By Lemma 5.4 the subspace E separates the points of M and so \sigma(M, E) is Hausdorff. By the Banach-Alaoglu Theorem (Theorem 1.4) the closed unit ball B of M is compact in the \sigma(M, M_*)-topology. Since \sigma(M, E) is weaker than \sigma(M, M_*), the identity map on B is \sigma(M, M_*)-\sigma(M, E)-continuous, and therefore a homeomorphism, by Lemma 7.1.1; as B is compact, it is even a uniform equivalence. The same holds for every closed ball, by scaling.

By Lemma 7.1.3, to prove that a uniformly bounded net (x_t) is \sigma-Cauchy, it is enough to check that for all \varepsilon > 0 and \varphi \in T there is a \Lambda such that \varphi(x_\lambda - x_\mu) \le \varepsilon for \lambda \ge \mu \ge \Lambda.

If (x_\lambda) is a uniformly bounded increasing net in M^s, and \varphi \in T, then (\varphi(x_\lambda)) is a uniformly bounded increasing net of real numbers and therefore is Cauchy. By the above paragraph, (x_\lambda) is \sigma-Cauchy and by the \sigma-compactness of B and the fact that (x_\lambda) is increasing there exists x \in B so that x_\lambda \to x in the \sigma-topology. Since for any \varphi \in T and any \lambda we have 0 \le \varphi(x - x_\lambda), by Lemma 5.3 we have 0 \le x - x_\lambda. Therefore \sup(x_\lambda) \le x. For a given \varphi \in T and \varepsilon = \varphi(x - \sup(x_\lambda)) one finds x_\lambda so that \varphi(x - x_\lambda) < \varphi(x - \sup(x_\lambda)), which is a contradiction; therefore x = \sup(x_\lambda).

By Corollary 3.2 and the fact that (u^{-1})^* = (u^*)^{-1}, the relation u^* x_\lambda u \le u^* x u is equivalent to x_\lambda \le x, and so \sup(u^* x_\lambda u) = u^* \sup(x_\lambda) u = u^* x u if u \in M is invertible.

If a \in M is not invertible, one can shift the spectrum horizontally to avoid zero, i.e. there is a c > 0 so that c1 + a is invertible. Indeed, observe that a - t1 = (a + c1) - (t + c)1 and therefore t \in \sigma(a) \iff t + c \in \sigma(a + c1). In particular, if we let t = -c here we find that 0 \in \sigma(a) + c \iff 0 \in \sigma(a + c1). By the invertible case above,

\begin{aligned} c^2 \varphi(x_\lambda) + c \varphi(a^* x_\lambda + x_\lambda a) + \varphi(a^* x_\lambda a) &= \varphi((c1 + a)^* x_\lambda (c1 + a)) \\ &\to \varphi((c1 + a)^* x (c1 + a)) \end{aligned}

for all \varphi \in T. It suffices to show that \varphi(a^* x_\lambda + x_\lambda a) \to \varphi(a^* x + x a) for all \varphi \in T, because then \varphi(a^* x_\lambda a) \to \varphi(a^* x a) by the above. We can use Cauchy-Schwarz (Lemma 7.1.2) for this as follows. If \alpha \le \beta, we have x_\beta - x_\alpha \ge 0, and since the net (x_\lambda) is uniformly bounded there exists M > 0 such that (using Lemma 3.3)

\begin{aligned} |\varphi(a^* (x_\beta - x_\alpha))| &= |\varphi(a^* (x_\beta - x_\alpha)^{1/2} (x_\beta - x_\alpha)^{1/2})| \\ &\le \varphi(a^* (x_\beta - x_\alpha) a)^{1/2} \varphi(x_\beta - x_\alpha)^{1/2} \\ &\le 2M \varphi(a^* a)^{1/2} \varphi(x_\beta - x_\alpha)^{1/2} \end{aligned}

and by the positivity of \varphi we have |\varphi(a^* (x_\beta - x_\alpha))| = |\varphi((x_\beta - x_\alpha) a)|. It follows that \varphi(a^* x_\lambda + x_\lambda a) \to \varphi(a^* x + x a), and the proof is finished.

By Lemma 5.4 there is a \sigma(M, M_*)-continuous positive linear functional \psi_0 on M such that \psi_0(p) \ne 0. Rescale this functional by a positive constant to obtain \psi such that \varphi(p) < \psi(p).

We proceed by contradiction. Suppose the conclusion does not hold. Then, for every nonzero subprojection p_1 of p there is a nonzero subprojection q \le p_1 such that \varphi(q) \ge \psi(q). In particular, (letting p_1 = q) there is a q \le p such that \varphi(q) \ge \psi(q). If (q_\alpha) is a chain of such nonzero projections, then q_\alpha \to \sup q_\alpha in the \sigma-topology by Theorem 7.1.4, and by Lemma 4.4 we know this supremum is a subprojection of p_1. Since \varphi is positive and normal, and \psi is positive and \sigma-continuous, we have \varphi(\sup q_\alpha) \ge \psi(\sup q_\alpha).

Therefore, by Zorn's Lemma, there is a maximal q_0 \le p such that \varphi(q_0) \ge \psi(q_0). We claim q_0 = p. If not, p - q_0 is a nonzero subprojection of p by Corollary 4.3, and there exists a nonzero projection q_1 \le p - q_0 such that \varphi(q_1) \ge \psi(q_1). But then q_0 is a proper subprojection of q_0 + q_1 and by linearity and positivity \varphi(q_0 + q_1) \ge \psi(q_0 + q_1), contradicting the maximality of q_0. Thus p = q_0 and so \varphi(p) \ge \psi(p), contradicting the hypothesis \varphi(p) < \psi(p).

Let x be on the unit sphere and let p be the supremum of the p_\alpha. By Theorem 7.1.4 and Lemma 4.4 we know p_\alpha converges in the \sigma-topology to p, and p is a projection. By Corollary 4.3 we know p - p_\alpha is a projection. By the C^*-property of the norm (\|x^* x\| = \|x\|^2), Lemma 3.3 and the positivity of \varphi we have \varphi(x^* x) \le \varphi(1). Now these facts together with Cauchy-Schwarz and the monotonicity of square roots (Lemma 7.1.2) give

\begin{aligned} |\varphi(x(p - p_\alpha))| &\le \varphi(x^* x)^{1/2} \varphi(p - p_\alpha)^{1/2} \\ &\le \varphi(1)^{1/2} \varphi(p - p_\alpha)^{1/2}. \end{aligned}

Hence, by the definition of the operator norm, \|\varphi(\cdot (p - p_\alpha))\| \le \varphi(1)^{1/2} \varphi(p - p_\alpha)^{1/2}. The right hand side converges to 0 with \alpha due to the normality of \varphi, and therefore \varphi(\cdot\, p_\alpha) converges to \varphi(\cdot\, p) in norm. Since the set of \sigma-continuous functionals on M is norm closed, it follows that \varphi(\cdot\, p) is \sigma-continuous. We obtain a maximal p_0 by Zorn's Lemma.

The functional \omega factors through a bounded functional on the GNS Hilbert space H_\psi of \psi, which by the Riesz representation theorem is \langle \eta, \cdot \rangle for some \eta \in H_\psi. Approximating \eta by the classes \hat y of elements y \in M, the functionals x \mapsto \psi(y^* x) converge in norm to \omega. These are \sigma-continuous by Lemma 2.2.6, and the \sigma-continuous functionals form a norm closed subspace of M^* (Proposition 2.2.2).

By Lemma 2.2.4, pMp is a W^*-algebra whose projections are exactly the projections q \le p of M. By Lemma 6.5 applied in pMp, every positive a \in pMp is a norm limit of nonnegative combinations of such projections, and the set where \varphi \le \psi is a norm closed convex cone.

The claim is obvious for the zero functional. Let \varphi be a nonzero positive normal linear functional. By Lemma 7.2.3 we have a maximal p_0 \in \mathcal{P}(M) such that M \ni x \mapsto \varphi(x p_0) is \sigma(M, M_*)-continuous. Assume for the purposes of finding a contradiction that p_0 \ne 1. By Lemma 7.2.1 there is a \sigma(M, M_*)-continuous positive functional \psi on M such that \varphi(1 - p_0) < \psi(1 - p_0). By Lemma 7.2.2 there is a nonzero subprojection p \le 1 - p_0 in M such that \varphi(q) < \psi(q) for every nonzero q \le p in M.

By Lemma 7.2.5, \varphi(p x^* x p) \le \psi(p x^* x p) for every x \in M. The functional \psi' = \psi(p \cdot p) is positive and \sigma-continuous by Lemma 2.2.6, and by the Cauchy–Schwarz inequality (Lemma 7.1.2)

|\varphi(xp)|^2 \le \varphi(1) \varphi(p x^* x p) \le \varphi(1) \psi'(x^* x).

Hence x \mapsto \varphi(xp) is \sigma-continuous by Lemma 7.2.4, and therefore so is x \mapsto \varphi(x(p_0 + p)) = \varphi(x p_0) + \varphi(xp). Since p_0 + p is a projection strictly larger than p_0, this contradicts the maximality of p_0, and therefore p_0 = 1 and the result follows.

(1) and (2) are equivalent by Proposition 2.2.2. (1) implies (3) by Theorem 7.1.4, (3) trivially implies (4), and (4) implies (1) by Theorem 7.2.6.

By Theorem 7.3.1, the positive \sigma(M, P)-continuous functionals are the positive \sigma(M, Q)-continuous functionals, since condition (4) does not mention the predual. On bounded sets, each ultraweak topology is the weak topology induced by its positive continuous functionals (Lemma 7.1.3), so the two topologies agree on the unit ball. By the Krein–Smulian theorem (Theorem 1.7), the identity of M is a homeomorphism \sigma(M, P) \to \sigma(M, Q). Hence P and Q are identified, via their isometric embeddings into M^*, with the same subspace of M^* (Proposition 2.2.2).