Finite-dimensional subfactors and lifting

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

The next lesson builds an injective subalgebra of a factor of type III₁ as the von Neumann algebra generated by a sequence of commuting matrix subalgebras. Each of them is first found in the asymptotic centralizer, by a limiting argument, and then pulled back to the factor. This lesson collects the tools.

For a matrix subalgebra \(F\) with a faithful state \(\psi_F\), Section 1 studies the conditional expectation onto the relative commutant of \(F\) induced by \(\psi_F\). A normal functional is unchanged by this expectation exactly when the matrix units of \(F\) scale it by the ratios of the weights of \(\psi_F\) (Lemma 1.1(3)); if they do so approximately, the functional moves by a controlled amount (Lemma 1.1(4)); and in the exact case the matrix units are eigenvectors of the modular group (Lemma 1.2). Section 2 completes partial isometries in factors of type III. Section 3 constructs two-by-two matrix units that scale a state by a prescribed ratio, directly from the Connes–Størmer transitivity theorem. Section 4 lifts matrix units from the asymptotic centralizer \(N_{\varphi,\omega}\) to matrix units of \(N\) whose sequences asymptotically commute with \(\varphi\). Section 5 approximates a unitary of a II₁ factor by a unitary of a subalgebra isomorphic to \(M_{2^p}(\mathbb C)\).

The scheme is that of Ando, Haagerup, Houdayer and Marrakchi [AHHM, Section 5]. Proposition 4.4 is the form, for the asymptotic centralizer, of the lifting lemma for matrix units that they quote from Connes [AHHM, Lemma 2.1].

We use: the conventions of Section 1, Lemmas 1.1 and 1.3, and Theorem 5.1 of Ultraproducts and the asymptotic centralizer, and its result (B8) on projections in factors; Lemma 1.1 (sequences in the asymptotic centralizer) and Lemmas 2.1 and 2.2 (approximately central projections, partial isometries between states) of Asymptotic centralizers of type III₁ factors; Corollary 1.3 (modular frequency) of Spectral shift maps; Lemma 2.3 and Corollary 2.4 of Haar unitaries, freeness and tracial ultraproducts; the Borel functional calculus of self-adjoint and of normal operators, Theorems 3.1 and 8.1 of the spectral theorem lesson; and the double commutant theorem, Section 4 of its lesson.

1. Matrix subalgebras and their expectations

Throughout, \(M\) is a von Neumann algebra and we use the bimodule notation of the ultraproduct lesson: for \(a\in M\) and \(\rho\in M_*\), \((a\rho)(x)=\rho(xa)\), \((\rho a)(x)=\rho(ax)\), \([a,\rho]=a\rho-\rho a\), and \(\|a\rho\|,\|\rho a\|\le\|a\|\|\rho\|\). For a hermitian functional \(\rho\) we have \((a\rho)^*=\rho a^*\), where \(\rho^*(x)=\overline{\rho(x^*)}\).

A matrix subalgebra of \(M\) is a \(*\)-subalgebra \(F\) containing \(1_M\) and spanned by a system of matrix units \((f_{ij})_{i,j=1}^k\): elements with \[ \begin{gathered} f_{ij}f_{kl}=\delta_{jk}f_{il},\qquad f_{ij}^*=f_{ji},\\ \sum_if_{ii}=1,\qquad f_{11}\ne0 . \end{gathered} \] Then every \(f_{ii}\ne0\) (because \(f_{1i}f_{ii}f_{i1}=f_{11}\)), the \(f_{ij}\) are linearly independent, and \(F\cong M_k(\mathbb C)\). Let \(\psi_F\) be a faithful state on \(F\). We say that the matrix units diagonalize \(\psi_F\), with weights \(h_1,\ldots,h_k\), if \(\psi_F(f_{ij})=\delta_{ij}h_i\); then \(h_i>0\) and \(\sum_ih_i=1\). In this lesson and the next, every matrix subalgebra comes with matrix units diagonalizing its state. For \(F=M_2(\mathbb C)\) and \(0<\lambda\), the state \(\omega_\lambda\) is the one with weights \[ \begin{gathered} h_1=\frac\lambda{1+\lambda},\\ h_2=\frac1{1+\lambda}, \end{gathered}\tag{1.1} \] and the normalized trace \(\tau\) of \(M_k(\mathbb C)\) has all weights equal to \(1/k\).

Put \[ \begin{gathered} E_F(x)=\sum_{i,j}h_i\,f_{ji}\,x\,f_{ij}\\ (x\in M), \end{gathered}\tag{1.2} \] and, for \(x\in M\) and indices \(a,b\), \[ x_{ab}=\sum_lf_{la}\,x\,f_{bl}. \tag{1.3} \]

Lemma 1.1. Let \(F\subset M\) be a matrix subalgebra with matrix units \((f_{ij})_{i,j=1}^k\) diagonalizing \(\psi_F\), with weights \(h_i\).

  1. Every \(x\in M\) has a unique expression \(x=\sum_{a,b}f_{ab}x_{ab}\) with \(x_{ab}\in F'\cap M\), and the coefficients are given by (1.3).
  2. \(E_F\) is a faithful normal conditional expectation of \(M\) onto \(F'\cap M\), with \(E_F(x)=\sum_ah_ax_{aa}\) and \(E_F(a)=\psi_F(a)1\) for \(a\in F\).
  3. For \(\rho\in M_*\) the following are equivalent: (a) \(\rho\circ E_F=\rho\); (b) \(f_{ij}\rho=(h_j/h_i)\,\rho f_{ij}\) for all \(i,j\). In this case \(\rho(ac)=\psi_F(a)\rho(c)\) for \(a\in F\) and \(c\in F'\cap M\).
  4. For every \(\rho\in M_*\), \[ \begin{aligned} &\|\rho\circ E_F-\rho\|\\ &\quad \le k\max_{i,j}\big\|f_{ij}\rho-(h_j/h_i)\,\rho f_{ij}\big\| . \end{aligned} \]
  5. If \(G\subset F'\cap M\) is a matrix subalgebra with matrix units \((g_{ab})\) diagonalizing \(\psi_G\), then \(F\vee G\) is a matrix subalgebra with matrix units \((f_{ij}g_{ab})\), which diagonalize the product state \(\psi_F\otimes\psi_G\), whose value at \(f_{ij}g_{ab}\) is \(\psi_F(f_{ij})\psi_G(g_{ab})\); and \(E_{F\vee G}=E_F\circ E_G=E_G\circ E_F\).
  6. If \(N\subset M\) is a von Neumann subalgebra containing \(F\) and \(E\colon M\to N\) is a conditional expectation, then \(E\circ E_F=E_F\circ E\), and \(E_F\) maps \(N\) onto \(F'\cap N\).

Proof. (1) For \(x\in M\) and the \(x_{ab}\) of (1.3), \(f_{cd}x_{ab}=f_{ca}xf_{bd}=x_{ab}f_{cd}\), so \(x_{ab}\in F'\cap M\); and \(\sum_{a,b}f_{ab}x_{ab}=\sum_{a,b,l}f_{ab}f_{la}xf_{bl}\), which is \(\sum_{a,b}f_{aa}xf_{bb}=x\). If \(\sum_{c,d}f_{cd}y_{cd}=0\) with \(y_{cd}\in F'\cap M\), then \(0=\sum_lf_{la}\big(\sum_{c,d}f_{cd}y_{cd}\big)f_{bl}\), which is \(\sum_lf_{ll}y_{ab}=y_{ab}\).

(2) By (1.3), \(\sum_jf_{ji}xf_{ij}=x_{ii}\), so \(E_F(x)=\sum_ih_ix_{ii}\in F'\cap M\). For \(c\in F'\cap M\), \(c_{ab}=\delta_{ab}c\), so \(E_F(c)=c\); and \(E_F(cxd)=cE_F(x)d\) for \(c,d\in F'\cap M\), directly from (1.2). The map is positive, unital and normal, being a finite sum of maps \(x\mapsto f_{ji}xf_{ij}\) with positive coefficients. If \(x\ge0\) and \(E_F(x)=0\), every term \(h_if_{ji}xf_{ij}\) vanishes, so \(x^{1/2}f_{ij}=0\) for all \(i,j\), and \(x^{1/2}=x^{1/2}\sum_jf_{jj}=0\). Finally \(f_{ji}f_{ab}f_{ij}=\delta_{ia}\delta_{bi}f_{jj}\) gives \(E_F(f_{ab})=\delta_{ab}h_a1=\psi_F(f_{ab})1\).

(3) Write \(x=\sum_{a,b}f_{ab}x_{ab}\). Then \(xf_{ij}=\sum_af_{aj}x_{ai}\) and \(f_{ij}x=\sum_bf_{ib}x_{jb}\), so by (2) \[ \begin{gathered} E_F(xf_{ij})=h_jx_{ji},\\ E_F(f_{ij}x)=h_ix_{ji}. \end{gathered} \] If \(\rho=\rho\circ E_F\), then \((f_{ij}\rho)(x)=\rho(xf_{ij})=h_j\rho(x_{ji})\) and \((\rho f_{ij})(x)=\rho(f_{ij}x)=h_i\rho(x_{ji})\), which is (b). Conversely, assume (b). With \((a\rho b)(x)=\rho(bxa)\), the definition (1.2) reads \(\rho\circ E_F=\sum_{i,j}h_i\,f_{ij}\rho f_{ji}\), and (b) gives \(f_{ij}\rho f_{ji}=(h_j/h_i)\rho f_{ij}f_{ji}=(h_j/h_i)\rho f_{ii}\). Hence \[ \rho\circ E_F=\sum_{i,j}h_j\,\rho f_{ii}=\rho\sum_if_{ii}=\rho . \] For the last statement, \(E_F(ac)=E_F(a)c=\psi_F(a)c\) by (2), so \(\rho(ac)=\rho(E_F(ac))=\psi_F(a)\rho(c)\).

(4) Let \(\delta\) be the maximum on the right. As in (3), \[ \rho\circ E_F-\rho=\sum_{i,j}h_i\Big(f_{ij}\rho-\frac{h_j}{h_i}\rho f_{ij}\Big)f_{ji}, \] and each term has norm at most \(h_i\delta\). Summing over \(j\) and then over \(i\) gives at most \(k\delta\sum_ih_i=k\delta\).

(5) The \(g_{ab}\) commute with the \(f_{ij}\), so the products \(f_{ij}g_{ab}\) satisfy the relations of matrix units indexed by pairs \((i,a)\), and \(f_{11}g_{11}\ne0\): otherwise \(f_{j1}f_{11}g_{11}f_{1j}=f_{jj}g_{11}=0\) for all \(j\) and \(g_{11}=0\). The formula (1.2) for \(F\vee G\) with the weights \(h_ik_a\) of the product state is \[ \begin{aligned} E_{F\vee G}(x)&=\sum_{i,j,a,b}h_ik_a\,f_{ji}g_{ba}\,x\,g_{ab}f_{ij}\\ &=E_F(E_G(x)), \end{aligned} \] and since the \(f\)'s and \(g\)'s commute, the same sum equals \(E_G(E_F(x))\).

(6) \(E\) is \(N\)-bimodular and the \(f_{ij}\) lie in \(N\), so \(E(f_{ji}xf_{ij})=f_{ji}E(x)f_{ij}\). For \(x\in N\), (1.2) gives \(E_F(x)\in N\cap F'\). \(\square\)

So \(E_F\) depends only on \(F\) and \(\psi_F\): by (1) and (2) it is the map \(\sum f_{ab}x_{ab}\mapsto\sum_{a,b}\psi_F(f_{ab})x_{ab}\), the slice by \(\psi_F\) under the identification of \(M\) with \(F\otimes(F'\cap M)\). By (3), \(\rho\circ E_F=\rho\) says that \(\rho\) is the product of \(\psi_F\) and of its restriction to \(F'\cap M\).

Lemma 1.2 (modular eigenvectors). Let \(\rho\) be a faithful normal state on \(M\) with \(\rho\circ E_F=\rho\). Then \[ \sigma^\rho_t(f_{ij})=(h_i/h_j)^{it}f_{ij}\qquad(t\in\mathbb R). \] Consequently \(\sigma^\rho_t(F)=F\) and \(\sigma^\rho_t(F'\cap M)=F'\cap M\); if all weights are equal, \(F\) lies in the centralizer \(M_\rho\).

Proof. By Lemma 1.1(3), \(f_{ij}\rho=e^{-s}\rho f_{ij}\) with \(s=\log(h_i/h_j)\). Corollary 1.3 of the lesson on spectral shift maps, whose Section 1 is stated for an arbitrary von Neumann algebra with a faithful normal state, gives \(\sigma^\rho_t(f_{ij})=e^{its}f_{ij}\). So \(\sigma^\rho_t\) maps \(F\) onto itself, and an element commuting with all \(f_{ij}\) is mapped to an element commuting with all \(\sigma^\rho_t(f_{ij})\), that is, with all \(f_{ij}\). \(\square\)

Lemma 1.3 (one unitary moves the remainder). Let \(T\subset M\) be a matrix subalgebra whose matrix units diagonalize the normalized trace, let \(\rho\) be a normal state with \(\rho\circ E_T=\rho\), and \(u\in T\) a unitary. For every \(z\in M\), \[ \|E_T(z)\|^2_\rho\le\|z\|^2_\rho-\tfrac14\|uzu^*-z\|^2_\rho , \] where \(\|y\|_\rho=\rho(y^*y)^{1/2}\).

Proof. By Lemma 1.1(3), with equal weights, \(f_{ij}\rho=\rho f_{ij}\), so \(a\rho=\rho a\) for every \(a\in T\). For \(c\in T'\cap M\), bimodularity and \(\rho\circ E_T=\rho\) give \(\rho(c^*(z-E_T(z)))\) equals \(\rho(c^*E_T(z)-c^*E_T(z))\), which is \(0\). With \(c=E_T(z)\) this gives \(\|z\|^2_\rho=\|E_T(z)\|^2_\rho+\|w\|^2_\rho\) for \(w=z-E_T(z)\). Since \(E_T(z)\) commutes with \(u\), \(uzu^*-z=uwu^*-w\). Finally \(\|uwu^*\|^2_\rho=\rho(uw^*wu^*)\), which is \(\rho(w^*wu^*u)=\|w\|^2_\rho\), because \(u\rho=\rho u\). So \(\|uzu^*-z\|_\rho\le2\|w\|_\rho\). \(\square\)

2. Partial isometries in factors of type III

In this section \(N\) is a countably decomposable factor of type III with a faithful normal state \(\varphi\). By (B8) of the ultraproduct lesson, every nonzero projection of \(N\) is equivalent to \(1\); in particular any two nonzero projections are equivalent.

Lemma 2.1 (small projections). Let \(e\in N\) be a nonzero projection, \(\rho\in N_*^+\) and \(\eta>0\). There is a nonzero projection \(r\le e\) with \(\rho(r)\le\eta\).

Proof. Let \(w\in N\) with \(w^*w=1\), \(ww^*=e\), and let \(g\in N\) be a projection different from \(0\) and \(1\) (a factor of type III is not \(\mathbb C\)). The projections \(wgw^*\) and \(w(1-g)w^*\) are orthogonal, nonzero (since \(w^*(wgw^*)w=g\)), and add up to \(e\); one of them has \(\rho\)-value at most \(\rho(e)/2\). Repeating this \(j\) times with \(2^{-j}\rho(e)\le\eta\) gives \(r\). \(\square\)

Lemma 2.2 (completing a partial isometry). Let \(p,q\in N\) be nonzero projections, \(v\in N\) a partial isometry with \(v^*v\le p\) and \(vv^*\le q\), and \(\eta>0\). There is a partial isometry \(x\in N\) with \(x^*x=p\), \(xx^*=q\) and \[ \begin{aligned} &\|x-v\|_\varphi+\|(x-v)^*\|_\varphi\\ &\quad \le4\big(\varphi(p-v^*v)+\varphi(q-vv^*)+\eta\big)^{1/2}. \end{aligned} \]

Proof. Put \(p_0=p-v^*v\) and \(q_0=q-vv^*\), projections. From \(v^*v\le p\) and \(vv^*\le q\) we get \(vp=v\) and \(qv=v\), hence \(vp_0=0\) and \(v^*q_0=0\).

Both defects nonzero. Choose \(s\in N\) with \(s^*s=p_0\), \(ss^*=q_0\), and put \(x=v+s\). Since \(s=q_0s=sp_0\), we have \(v^*s=v^*q_0s=0\) and \(vs^*=vp_0s^*=0\). Hence \(x^*x=v^*v+s^*s=p\) and \(xx^*=vv^*+ss^*=q\). Moreover \(\|x-v\|_\varphi=\varphi(p_0)^{1/2}\) and \(\|(x-v)^*\|_\varphi=\varphi(q_0)^{1/2}\), whose sum is at most \(2(\varphi(p_0)+\varphi(q_0))^{1/2}\).

Both defects zero. Take \(x=v\).

Exactly one defect zero. Suppose \(p_0=0\ne q_0\). Then \(v^*v=p\ne0\). Lemma 2.1 in \(N\), for the normal positive functional \(r\mapsto\varphi(r)+\varphi(vrv^*)\), gives a nonzero projection \(r\le p\) with \(\varphi(r)+\varphi(vrv^*)\le\eta\). The partial isometry \(v'=v(p-r)\) has \(v'^*v'=p-r\) and \(v'v'^*=vv^*-vrv^*\), so its defects are \(r\ne0\) and \(q_0+vrv^*\ne0\). The first case gives \(x=v'+s\) with \(x^*x=p\), \(xx^*=q\), \(s^*s=r\) and \(ss^*=q_0+vrv^*\). Now \(x-v=s-vr\), so \[ \begin{aligned} &\|x-v\|_\varphi\le\varphi(r)^{1/2}+\varphi(rv^*vr)^{1/2}\\ &\quad \le2\eta^{1/2},\\ &\|(x-v)^*\|_\varphi\le\varphi(q_0+vrv^*)^{1/2}\\ &\quad +\varphi(vrv^*)^{1/2}\le(\varphi(q_0)+\eta)^{1/2}+\eta^{1/2}, \end{aligned} \] and the sum is at most \(4(\varphi(q_0)+\eta)^{1/2}\). If \(q_0=0\ne p_0\), apply this case to \(v^*\), with \(p\) and \(q\) exchanged, and take the adjoint of the result. \(\square\)

3. Matrix units with a prescribed ratio

Proposition 3.1. Let \(N\) be a factor of type III₁ with separable predual, \(\varphi\) a faithful normal state on \(N\), \(\lambda>0\) and \(\delta>0\). There are a projection \(e\in N\) and a partial isometry \(x\in N\) with \[ \begin{gathered} x^*x=1-e,\qquad xx^*=e,\\ \|[e,\varphi]\|\le\delta,\qquad \|x\varphi-\lambda\varphi x\|\le\delta . \end{gathered} \] The elements \(f_{11}=1-e\), \(f_{22}=e\), \(f_{21}=x\), \(f_{12}=x^*\) are matrix units of a matrix subalgebra \(F\cong M_2(\mathbb C)\) of \(N\), which diagonalize \(\omega_\lambda\), and \[ \|\varphi\circ E_F-\varphi\|\le2\max(1,\lambda^{-1})\,\delta . \]

Proof. Let \(0<\varepsilon<\min(1,\lambda)/(2(1+\lambda))\). By Lemma 2.1 of the asymptotic centralizer lesson there is a projection \(e\) with \(\|[e,\varphi]\|<\varepsilon\) and \(|\varphi(e)-1/(1+\lambda)|<\varepsilon\). Then \(\varphi(e)>1/(2(1+\lambda))\) and \(\varphi(1-e)>\lambda/(1+\lambda)-\varepsilon\), which is greater than \(\lambda/(2(1+\lambda))\), so \(e\) and \(1-e\) are nonzero.

Apply Lemma 2.2 of the asymptotic centralizer lesson to the projections \(p=1-e\), \(q=e\), the states \(\chi_1=(1-e)\varphi(1-e)/\varphi(1-e)\), \(\chi_2=e\varphi e/\varphi(e)\) (so \(\chi_1(p)=\chi_2(q)=1\)), and \(\varepsilon\). It gives a partial isometry \(v\) with \(v^*v\le1-e\), \(vv^*\le e\), \(\chi_1(1-e-v^*v)\le\varepsilon\), \(\chi_2(e-vv^*)\le\varepsilon\) and \(\|v\chi_1-\chi_2v\|\le6\sqrt\varepsilon\).

Since \(v=vp\), \(v\varphi-v(p\varphi p)=vp\varphi(1-p)\), and \(p\varphi(1-p)=[p,\varphi](1-p)\); so \(\|v\varphi-v(p\varphi p)\|\le\|[p,\varphi]\|=\|[e,\varphi]\|\). Likewise \(\|\varphi v-(q\varphi q)v\|\le\|[e,\varphi]\|\). Further, \[ \begin{aligned} &v(p\varphi p)-\lambda(q\varphi q)v=\varphi(p)\big(v\chi_1-\chi_2v\big)\\ &\quad +\big(\varphi(p)-\lambda\varphi(q)\big)\chi_2v , \end{aligned} \] and \(|\varphi(p)-\lambda\varphi(q)|=|1-(1+\lambda)\varphi(e)|\), which is less than \((1+\lambda)\varepsilon\). Altogether \[ \begin{aligned} &\|v\varphi-\lambda\varphi v\|\le(1+\lambda)\varepsilon+6\sqrt\varepsilon\\ &\quad +(1+\lambda)\varepsilon\le(2\lambda+8)\sqrt\varepsilon . \end{aligned} \] Lemma 2.2 of this lesson, with \(\eta=\varepsilon\), gives a partial isometry \(x\) with \(x^*x=1-e\), \(xx^*=e\), and \(\|x-v\|_\varphi+\|(x-v)^*\|_\varphi\le4(3\varepsilon)^{1/2}\), because \(\varphi(p-v^*v)=\varphi(p)\chi_1(p-v^*v)\le\varepsilon\) and likewise for \(q\). By Lemma 1.1(b),(c) of the ultraproduct lesson, \(\|(x-v)\varphi\|\le\|x-v\|_\varphi\) and \(\|\varphi(x-v)\|\le\|(x-v)^*\|_\varphi\), so \[ \begin{aligned} &\|x\varphi-\lambda\varphi x\|\le(2\lambda+8)\sqrt\varepsilon\\ &\quad +4\sqrt3\max(1,\lambda)\sqrt\varepsilon . \end{aligned} \] Choosing \(\varepsilon\) small makes this and \(\|[e,\varphi]\|<\varepsilon\) at most \(\delta\).

The relations of matrix units follow from \(x^*x=1-e\), \(xx^*=e\). The weights of \(\omega_\lambda\) in (1.1) have \(h_1/h_2=\lambda\). For Lemma 1.1(4) we need the four quantities \(\|f_{ij}\varphi-(h_j/h_i)\varphi f_{ij}\|\): for \(f_{11},f_{22}\) they are \(\|[e,\varphi]\|\le\delta\); for \(f_{21}=x\) it is \(\|x\varphi-\lambda\varphi x\|\le\delta\); for \(f_{12}=x^*\) it is \(\|x^*\varphi-\lambda^{-1}\varphi x^*\|=\lambda^{-1}\|(\varphi x-\lambda x\varphi)^*\|\), which is at most \(\lambda^{-1}\delta\), since \((a\varphi)^*=\varphi a^*\). Lemma 1.1(4) with \(k=2\) finishes the proof. \(\square\)

4. Lifting from the asymptotic centralizer

In this section \(N\) is a countably decomposable factor of type III with a faithful normal state \(\varphi\), \(\omega\) is a free ultrafilter, \(A=A_{\varphi,\omega}\) is the \(C^*\)-algebra of bounded sequences \((x_m)\) in \(N\) with \(\lim_\omega\|[x_m,\varphi]\|=0\), \(\mathcal I_\omega\subset A\) the ideal of sequences tending to \(0\) \(*\)-strongly along \(\omega\), and \(N_{\varphi,\omega}=A/\mathcal I_\omega\) with its faithful normal trace \(\tau\), as in Section 1 of the asymptotic centralizer lesson. A sequence in \(A\) represents its class. Two sequences that agree on a set belonging to \(\omega\) differ by an element of \(\mathcal I_\omega\). A self-adjoint sequence \((y_m)\) lies in \(\mathcal I_\omega\) as soon as \(\lim_\omega\|y_m\|_\varphi=0\), by Lemma 1.3 of the ultraproduct lesson; if \((y_m)\in\mathcal I_\omega\) and \((x_m)\in A\), then \((x_my_m)\in\mathcal I_\omega\) and \((x_m+y_m)\in A\), by Theorem 5.1(1),(5) there.

Lemma 4.1 (projections under a projection). Let \(P\le F\) be projections of \(N_{\varphi,\omega}\), and \((f_m)\in A\) a sequence of projections representing \(F\). Then \(P\) has a representative \((p_m)\in A\) consisting of projections with \(p_m\le f_m\) for all \(m\).

Proof. Let \((y_m)\in A\) represent \(P\). Then \((f_my_mf_m)\) represents \(FPF=P\), and so does its self-adjoint part \((a_m)\), \(a_m=\frac12f_m(y_m+y_m^*)f_m\). Let \(g(t)=\min(\max(t,0),1)\). By Lemma 1.1(3) of the asymptotic centralizer lesson, \((g(a_m))\in A\) represents \(g(P)=P\). Since \(g(0)=0\), \(g\) is a uniform limit on \([-\|a_m\|,\|a_m\|]\) of polynomials without constant term, and these map \(a_m=f_ma_mf_m\) into \(f_mNf_m\); so \(x_m=g(a_m)\) satisfies \(0\le x_m\le f_m\). Put \(p_m=1_{[1/2,1]}(x_m)\). Then \(p_m=x_mk(x_m)\) with the bounded Borel function \(k(t)=t^{-1}1_{[1/2,1]}(t)\), so \((1-f_m)p_m=0\), that is \(p_m\le f_m\). The proof of Lemma 1.1(4) of the asymptotic centralizer lesson now applies verbatim: \(\|x_m-p_m\|_\varphi\le2\|x_m-x_m^2\|_\varphi\to_\omega0\), so \((p_m)\) belongs to \(A\) and represents \(P\). \(\square\)

Lemma 4.2 (partitions of unity). Let \(P_1,\ldots,P_k\) be projections of \(N_{\varphi,\omega}\) with \(\sum_iP_i=1\). There are representatives \((p_{i,m})_m\in A\) of the \(P_i\), consisting of projections, with \(\sum_ip_{i,m}=1\) for every \(m\).

Proof. Lemma 4.1 with \(F=1\) and \(f_m=1\) gives \((p_{1,m})\). If \(p_{1,m},\ldots,p_{j,m}\) have been found for some \(j<k-1\), mutually orthogonal, then \(f_m=1-\sum_{i\le j}p_{i,m}\) is a projection, \((f_m)\in A\) represents \(1-\sum_{i\le j}P_i\ge P_{j+1}\), and Lemma 4.1 gives \(p_{j+1,m}\le f_m\). Finally \(p_{k,m}=1-\sum_{i<k}p_{i,m}\) represents \(P_k\). \(\square\)

Lemma 4.3 (partial isometries). Let \(V\in N_{\varphi,\omega}\) be a nonzero partial isometry, and let \((p_m),(q_m)\in A\) be sequences of projections representing \(V^*V\) and \(VV^*\). There are a set \(W\in\omega\) and a representative \((x_m)\in A\) of \(V\) such that, for \(m\in W\), \(x_m\) is a partial isometry with \(x_m^*x_m=p_m\) and \(x_mx_m^*=q_m\).

Proof. Let \((y_m)\in A\) represent \(V\). Then \(y_m'=q_my_mp_m\) represents \(QVP=V\), where \(P=V^*V\), \(Q=VV^*\), and \(a_m=y_m'^*y_m'\in p_mNp_m\) is positive and represents \(P\); so \((a_m-p_m)\in\mathcal I_\omega\). Let \(h(t)=t^{-1/2}1_{[1/4,\infty)}(t)\) for \(t\ge0\), and \(w_m=y_m'h(a_m)\). As in the previous proof, \(h(a_m)=a_mk(a_m)\) with \(k(t)=t^{-3/2}1_{[1/4,\infty)}(t)\), so \(h(a_m)=p_mh(a_m)p_m\). Then \[ \begin{aligned} &w_m^*w_m=h(a_m)a_mh(a_m)\\ &\quad =1_{[1/4,\infty)}(a_m)=:p_m'\le p_m , \end{aligned} \] so \(w_m\) is a partial isometry, and \(q_m'=w_mw_m^*\le q_m\) because \(q_mw_m=w_m\).

For \(t\ge0\) one checks \(1_{[0,1/4)}(t)\le\frac43|t-1|\) and \(|h(t)-1|\le2|t-1|\). Since \(p_m\) commutes with \(a_m\) and \(p_m(a_m-1)p_m=a_m-p_m\), the functional calculus gives \[ \begin{aligned} &(p_m-p_m')^2=p_m1_{[0,1/4)}(a_m)^2p_m\\ &\quad \le\tfrac{16}9(a_m-p_m)^2,\\ &(h(a_m)-p_m)^2=p_m(h-1)^2(a_m)p_m\\ &\quad \le4(a_m-p_m)^2 . \end{aligned} \] Applying \(\varphi\), the self-adjoint sequences \((p_m-p_m')\) and \((h(a_m)-p_m)\) lie in \(\mathcal I_\omega\). Hence \(w_m-y_m'=y_m'(h(a_m)-p_m)\) defines an element of \(\mathcal I_\omega\), \((w_m)\in A\) represents \(V\), and \((q_m')\) represents \(VV^*=Q\); so \((q_m-q_m')\in\mathcal I_\omega\) and \(\lim_\omega\varphi(q_m-q_m')=0\). Also \(\lim_\omega\varphi(p_m-p_m')=0\).

Since \(V\ne0\), \(\lim_\omega\varphi(p_m)=\tau(P)>0\) and \(\lim_\omega\varphi(q_m)=\tau(Q)>0\), so \(W=\{m:p_m\ne0,\ q_m\ne0\}\in\omega\). For \(m\in W\), Lemma 2.2 with \(\eta=1/m\) gives a partial isometry \(x_m\) with \(x_m^*x_m=p_m\), \(x_mx_m^*=q_m\) and \[ \begin{aligned} &\|x_m-w_m\|_\varphi+\|(x_m-w_m)^*\|_\varphi\\ &\quad\le4\big(\varphi(p_m-p_m')+\varphi(q_m-q_m')\\ &\qquad\quad+1/m\big)^{1/2}\to_\omega0 . \end{aligned} \] For \(m\notin W\) put \(x_m=0\). Then \((x_m-w_m)\in\mathcal I_\omega\), so \((x_m)\in A\) represents \(V\). \(\square\)

Proposition 4.4 (lifting matrix units). Let \(\Theta\colon M_k(\mathbb C)\to N_{\varphi,\omega}\) be a unital \(*\)-homomorphism. There are unital \(*\)-homomorphisms \(\theta_m\colon M_k(\mathbb C)\to N\), \(m\ge1\), such that for every \(a\in M_k(\mathbb C)\) the sequence \((\theta_m(a))_m\) belongs to \(A\) and represents \(\Theta(a)\). In particular \(\lim_\omega\|[\theta_m(a),\varphi]\|=0\).

Proof. Let \((\varepsilon_{ij})\) be the matrix units of \(M_k(\mathbb C)\) and \(E_{ij}=\Theta(\varepsilon_{ij})\); \(\Theta\) is injective because \(M_k(\mathbb C)\) is simple, so \(E_{i1}\ne0\). Lemma 4.2 gives projections \(e_{i,m}\) with \(\sum_ie_{i,m}=1\), \((e_{i,m})_m\) representing \(E_{ii}\). For \(i\ge2\), Lemma 4.3 applied to \(V=E_{i1}\), \((p_m)=(e_{1,m})\), \((q_m)=(e_{i,m})\) gives \(W_i\in\omega\) and partial isometries \(x_{i,m}\), \(m\in W_i\), with \(x_{i,m}^*x_{i,m}=e_{1,m}\), \(x_{i,m}x_{i,m}^*=e_{i,m}\), whose sequence represents \(E_{i1}\). Put \(x_{1,m}=e_{1,m}\) and \(W=\bigcap_iW_i\in\omega\). For \(m\in W\), the elements \(x_{i,m}x_{j,m}^*\) are matrix units: \(x_{j,m}^*x_{k,m}=x_{j,m}^*e_{j,m}e_{k,m}x_{k,m}=\delta_{jk}e_{1,m}\), so \((x_{i,m}x_{j,m}^*)(x_{k,m}x_{l,m}^*)=\delta_{jk}x_{i,m}x_{l,m}^*\), and \(\sum_ix_{i,m}x_{i,m}^*=\sum_ie_{i,m}=1\). Let \(\theta_m\) be the unital \(*\)-homomorphism with \(\theta_m(\varepsilon_{ij})=x_{i,m}x_{j,m}^*\). For \(m\notin W\) let \(\theta_m\) be any unital \(*\)-homomorphism \(M_k(\mathbb C)\to N\); one exists because \(1\) is a sum of \(k\) nonzero orthogonal projections (Lemma 2.1, applied repeatedly), which are equivalent (B8). The sequence \((\theta_m(\varepsilon_{ij}))\) agrees on \(W\) with \((x_{i,m}x_{j,m}^*)\in A\), which represents \(E_{i1}E_{1j}=E_{ij}\). Linearity gives the statement for every \(a\). \(\square\)

5. Dyadic matrix subalgebras

Lemma 5.1. Let \(Q\) be a II₁ factor with faithful normal tracial state \(\tau\), \(\|y\|_\tau=\tau(y^*y)^{1/2}\), \(U\in Q\) a unitary and \(\delta>0\). There are \(p\ge0\), a unital \(*\)-homomorphism \(\Theta\colon M_{2^p}(\mathbb C)\to Q\) and a unitary \(c\in M_{2^p}(\mathbb C)\) with \(\|U-\Theta(c)\|_\tau\le\delta\).

Proof. Finite spectrum. Choose \(K\) with \(2\pi/K\le\delta/2\), and let \(e_l=1_{\Delta_l}(U)\) for the arcs \(\Delta_l=\{e^{i\theta}:2\pi l/K\le\theta<2\pi(l+1)/K\}\), \(0\le l<K\). By the Borel functional calculus of the normal operator \(U\) (Theorem 8.1 of the spectral theorem lesson) the \(e_l\) are mutually orthogonal projections with sum \(1\), they commute with every operator commuting with \(U\) and \(U^*\), hence lie in \(Q''=Q\), and \(U_1=\sum_le^{2\pi il/K}e_l\) satisfies \(\|U-U_1\|\le2\pi/K\), because \(|z-e^{2\pi il/K}|\le2\pi/K\) on \(\Delta_l\).

Dyadic traces. Choose \(p\) with \(K2^{-p}\le\delta^2/16\). For each \(l\) choose a projection \(f_l\le e_l\) with \(\tau(f_l)=2^{-p}\lfloor2^p\tau(e_l)\rfloor\): by Corollary 2.4 of the lesson on Haar unitaries there is a projection of this trace, and by (B8) of the ultraproduct lesson it is equivalent to a subprojection of \(e_l\), which has the same trace. Put \(r=1-\sum_lf_l\), a projection with \(\tau(r)=\sum_l(\tau(e_l)-\tau(f_l))\le K2^{-p}\), and \(U_2=\sum_le^{2\pi il/K}f_l+r\). Then \(U_1-U_2=\sum_l(e^{2\pi il/K}-1)(e_l-f_l)\), a combination of orthogonal projections with coefficients of modulus at most \(2\), so \(\|U_1-U_2\|_\tau\le2\tau(r)^{1/2}\le\delta/2\). Hence \(\|U-U_2\|_\tau\le\|U-U_1\|+\|U_1-U_2\|_\tau\), which is at most \(\delta\).

A matrix subalgebra. The projections \(f_0,\ldots,f_{K-1},r\) add up to \(1\) and have traces in \(2^{-p}\mathbb Z\). Splitting each of them, by the same two facts, into projections of trace exactly \(2^{-p}\), we obtain mutually orthogonal projections \(g_1,\ldots,g_{2^p}\) of trace \(2^{-p}\) with sum \(1\), such that each \(f_l\) and \(r\) is a sum of some of them. By Lemma 2.3 of the lesson on Haar unitaries there are \(w_i\in Q\) with \(w_i^*w_i=g_1\), \(w_iw_i^*=g_i\) (\(w_1=g_1\)). The elements \(w_iw_j^*\) are matrix units with \(\sum_iw_iw_i^*=1\), and \(\Theta(\varepsilon_{ij})=w_iw_j^*\) defines a unital \(*\)-homomorphism. The unitary \(U_2\) is \(\Theta(c)\) for the diagonal unitary \(c\) whose \(i\)-th entry is \(e^{2\pi il/K}\) if \(g_i\le f_l\), and \(1\) if \(g_i\le r\). \(\square\)

6. Exercises

Exercise 6.1. Let \(M=M_2(\mathbb C)\), \(F=M\) with its standard matrix units, and \(\psi_F=\omega_\lambda\). Compute \(E_F\), and check Lemma 1.1(3) for \(\rho=\omega_\lambda\) by computing \(e_{21}\omega_\lambda\) and \(\omega_\lambda e_{21}\).

Exercise 6.2. In the setting of Exercise 6.1, let instead \(\psi_F=\tau\) be the normalized trace and \(\rho=\omega_\lambda\). Compute \(\|\rho\circ E_F-\rho\|\) and \(\max_{i,j}\|[e_{ij},\rho]\|\), and compare with Lemma 1.1(4).

Exercise 6.3. Show that Lemma 2.2 fails in \(M_2(\mathbb C)\): for \(p=1\), \(q=e_{11}\) and \(v=0\) there is no partial isometry \(x\) with \(x^*x=p\) and \(xx^*=q\).

Exercise 6.4. Let \(\rho\) be a faithful normal state on \(M\) with \(\rho\circ E_F=\rho\). Show that \(\rho(f_{ii})=h_i\), and that \(\sigma^\rho_t(x)=h^{it}xh^{-it}\) for \(x\in F\), where \(h=\sum_ih_if_{ii}\).

Exercise 6.5. Show that the unitary \(U\) in Lemma 5.1 cannot always be taken inside a finite-dimensional subalgebra: a Haar unitary of \(Q\) (Corollary 2.4 of the lesson on Haar unitaries), that is, a unitary with \(\tau(U^j)=0\) for all \(j\ne0\), lies in no finite-dimensional \(*\)-subalgebra.

7. Solutions

6.1. With weights \(h_1=\lambda/(1+\lambda)\), \(h_2=1/(1+\lambda)\), \(F'\cap M=\mathbb C1\) and \(E_F(x)=\sum_{i,j}h_ie_{ji}xe_{ij}\), which is \((h_1x_{11}+h_2x_{22})1=\omega_\lambda(x)1\). For \(x=(x_{ab})\), \((e_{21}\omega_\lambda)(x)=\omega_\lambda(xe_{21})=h_1x_{12}\) and \((\omega_\lambda e_{21})(x)=\omega_\lambda(e_{21}x)=h_2x_{12}\); so \(e_{21}\omega_\lambda=(h_1/h_2)\,\omega_\lambda e_{21}=\lambda\,\omega_\lambda e_{21}\), which is (b) for \((i,j)=(2,1)\).

6.2. Here \(E_F(x)=\tau(x)1\), so \(\rho\circ E_F=\tau\) and \(\|\rho\circ E_F-\rho\|=\|\tau-\omega_\lambda\|\), which is \(|h_1-\tfrac12|+|h_2-\tfrac12|=\frac{|1-\lambda|}{1+\lambda}\). For \(e_{12}\): \([e_{12},\rho](x)=\rho(xe_{12})-\rho(e_{12}x)\), which is \((h_2-h_1)x_{21}\), of norm \(|h_2-h_1|=\frac{|1-\lambda|}{1+\lambda}\); the same holds for \(e_{21}\), and the diagonal units commute with \(\rho\). So the maximum is \(\frac{|1-\lambda|}{1+\lambda}\), and Lemma 1.1(4) gives the bound \(2\frac{|1-\lambda|}{1+\lambda}\), twice the true value.

6.3. \(x^*x=1\) makes \(x\) an isometry of \(\mathbb C^2\), hence unitary, so \(xx^*=1\ne e_{11}\). In \(M_2(\mathbb C)\) projections of different trace are not equivalent; Lemma 2.2 uses that all nonzero projections of a type III factor are equivalent.

6.4. \(\rho(f_{ii})=\rho(E_F(f_{ii}))=\psi_F(f_{ii})=h_i\). By Lemma 1.2, \(\sigma^\rho_t(f_{ij})=(h_i/h_j)^{it}f_{ij}=h^{it}f_{ij}h^{-it}\), since \(h^{it}f_{ij}h^{-it}=h_i^{it}h_j^{-it}f_{ij}\); both sides are linear in \(x\in F\).

6.5. A unitary in a finite-dimensional \(*\)-subalgebra has finite spectrum, so \(U=\sum_lz_lq_l\) with distinct \(z_l\) of modulus \(1\) and nonzero orthogonal projections \(q_l\) adding up to \(1\). Put \(t_l=\tau(q_l)>0\), so that \(\tau(U^j)=\sum_lz_l^jt_l\). Expanding the square, \[ \begin{aligned} &\frac1J\sum_{j=0}^{J-1}\Big|\sum_lz_l^jt_l\Big|^2\\ &\quad =\sum_{l,l'}t_lt_{l'}\,\frac1J\sum_{j=0}^{J-1}(z_l\bar z_{l'})^j\\ &\quad \longrightarrow\sum_lt_l^2>0\\ &(J\to\infty), \end{aligned} \] because \(\frac1J\sum_{j<J}z^j=\frac{1-z^J}{J(1-z)}\to0\) for \(|z|=1\), \(z\ne1\). For a Haar unitary the left side equals \(\frac1J|\tau(U^0)|^2=\frac1J\to0\), a contradiction.

References