Semifinite bottom factors

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

The previous lesson proved the main theorem of [OAI] when the smaller algebra \(N\) is a factor of type III₁. This lesson treats semifinite factors \(N\). Again \(N\subset M\) are von Neumann algebras with separable preduals and a faithful normal conditional expectation, and we find an approximately finite-dimensional \(P\subset N\) with a faithful normal conditional expectation of \(N\) onto \(P\) and \[ P'\cap c(M)=N'\cap c(M) \] (Theorem 4.1). Here the modular theory is much simpler. If \(N\) is of type II₁, its trace extends to a state of \(M\) whose modular group fixes \(N\), and the relative commutant in the core of a subalgebra fixed by the modular group is the crossed product of its relative commutant in \(M\) (Lemma 2.2 of Core commutants). So it suffices to find \(P\) with \(P'\cap M=N'\cap M\), which is a theorem of Popa, recalled in [AHHM, Introduction]; Section 2 proves it with the tools of the previous two lessons. For \(N\) of type II∞ we pass to a corner of type II₁ and use a state whose modular group acts on matrix units by explicit phases (Section 3). For \(N\) of type I one can take \(P=N\). In [OAI] these cases are obtained from Marrakchi's weak Dixmier property [M1, Proposition 4.11] and [M1, Theorem D].

We use: Lemma 3.1 (averaging over state-preserving automorphisms) of The relative bicentralizer; Section 2 of Core commutants (the regular realization of the core, the independence of relative commutants from the state, and Lemmas 2.1 and 2.2), whose proofs use only the modular group of a faithful normal state of \(M\); Corollary 1.3 of Spectral shift maps; Lemmas 1.1, 1.3 and 5.1 of Finite-dimensional subfactors and lifting; result (B6) (the centralizer is the fixed-point algebra of the modular group) and (B8) of Ultraproducts and the asymptotic centralizer; result (B1) of Full factors; the decomposition of the unit of a properly infinite algebra into equivalent projections, Proposition 15.3(3) of the projections lesson; Corollary 5.2(b),(c) of Injective von Neumann algebras; the modular expectation theorem (ME.1)–(ME.3) of Conditional expectations from modular invariance; the Schwarz inequality for conditional expectations, CE-006; and the double commutant theorem, Section 4 of its lesson.

1. Setting

Throughout, \(N\subset M\) are von Neumann algebras with separable preduals and the same unit, and \(E\colon M\to N\) is a faithful normal conditional expectation. For a faithful normal state \(\varphi\) on \(N\) put \(\bar\varphi=\varphi\circ E\), and let \(c(M)\) be the core of \(M\) in the regular realization of Section 2 of the lesson on core commutants, built with \(\sigma=\sigma^{\bar\varphi}\): \(\pi(x)\) acts on \(L^2(\mathbb R,L^2(M,\bar\varphi))\) by \((\pi(x)f)(s)=\sigma_{-s}(x)f(s)\), and \(\lambda(t)\) by translation. We identify \(M\) with \(\pi(M)\). Relative commutants of subalgebras of \(M\) in \(c(M)\) do not depend on the state, so each case may use its own \(\varphi\).

For a von Neumann subalgebra \(Q_0\subset M\) fixed pointwise by \(\sigma\), Lemma 2.2 of that lesson gives \[ \begin{aligned} &Q_0'\cap c(M)\\ &\quad =(Q_0'\cap M)\rtimes\mathbb R , \end{aligned}\tag{1.1} \] the von Neumann algebra generated by \(\pi(Q_0'\cap M)\) and \(\lambda(\mathbb R)\). Every element of \((Q_0'\cap M)\rtimes\mathbb R\) commutes with \(\pi(Q_0)\): for \(q\in Q_0\), \(\pi(q)=1\otimes q\) commutes with \(\lambda(t)\) and with \(\pi(y)\), \(y\in Q_0'\cap M\), whose values \(\sigma_{-s}(y)\) lie in \(\sigma_{-s}(Q_0)'\cap M=Q_0'\cap M\).

Lemma 1.1 (centralizer criterion). Let \(\varphi\) be a faithful normal state on \(N\), and \(P\subset N\) a von Neumann subalgebra with \(\sigma^\varphi_t(P)=P\) for all \(t\). Write \(N_\varphi\) and \(P_\varphi=P\cap N_\varphi\) for the centralizers. If \(P_\varphi'\cap M\subset N_\varphi'\cap M\) and \(N\) is generated by \(N_\varphi\cup P\), then \(P'\cap c(M)=N'\cap c(M)\).

Proof. The inclusion \(\supset\) is clear. The modular group \(\sigma=\sigma^{\bar\varphi}\) restricts to \(\sigma^\varphi\) on \(N\) (modular expectation theorem), so \(N_\varphi\) and \(P_\varphi\) are fixed by \(\sigma\). Let \(z\in P'\cap c(M)\). By (1.1) and the hypothesis, \[ \begin{aligned} &z\in P_\varphi'\cap c(M)=(P_\varphi'\cap M)\rtimes\mathbb R\\ &\quad \subset(N_\varphi'\cap M)\rtimes\mathbb R=N_\varphi'\cap c(M); \end{aligned} \] the middle inclusion holds because the crossed product of the smaller \(\sigma\)-invariant algebra \(P_\varphi'\cap M\) is generated by part of the generators of the other. So \(z\) commutes with \(\pi(N_\varphi)\) and with \(\pi(P)\). The normal \(*\)-homomorphism \(\pi\) maps the von Neumann algebra \(N\) generated by \(N_\varphi\cup P\) onto the von Neumann algebra generated by \(\pi(N_\varphi)\cup\pi(P)\), so \(z\) commutes with \(\pi(N)\). \(\square\)

2. Finite factors

In this section \(N\) is a factor of type II₁ with its faithful normal tracial state \(\tau\), and \(\bar\tau=\tau\circ E\). For \(x\in N\) and \(y\in M\), \(\bar\tau(xy)=\tau(xE(y))=\tau(E(y)x)=\bar\tau(yx)\), so \(N\) lies in the centralizer \(M_{\bar\tau}\) and is fixed by \(\sigma^{\bar\tau}\) (B6). In particular \(\bar\tau(uyu^*)=\bar\tau(y)\) for \(u\in\mathcal U(N)\), \(\|uy\|_{\bar\tau}=\|yu^*\|_{\bar\tau}=\|y\|_{\bar\tau}\), and for a matrix subalgebra \(T\subset N\) with its trace, \(\bar\tau\circ E_T=\bar\tau\) (Lemma 1.1(3) of the lesson on finite-dimensional subfactors, since \(f\bar\tau=\bar\tau f\) for \(f\in T\)). The automorphisms \(\operatorname{Ad}u\), \(u\in\mathcal U(N)\), preserve \(\bar\tau\), and their common fixed points form \(N'\cap M\), because the unitaries span \(N\). Lemma 3.1 of the first lesson gives the \(\bar\tau\)-preserving conditional expectation \(E'\) of \(M\) onto \(N'\cap M\), and for every \(z\in M\) with \(E'(z)=0\) a unitary \(u\in N\) with \(\|uzu^*-z\|_{\bar\tau}\ge\|z\|_{\bar\tau}\).

Lemma 2.1 (one step). Let \(R\subset N\) be a matrix subalgebra with its trace, and \(z\in R'\cap M\) with \(E'(z)=0\). There is a matrix subalgebra \(T\subset R'\cap N\) isomorphic to \(M_{2^p}(\mathbb C)\), \(p\ge0\), with \[ \|E_T(z)\|^2_{\bar\tau}\le\tfrac{15}{16}\|z\|^2_{\bar\tau}. \]

Proof. If \(z=0\) take \(T=\mathbb C1\). Put \(N_1=R'\cap N\) and \(M_1=R'\cap M\). By Lemma 1.1(1) of the lesson on finite-dimensional subfactors, \(N_1\) is isomorphic to a corner of \(N\), hence a factor of type II₁, and \(N\) is generated by \(R\) and \(N_1\); so \(N_1'\cap M_1=N'\cap M\). The automorphisms \(\operatorname{Ad}u\), \(u\in\mathcal U(N_1)\), of \(M_1\) preserve \(\bar\tau\) and have fixed-point algebra \(N_1'\cap M_1=N'\cap M\); the restriction of \(E'\) to \(M_1\) is the \(\bar\tau\)-preserving expectation onto it. Lemma 3.1 of the first lesson, applied in \(M_1\), gives a unitary \(u\in N_1\) with \(\|uzu^*-z\|_{\bar\tau}\ge\|z\|_{\bar\tau}\).

The restriction of \(\tau\) to \(N_1\) is its normalized trace (B8). Lemma 5.1 of the lesson on finite-dimensional subfactors gives unitaries \(u_j\in N_1\), each in a matrix subalgebra \(T_j\subset N_1\) isomorphic to some \(M_{2^{p_j}}(\mathbb C)\), with \(\|u_j-u\|_{\bar\tau}\le1/j\). Let \(\Omega\) be the vector of \(\bar\tau\) in \(L^2(M,\bar\tau)\) and \(D_j=u_j-u\). For \(y'\) in the commutant of \(M\), \(\|D_jy'\Omega\|=\|y'D_j\Omega\|\to0\); these vectors are dense and \(\|D_j\|\le2\), so \(D_j\to0\) strongly and \(\|D_jz\|_{\bar\tau}\to0\). Also \(\|zD_j^*\|^2_{\bar\tau}=\bar\tau(z^*zD_j^*D_j)\), which is \(\langle D_j^*D_j\Omega,z^*z\Omega\rangle\le2\|z\|^2\|D_j\Omega\|\to0\), because \(D_j\in M_{\bar\tau}\). Since \(u_jzu_j^*-uzu^*=D_jzu_j^*+uzD_j^*\), we get \(\|u_jzu_j^*-z\|_{\bar\tau}\ge\frac12\|z\|_{\bar\tau}\) for \(j\) large. Let \(T=T_j\). As \(\bar\tau\circ E_T=\bar\tau\), Lemma 1.3 of the lesson on finite-dimensional subfactors gives \(\|E_T(z)\|^2_{\bar\tau}\le\|z\|^2_{\bar\tau}-\frac14\cdot\frac14\|z\|^2_{\bar\tau}\). \(\square\)

Theorem 2.2 (Popa). Let \(N\) be a factor of type II₁. There are mutually commuting matrix subalgebras \(T_1,T_2,\ldots\) of \(N\), with \(T_n\cong M_{2^{p_n}}(\mathbb C)\), such that the von Neumann algebra \(T\) they generate satisfies \(T'\cap M=N'\cap M\).

Proof. By (B1) of the lesson on full factors, the unit ball of \(M\) contains a sequence \((a_j)_{j\ge1}\) that is \(*\)-strongly dense. Let \(k(0),k(1),\ldots\) be a sequence of positive integers in which every positive integer occurs infinitely often, and \(x_n=a_{k(n)}\). Put \(T^{(0)}=\mathbb C1\). If \(T_1,\ldots,T_n\) are constructed, let \(T^{(n)}=T_1\vee\cdots\vee T_n\), a matrix subalgebra with its trace (Lemma 1.1(5) of the lesson on finite-dimensional subfactors), and \[ y_n=E_{T^{(n)}}(x_n),\qquad z_n=y_n-E'(y_n). \] Then \(z_n\in T^{(n)\prime}\cap M\), because \(E'(y_n)\in N'\cap M\subset T^{(n)\prime}\cap M\); moreover \(E'(z_n)=0\) and \(\|z_n\|\le2\). Lemma 2.1 with \(R=T^{(n)}\) gives \(T_{n+1}\subset T^{(n)\prime}\cap N\) with \(\|E_{T_{n+1}}(z_n)\|^2_{\bar\tau}\le\frac{15}{16}\|z_n\|^2_{\bar\tau}\).

The expectations \(E_{T^{(n)}}\), \(E_{T_n}\) and \(E'\) preserve \(\bar\tau\), so by the Schwarz inequality they do not increase \(\|\cdot\|_{\bar\tau}\). Let \(w\) be in the unit ball of \(T'\cap M\) with \(E'(w)=0\). Choose \(n_1<n_2<\cdots\) with \(\|x_{n_k}-w\|_{\bar\tau}\to0\): choose \(j_k\) with \(\|a_{j_k}-w\|_{\bar\tau}<1/k\), which the \(*\)-strong density allows, and \(n_k>n_{k-1}\) with \(k(n_k)=j_k\). Since \(E_{T^{(n)}}(w)=w\), \(\|y_{n_k}-w\|_{\bar\tau}\le\|x_{n_k}-w\|_{\bar\tau}\to0\) and \(\|E'(y_{n_k})\|_{\bar\tau}=\|E'(y_{n_k}-w)\|_{\bar\tau}\to0\), so \(\|z_{n_k}-w\|_{\bar\tau}\to0\); since \(E_{T_{n+1}}(w)=w\), also \(\|E_{T_{n_k+1}}(z_{n_k})-w\|_{\bar\tau}\to0\). Letting \(k\to\infty\) in the estimate of Lemma 2.1 gives \(\|w\|^2_{\bar\tau}\le\frac{15}{16}\|w\|^2_{\bar\tau}\), so \(w=0\).

Now let \(x\in T'\cap M\), \(x\ne0\). Then \(E'(x)\in N'\cap M\subset T'\cap M\), so \(w=(x-E'(x))/(2\|x\|)\) is as above, and \(x=E'(x)\in N'\cap M\). The reverse inclusion holds because \(T\subset N\). \(\square\)

The algebra \(T\) is generated by the increasing sequence of matrix subalgebras \(T^{(n)}\), all containing \(1\). Its relative commutant in \(N\) is \(N'\cap N=\mathbb C1\).

3. Infinite semifinite factors

In this section \(N\) is a factor of type II∞. It is properly infinite and countably decomposable, and it has a nonzero finite projection. By Proposition 15.3(3) of the projections lesson, \(1=\sum_{i\ge1}e_{ii}\) with mutually orthogonal projections \(e_{ii}\) equivalent to a finite projection; choosing partial isometries \(w_i\) with \(w_i^*w_i=e_{11}\), \(w_iw_i^*=e_{ii}\) (\(w_1=e_{11}\)) and putting \(e_{ij}=w_iw_j^*\) gives matrix units \((e_{ij})_{i,j\ge1}\) in \(N\). Let \(e=e_{11}\), \(N_0=eNe\) and \(M_0=eMe\). Since \(N\) is a factor without minimal projections and \(e\) is finite, \(N_0\) is a factor of type II₁ (B8); let \(\tau_0\) be its trace. The expectation \(E\) maps \(M_0\) onto \(N_0\), and \(E_0=E|_{M_0}\) is a faithful normal conditional expectation.

For \(x\in M\) put \(x_{ij}=e_{1i}xe_{j1}\in M_0\). With \(q_n=\sum_{i\le n}e_{ii}\), the elements \(q_nxq_n=\sum_{i,j\le n}e_{i1}x_{ij}e_{1j}\) converge strongly to \(x\), and \(x\in N\) exactly when all \(x_{ij}\in N_0\). Fix \(\lambda_i=2^{-i}\) and define \[ \varphi(x)=\sum_{i\ge1}\lambda_i\,\tau_0(x_{ii})\qquad(x\in N). \]

Lemma 3.1.

  1. \(\varphi\) is a faithful normal state on \(N\), and \(e_{ij}\varphi=(\lambda_j/\lambda_i)\,\varphi e_{ij}\); hence \(\sigma^\varphi_t(e_{ij})=(\lambda_i/\lambda_j)^{it}e_{ij}\).
  2. \(N_0\subset N_\varphi\), and \(N_\varphi\) consists of the \(x\in N\) with \(x_{ij}=0\) for \(i\ne j\).
  3. \(N\) is generated by \(N_\varphi\) and the \(e_{ij}\).

Proof. (1) \(\varphi\) is a norm-convergent sum of normal positive functionals with total mass \(\sum_i\lambda_i=1\). If \(x\ge0\) and \(\varphi(x)=0\), then \(x_{ii}=(x^{1/2}e_{i1})^*(x^{1/2}e_{i1})=0\) for all \(i\), so \(x^{1/2}e_{ii}=x^{1/2}e_{i1}e_{1i}=0\) and \(x=0\). For \(y\in N\), \((e_{ij}y)_{kk}=\delta_{ki}y_{jk}\) and \((ye_{ij})_{kk}=\delta_{kj}y_{ki}\), so \[ \begin{gathered} (\varphi e_{ij})(y)=\varphi(e_{ij}y)=\lambda_i\tau_0(y_{ji}),\\ (e_{ij}\varphi)(y)=\varphi(ye_{ij})=\lambda_j\tau_0(y_{ji}). \end{gathered} \] Corollary 1.3 of the lesson on spectral shift maps, with \(e^{-s}=\lambda_j/\lambda_i\), gives the modular group.

(2) For \(a\in N_0\) and \(y\in N\), \((ay)_{kk}=\delta_{k1}ay_{11}\) and \((ya)_{kk}=\delta_{k1}y_{11}a\), so \(\varphi(ay)=\varphi(ya)\) because \(\tau_0\) is a trace; thus \(N_0\subset N_\varphi\), and by (1) also \(e_{ii}\in N_\varphi\). Let \(x\in N\). By (1) and \(N_0\subset N_\varphi\), \(\sigma^\varphi_t(e_{ii}xe_{jj})=\sigma^\varphi_t(e_{i1}x_{ij}e_{1j})\), which is \((\lambda_i/\lambda_j)^{it}e_{ii}xe_{jj}\). If \(x\in N_\varphi\), the left side is \(e_{ii}xe_{jj}\), so \(e_{ii}xe_{jj}=0\) for \(i\ne j\). Conversely, if \(x_{ij}=0\) for \(i\ne j\), then \(x\) is the strong limit of the \(\sigma^\varphi\)-fixed elements \(q_nxq_n\), so it is fixed.

(3) Each \(q_nxq_n=\sum_{i,j\le n}e_{i1}x_{ij}e_{1j}\) with \(x_{ij}\in N_0\subset N_\varphi\) lies in the algebra generated by \(N_\varphi\) and the \(e_{ij}\), and \(q_nxq_n\to x\) strongly. \(\square\)

Theorem 3.2. Let \(N\) be a factor of type II∞ and \(\varphi\) the state above. There is a \(\sigma^\varphi\)-invariant, approximately finite-dimensional von Neumann subalgebra \(P\subset N\) with \(P'\cap M=N'\cap M\) and \(P'\cap c(M)=N'\cap c(M)\).

Proof. Theorem 2.2, applied to \(N_0\subset M_0\) with \(E_0\), gives an increasing sequence of matrix subalgebras \(R_n\subset N_0\), all containing \(e\), generating \(P_0\) with \(P_0'\cap M_0=N_0'\cap M_0\). Let \(P\) be the von Neumann algebra generated by the \(e_{ij}\) and the elements \(e_{i1}ae_{1j}\), \(a\in P_0\). These generators are eigenvectors of \(\sigma^\varphi\) by Lemma 3.1(1),(2), so \(P\) is \(\sigma^\varphi\)-invariant.

Approximately finite-dimensional. The sets \(A_n\) of the elements \(\sum_{i,j\le n}e_{i1}r_{ij}e_{1j}+c(1-q_n)\) with \(r_{ij}\in R_n\) and \(c\in\mathbb C\) are finite-dimensional \(*\)-subalgebras containing \(1\). They increase, because \(R_n\subset R_{n+1}\) and \(1-q_n=e_{n+1,1}ee_{1,n+1}+(1-q_{n+1})\). Their union contains all generators of \(P\) with indices at most \(n\) and \(a\in R_n\), and the strong closure of \(\bigcup_nR_n\) is \(P_0\); so the union generates \(P\).

Centralizers. The projections \(e_{ii}\) and the elements \(e_{i1}ae_{1i}\), \(a\in P_0\), lie in \(P\cap N_\varphi=P_\varphi\). Let \(x\in M\) commute with all of them. Commuting with the \(e_{ii}\) gives \(x_{ij}=0\) for \(i\ne j\), and \(e_{i1}ae_{1i}x=xe_{i1}ae_{1i}\) gives \(ax_{ii}=x_{ii}a\); so \(x_{ii}\in P_0'\cap M_0=N_0'\cap M_0\). Such an \(x\) commutes with every element of \(N_\varphi\), which by Lemma 3.1(2) is the strong limit of block diagonal sums \(\sum_{i\le n}e_{i1}b_{ii}e_{1i}\) with \(b_{ii}\in N_0\). Hence \(P_\varphi'\cap M\subset N_\varphi'\cap M\). By Lemma 3.1(3), \(N\) is generated by \(N_\varphi\) and \(P\), and Lemma 1.1 gives \(P'\cap c(M)=N'\cap c(M)\). Intersecting with \(M\) gives \(P'\cap M=N'\cap M\). \(\square\)

4. The theorem

Theorem 4.1 (OpenAI, for semifinite bottom factors). Let \(N\subset M\) be von Neumann algebras with separable preduals and a faithful normal conditional expectation \(E\colon M\to N\), and suppose that \(N\) is a semifinite factor. There are an approximately finite-dimensional, hence injective, von Neumann subalgebra \(P\subset N\) and a faithful normal conditional expectation of \(N\) onto \(P\) such that \[ \begin{gathered} P'\cap c(M)=N'\cap c(M),\\ P'\cap M=N'\cap M . \end{gathered} \]

Proof. Type I. A factor with a minimal projection is isomorphic to some \(B(H)\) (Corollary 10.4 of the projections lesson), and \(H\) is separable because \(N_*\) is. Take \(P=N\), with the identity as expectation. It is approximately finite-dimensional: if \((\xi_k)\) is an orthonormal basis and \(q_n\) the projection onto the span of \(\xi_1,\ldots,\xi_n\), the algebras \(q_nB(H)q_n+\mathbb C(1-q_n)\) increase and generate \(B(H)\). (Injectivity also follows from Corollary 5.2(b) of the lesson on injective von Neumann algebras.)

Type II₁. Let \(P=T\) from Theorem 2.2. Use the state \(\tau\): the algebras \(P\subset N\) lie in \(M_{\bar\tau}\), so they are fixed by \(\sigma^{\bar\tau}\), and (1.1) gives \[ \begin{aligned} &P'\cap c(M)=(P'\cap M)\rtimes\mathbb R\\ &\quad =(N'\cap M)\rtimes\mathbb R=N'\cap c(M). \end{aligned} \] The \(\tau\)-preserving conditional expectation of \(N\) onto \(P\) exists by the modular expectation theorem, since \(\sigma^\tau\) is trivial.

Type II∞. Theorem 3.2, with the \(\varphi\)-preserving conditional expectation onto the \(\sigma^\varphi\)-invariant \(P\).

In all cases \(P\) is approximately finite-dimensional, hence injective by Corollary 5.2(c) of the lesson on injective von Neumann algebras. \(\square\)

With Theorem 5.2 of the previous lesson, this proves the main theorem of [OAI] for every bottom factor that is semifinite or of type III₁. Factors of type III\(_\lambda\), \(0\le\lambda<1\), and bottom algebras with nontrivial centre are not treated in this course.

5. Exercises

Exercise 5.1. In Theorem 2.2, show that \(T\) is a factor and that \(T'\cap N=\mathbb C1\).

Exercise 5.2. In Section 3, show that \(\varphi\) is not a trace on \(N\), and that \(N_\varphi\) is not a factor.

Exercise 5.3. Show that in Theorem 3.2 the equality \(P'\cap M=N'\cap M\) also follows directly from \(P_0'\cap M_0=N_0'\cap M_0\), without the core.

Exercise 5.4. Let \(N\) be of type II₁ and suppose \(N\subset M\) is irreducible, \(N'\cap M=\mathbb C1\). Show that \(T'\cap c(M)=\lambda(\mathbb R)''\), the algebra generated by the translations.

6. Solutions

5.1. The centre of \(T\) commutes with \(T\) and lies in \(N\), so it lies in \(T'\cap N=T'\cap M\cap N\), which is \(N'\cap M\cap N=\mathbb C1\).

5.2. \(\varphi(e_{12}e_{21})=\varphi(e_{11})=\lambda_1\) while \(\varphi(e_{21}e_{12})=\varphi(e_{22})=\lambda_2\ne\lambda_1\). The projections \(e_{ii}\) lie in \(N_\varphi\) and commute with every element of \(N_\varphi\), which is block diagonal by Lemma 3.1(2); they are central in \(N_\varphi\) and different from \(0\) and \(1\).

5.3. Let \(x\in M\) commute with \(P\). Since \(x\) commutes with \(e_{j1}\), \(x_{ij}=e_{1i}xe_{j1}=e_{1i}e_{j1}x=\delta_{ij}e_{11}x\), and multiplying on the right by \(e_{11}\) gives \(x_{ij}=\delta_{ij}x_{11}\). For \(a\in P_0\) the element \(e_{11}ae_{11}=a\) lies in \(P\), so \(x_{11}\) commutes with \(P_0\) and lies in \(P_0'\cap M_0=N_0'\cap M_0\). Then \(x=\sum_ie_{i1}x_{11}e_{1i}\) (strong limit of \(q_nxq_n\)) commutes with every \(e_{i1}be_{1j}\), \(b\in N_0\), since \(x_{11}b=bx_{11}\); these elements and the \(e_{ij}\) generate \(N\).

5.4. By (1.1), \(T'\cap c(M)=(T'\cap M)\rtimes\mathbb R=(\mathbb C1)\rtimes\mathbb R\), the algebra generated by \(\pi(\mathbb C1)\) and \(\lambda(\mathbb R)\).

References