# Conditional expectations and type III

A normal conditional expectation can detect a finite part of the larger algebra and force a finite part in the smaller one. A family of expectations that detects every nonzero positive element therefore transfers type III from the smaller algebra to the larger algebra. The family need not contain a faithful member.

Let \(N\subseteq M\) be von Neumann algebras. A retraction \(E:M\to N\) is a complex-linear map fixing \(N\) pointwise. By The contractive retraction theorem, contractivity implies positivity and the identities
\[
 \begin{gathered}
 E(n_1xn_2)\\
 =n_1E(x)n_2,\\
 E(x^*)=E(x)^*.
 \end{gathered}
 \tag{XT.1}
\]
Here \(n_1,n_2\in N\) and \(x\in M\). Such a map is a conditional expectation. If \(N\ne0\), its norm is one. Normality means ultraweak continuity; it is an additional hypothesis throughout this lesson.

A projection \(e\) is finite if every partial isometry with initial projection \(e\) and final projection at most \(e\) has final projection equal to \(e\). An algebra is type III if it has no nonzero finite projection. We allow the zero algebra with this vacuous convention. Neither algebra is assumed to be a factor or sigma-finite.

## The normal functional that carries the estimate

The sigma-strong topology on a von Neumann algebra \(A\) is generated by
\[
 p_\varphi(x)=\varphi(x^*x)^{1/2},
 \qquad \varphi\in A_*^+.
\]
The sigma-strong seminorm theorem identifies these with the usual square-summable vector seminorms and proves their agreement with strong operator convergence on norm-bounded sets.

**Lemma.** Let \(\Phi:A\to B\) be a positive normal bounded map. If \(c_\alpha\to0\) sigma-strongly in \(A\), then, for every \(\psi\in B_*^+\),
\[
 \psi\bigl(\Phi(c_\alpha^*c_\alpha)\bigr)
 \longrightarrow0.
 \tag{XT.2}
\]

**Proof.** The composition \(\psi\circ\Phi\) is positive and ultraweakly continuous, hence belongs to \(A_*^+\). The left side of (XT.2) is precisely \(p_{\psi\circ\Phi}(c_\alpha)^2\). \(\square\)

We will also use two consequences of the same seminorm description. First, sigma-strong convergence on bounded subsets of a von Neumann subalgebra is inherited from its concrete representation in the larger algebra. Indeed, both intrinsic topologies agree there with strong operator convergence. If the subalgebra has identity \(r\), its elements act by zero on \((1-r)H\); testing vectors in \(rH\) or all of \(H\) therefore gives the same strong topology.

Second, for a projection \(e\in N\) and \(\psi\in(eNe)_*^+\), the formula
\[
 \widetilde\psi(n)=\psi(ene)
\]
defines an element of \(N_*^+\). This is the positive normal extension proved in The inherited predual of a corner. Thus every seminorm on a corner can be tested after composing its compression with an expectation.

## Detecting one finite-trace projection

**Lemma.** Suppose \(z\in Z(M)\) is a central projection and \(Mz\) has a faithful normal semifinite trace \(\tau\). Let \(q\in Mz\) be a projection with \(0<\tau(q)<\infty\). If a normal contractive retraction \(E:M\to N\) satisfies \(E(q)\ne0\), then \(N\) has a nonzero finite projection.

**Proof.** The positive element \(E(q)\) is nonzero. Set \(\delta=\|E(q)\|/2>0\) and take its spectral projection in \(N\),
\[
 e=1_{[\delta,\infty)}(E(q)).
\]
The bounded spectral calculus gives \(e\ne0\) and
\[
 e\leq\delta^{-1}E(q).
 \tag{XT.3}
\]
For \(x\in eNe\), positivity and the module identity (XT.1) imply
\[
 \begin{gathered}
 xx^*=xex^*\\
 \leq\delta^{-1}xE(q)x^*\\
 =\delta^{-1}E(xqx^*).
 \end{gathered}
 \tag{XT.4}
\]
We will prove that adjoint is sigma-strongly continuous at zero on the unit ball of \(eNe\).

Let \((x_\alpha)\) be any net in that ball converging sigma-strongly to zero. It also converges strongly to zero in the ambient representation. Since \(z\) is central, \(zx_\alpha\in Mz\), and this bounded net converges strongly, hence sigma-strongly, to zero in \(Mz\). The projection \(q\) is square integrable for \(\tau\). Applying The square-integrable factor theorem inside \(Mz\) gives
\[
 \begin{gathered}
 c_\alpha=q(zx_\alpha)^*=qx_\alpha^*,\\
 c_\alpha\longrightarrow0
 \quad\hbox{sigma-strongly}.
 \end{gathered}
\]
The equality uses centrality of \(z\) and \(qz=q\). The conclusion initially holds in \(Mz\); it also holds in \(M\), because the net is norm bounded and the strong topology is inherited as in XT-01. In particular,
\[
 c_\alpha^*c_\alpha=x_\alpha qx_\alpha^*.
\]

Fix \(\psi\in(eNe)_*^+\) and extend it to \(\widetilde\psi\in N_*^+\) by compression. Applying it to (XT.4), then using XT-01, gives
\[
 \begin{gathered}
 0\leq\psi(x_\alpha x_\alpha^*)\\
 \leq\delta^{-1}\widetilde\psi
       \bigl(E(c_\alpha^*c_\alpha)\bigr)\\
 \longrightarrow0.
 \end{gathered}
\]
The first quantity is \(p_\psi(x_\alpha^*)^2\). Since \(\psi\) was arbitrary, \(x_\alpha^*\to0\) sigma-strongly in \(eNe\). This proves the asserted continuity for arbitrary nets in its unit ball.

If \(e\) were infinite in \(N\), Adjoint in an infinite corner would give a sequence in this very unit ball converging sigma-strongly to zero whose adjoints do not. Thus \(e\) is finite, as required. \(\square\)

The trace is used only on \(Mz\). It has not been extended by zero and then treated as faithful on all of \(M\). Normality of \(E\) is used precisely to make \(\widetilde\psi\circ E\) a normal functional in the convergence estimate.

## Type III transfer through a separating family

Call a family \((E_i)_{i\in I}\) of normal contractive retractions \(M\to N\) **separating** if every nonzero \(b\in M_+\) has \(E_i(b)\ne0\) for some \(i\). Equivalently,
\[
 \begin{gathered}
 x\ne0\quad\Longrightarrow\\
 E_i(x^*x)\ne0\\
 \text{for some }i.
 \end{gathered}
 \tag{XT.5}
\]
For this equivalence use \(x=b^{1/2}\) in one direction and \(x^*x\ne0\) for \(x\ne0\) in the other. The square root and the latter implication follow from the C*-identity and bounded positive calculus.

**Theorem.** If such a separating family exists and \(N\) is type III, then \(M\) is type III.

**Proof.** Suppose instead that \(M\) contains a nonzero finite projection \(p\). We first construct the finite-trace projection required by XT-02, without assuming that a whole finite algebra has a faithful finite normal trace.

The corner \(pMp\) is finite: an isometry in that corner would otherwise be a partial isometry in \(M\) contradicting finiteness of \(p\). The earlier programme finite-trace separation theorem, Theorem 4.7, therefore supplies a finite normal trace \(\theta\) on \(pMp\) with \(\theta(p)>0\).

Let \(q=s(\theta)\) be its support in \(pMp\). The support of a normal trace, applied to that corner, shows that \(q\ne0\), that \(q\) is central in \(pMp\), and that \(\theta\) restricts to a faithful finite normal trace on \(qMq\). Moreover,
\[
 0<\theta(q)=\theta(p)<\infty.
\]
We do not require \(q\) to be central in \(M\).

By the earlier programme corner-trace extension theorem, Theorem 6.2(1)–(3), this trace extends to a faithful normal semifinite trace \(\tau\) on \(Mz\), where \(z=c_M(q)\) is the central carrier of \(q\) in \(M\). The restriction identity gives
\[
 \begin{gathered}
 0<\tau(q)<\infty,\\
 \tau(q)=\theta(q).
 \end{gathered}
 \tag{XT.6}
\]
Thus \(q\) is a nonzero positive element to which the separating hypothesis applies. Choose \(i\) with \(E_i(q)\ne0\). Lemma XT-02 now produces a nonzero finite projection in \(N\), contrary to its being type III. Hence no such \(p\) exists, proving the theorem. \(\square\)

The two programme theorems are proved in *Traces on von Neumann algebras*, by Claude Opus 5.5 (Anthropic), September–October 2026, CC0. Only one finite normal trace on one nonzero corner is selected. No sum or average over the family \(I\), countable subfamily, or faithful normal state is needed.

If a common identity was not assumed initially, separation forces it. Indeed, writing \(r\) for the identity of \(N\), the retraction theorem gives \(E_i(1_M)=r\), hence \(E_i(1_M-r)=0\) for every \(i\). Separation implies \(1_M-r=0\). If \(N=0\), every map is zero, so separation forces \(M=0\); this is the vacuous case of the theorem. For a nonzero range all the retractions have norm one, giving the norm-one-projection formulation as well.

## Families, composition and the separation hypothesis

**A separating family with no faithful member.** Let \(A\ne0\) be a von Neumann algebra, let \(M=A\oplus A\), and let \(N=\{(a,a):a\in A\}\). Define
\[
 \begin{gathered}
 E_1(a,b)=(a,a),\\
 E_2(a,b)=(b,b).
 \end{gathered}
\]
Both maps fix \(N\), preserve positivity, and have norm one because the direct-sum norm is \(\|(a,b)\|=\max(\|a\|,\|b\|)\). Coordinatewise ultraweak convergence proves normality. Neither map is faithful: \(E_1(0,1)=0\) and \(E_2(1,0)=0\). But a nonzero positive pair has at least one nonzero coordinate, so the pair of maps is separating. If \(A\) is type III, its diagonal copy \(N\) is type III, and XT-03 proves that \(A\oplus A\) is type III. This uses a family that cannot be replaced by either of its members.

**Separation cannot be omitted.** Fix any nonzero type III algebra \(A\subseteq B(H)\), with its faithful unital concrete representation. Put
\[
 \begin{gathered}
 M=A\oplus B(H),\\
 N=\{(a,a):a\in A\},\\
 E(a,b)=(a,a).
 \end{gathered}
\]
The graph algebra \(N\) is a von Neumann subalgebra: coordinatewise ultraweak limits preserve the equality of its two represented coordinates. Projection onto the first coordinate identifies it normally with \(A\), so it is type III. The displayed map is a normal norm-one retraction by the same coordinate and norm calculation as before.

However, for a rank-one projection \(f\in B(H)\), the projection \((0,f)\) is nonzero and finite in \(M\): its corner is \(\mathbb C(0,f)\), in which no proper isometry exists. Hence \(M\) is not type III. Precisely the missing hypothesis is visible in
\[
 E(0,f)=0.
\]

**Composing separating families.** Let \(P\subseteq N\subseteq M\), with separating families \(E_i:M\to N\) and \(F_j:N\to P\) of normal contractive retractions. Their composites \(F_jE_i\) are normal contractions and fix \(P\) pointwise. Positivity follows by composition. For \(0\ne b\in M_+\), choose \(i\) with \(E_i(b)\ne0\), then \(j\) with \(F_j(E_i(b))\ne0\). Thus the composite family is separating. Applying XT-03 proves that type III of \(P\) implies type III of \(M\). The index sets may be arbitrary; the proof selects only the two indices needed for the given positive element.

The type-transfer theorem corresponds to Lemma V.2.29 of M. Takesaki, *Theory of Operator Algebras I*, printed p. 327. Its analytic mechanism is the bounded-set adjoint continuity and infinite-corner discontinuity established in the preceding lesson.