unitaries whose von Neumann algebras form a free family generating \(\mathcal V\). By associativity, the family \[ \begin{gathered} \mathcal S_J,\qquad W^*(x^l_{pp}),\qquad W^*(|\gamma^l_{pq}|),\\ W^*(w^l_{pq})\ \ ((l,p)\neq(1,1)),\\ W^*(v_2),\ldots,W^*(v_k) \end{gathered} \] is free. Let \(\mathcal A\) be the von Neumann algebra generated by all members except \(W^*(v_2),\ldots,W^*(v_k)\). By grouping, \(\mathcal A,W^*(v_2),\ldots,W^*(v_k)\) form a free family. Put \(B_1=W^*\big(u,\ x^l_{11},\ w^l_{pq}\big)\) and, for \(2\le q\le k\), \(R_q=W^*\big(u,\ x^l_{qq},\ |\gamma^l_{pq}|\ (p<q)\big)\), all contained in \(\mathcal A\). By Lemma 2.3, \(B_1,v_2R_2v_2^*,\ldots,v_kR_kv_k^*\) form a free family. Inside \(B_1\), the algebras \(W^*(u)\subseteq\mathcal S_J\), \(W^*(x^l_{11})\) and \(W^*(w^l_{pq})\) are members of the free family above, hence free. Inside \(R_q\), the algebras \(W^*(u)\), \(W^*(x^l_{qq})\) and \(W^*(|\gamma^l_{pq}|)\) are free, and the trace-preserving automorphism \(\operatorname{Ad}v_q\) keeps them free. Each generator (i)-(iv) lies in exactly one of these: \(u\), \(x^l_{11}\) and the \(w^l_{pq}\) in \(B_1\), and \(v_quv_q^*\), \(v_qx^l_{qq}v_q^*\), \(v_q|\gamma^l_{pq}|v_q^*\) in \(v_qR_qv_q^*\). By associativity, the von Neumann algebras of the \(1+k^2(n-1)\) generators of \(\mathcal C\) form a free family.

Step 5: Haar generators. The unitaries \(v_puv_p^*\) and \(w^l_{pq}\) are Haar unitaries. The elements \(v_px^l_{pp}v_p^*\) are semicircular, and \(v_q|\gamma^l_{pq}|v_q^*\) has the distribution of \(|\gamma^l_{pq}|\), 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 generator's von Neumann algebra is generated by a Haar unitary, and by Lemma 1.3 there \((\mathcal C,\varphi)\cong L(\mathbb F_{1+k^2(n-1)})\). Hence \((P_1NP_1,k\psi)\cong L(\mathbb F_{1+k^2(n-1)})\). \(\square\)

For \(k=2\) the construction is the one of the previous lesson, up to the choice of \(h\), and \(1+4(n-1)=4n-3\).

4. Rational corners and the fundamental group

Corollary 4.1. Let \(R=L(\mathbb F_2)\) with trace \(\tau\).

  1. For every projection \(p\in R\) whose trace \(\tau(p)\) is a positive rational number, \(pRp\cong R\) by an isomorphism carrying \(\frac1{\tau(p)}\tau\) to \(\tau\).
  2. For every integer \(m\ge1\), \(M_m(\mathbb C)\otimes R\cong R\) by an isomorphism carrying \(\operatorname{tr}_m\otimes\tau\) to \(\tau\).
  3. For all integers \(m,k\ge1\) and a projection \(p\in R\) of trace \(\frac1k\), \(M_m(\mathbb C)\otimes pRp\cong R\).

The same holds for every \(L(\mathbb F_n)\), \(n\ge2\), since these are all isomorphic to \(R\) (Theorem 3.1 of the previous lesson).

Proof. A corner of trace \(\frac1k\). For \(k\ge2\) and a projection \(q\in R\) of trace \(\frac1k\), Theorem 3.1 gives \(qRq\cong L(\mathbb F_{1+k^2})\), and Theorem 3.1 of the previous lesson gives \(L(\mathbb F_{1+k^2})\cong R\); so \(qRq\cong R\). For \(k=1\), \(q=1\).

(2) Let \(m\ge2\). By Lemma 2.2(2), \(1\) is the sum of \(m\) mutually orthogonal projections of trace \(\frac1m\); let \(q\) be one of them. By Lemma 2.2(1) and the first step, \(R\cong M_m(\mathbb C)\otimes qRq\cong M_m(\mathbb C)\otimes R\).

(1) Write \(\tau(p)=\frac mk\) with \(1\le m\le k\). By Lemma 2.2(2), \(p\) is the sum of \(m\) mutually orthogonal projections of trace \(\frac1k\); let \(q\) be one of them. By Lemma 2.2(1), the first step and (2), \(pRp\cong M_m(\mathbb C)\otimes qRq\), which is isomorphic to \(M_m(\mathbb C)\otimes R\cong R\).

(3) By the first step and (2), \(M_m(\mathbb C)\otimes pRp\cong M_m(\mathbb C)\otimes R\cong R\).

All isomorphisms carry the normalized traces to each other. \(\square\)

Murray and von Neumann defined the fundamental group of a II₁ factor \(R\) as the set of numbers \(t>0\) for which the amplification \(R_t\) is isomorphic to \(R\); for \(t=\frac mk\), \(R_t\) can be taken to be \(M_m(\mathbb C)\otimes pRp\) with \(\tau(p)=\frac1k\). Corollary 4.1(3) says that the fundamental group of \(L(\mathbb F_2)\) contains every positive rational number. The equality of this fundamental group with \(\mathbb R_{>0}\), [OAI, Corollary 6.1], uses the interpolated free group factors and is not proved in this course.

5. Exercises

Exercise 5.1. List the ten generators of the corner in Theorem 3.1 for \(n=2\) and \(k=3\).

Exercise 5.2. Check directly that \(\psi(h^3)=0\) for \(k=3\), and compute \(\psi(P_1hP_1)\).

Exercise 5.3. Use Corollary 4.1 to show that for projections \(p,q\in L(\mathbb F_2)\) with \(\tau(p)=\frac23\) and \(\tau(q)=\frac35\), the corners \(pL(\mathbb F_2)p\) and \(qL(\mathbb F_2)q\) are isomorphic.

Exercise 5.4. Show that Theorem 3.1 for \(k\) and \(k'\) implies the formula for \(kk'\) (compare Exercise 5.1 of the previous lesson): \[ \begin{aligned} &1+k'^2\big((1+k^2(n-1))-1\big)\\ &\quad=1+(kk')^2(n-1). \end{aligned} \]

6. Solutions

5.1. With \(v_2=v^1_{12}\), \(v_3=v^1_{13}\): (i) \(u\), \(v_2uv_2^*\), \(v_3uv_3^*\); (ii) \(x^1_{11}\), \(v_2x^1_{22}v_2^*\), \(v_3x^1_{33}v_3^*\); (iii) \(v_2|\gamma^1_{12}|v_2^*\), \(v_3|\gamma^1_{13}|v_3^*\), \(v_3|\gamma^1_{23}|v_3^*\); (iv) \(w^1_{23}=v_2v^1_{23}v_3^*\).

5.2. \(h^3=\operatorname{diag}(e^{2\pi ia},\omega^3e^{2\pi ia},\omega^6e^{2\pi ia})\), which is \(1\otimes u\) since \(\omega^3=1\), and \(\psi(1\otimes u)=\varphi(u)=0\). Next \(P_1hP_1=E_{11}\otimes e^{2\pi ia/3}\), so \(\psi(P_1hP_1)=\frac13\int_0^1e^{2\pi it/3}dt\), which is \(\frac13\cdot\frac{3}{2\pi i}(e^{2\pi i/3}-1)\neq0\); \(h\) is centred, but its corner is not.

5.3. By Corollary 4.1(1) both corners are isomorphic to \(L(\mathbb F_2)\), by isomorphisms carrying the normalized traces to \(\tau\).

5.4. Let \(p\le q\) with \(\tau(q)=\frac1k\) and \(\tau(p)=\frac1{kk'}\) (Lemma 2.2(2) inside \(qRq\)). Then \(pRp=p(qRq)p\), where \(qRq\cong L(\mathbb F_{1+k^2(n-1)})\) and \(p\) has normalized trace \(\frac1{k'}\) in \(qRq\). Theorem 3.1 with \(k'\) gives \(L(\mathbb F_{N'})\) with \(N'=1+k'^2\big(k^2(n-1)\big)\), that is, \(N'=1+(kk')^2(n-1)\); this is Theorem 3.1 for \(kk'\).

References