By Lemma 38 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 \(\varphi (p){\lt}\psi (p)\).

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, \(q_{\alpha }\to \sup q_{\alpha }\) in the \(\sigma \)–topology by Theorem 45, and by Lemma 34 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 32, 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 \(\varphi (p){\lt}\psi (p)\).

Let \(x\) be on the unit sphere and let \(p\) be the supremum of the \(p_{\alpha }\). By Theorem 45 and Lemma 34 we know \(p_{\alpha }\) converges in \(\sigma \)–topology to \(p\), and \(p\) is a projection. By Corollary 32 we know \(p-p_\alpha \) is a projection. By the \(C^{*}\)–property of the norm (\(\| x^{*}x\| =\| x\| ^{2}\)), Lemma 29 and the positivity of \(\varphi \) we have \(\varphi (x^{*}x)\le \varphi (1)\). Now these facts together with Cauchy-Schwartz and the monotonicity of square roots,

Hence, by the definition of 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 \(\sigma (\cdot p)\) is \(\sigma \)–continuous. We obtain a maximal \(p_{0}\) by Zorn’s Lemma.

The claim is obvious for the zero functional. Let \(\varphi \) be a nonzero positive normal linear functional. By Lemma 48 we have a maximal \(p_0\in \mathcal{P}(M)\) such that \(M\ni x \mapsto \varphi (xp_0)\) is \(\sigma (M,M_{*})\)–continuous. Assume for the purposes of finding a contradiction that \(p_0\ne 1\). By Lemma 46 there is a \(\sigma (M,M_{*})\)–continuous positive functional \(\psi \) on \(M\) such that \(\varphi (1-p_0){\lt}\psi (1-p_0)\). By Lemma 47 there is a nonzero subprojection \(p\le 1-p_0\) in \(M\) such that \(\varphi (q){\lt} \psi (q)\) for every nonzero \(q\le p\) in \(M\). Let \(x\in pMp\) be on the unit sphere. Then \(x^{*}x\) is positive and hence normal, so the \(C^{*}\)–subalgebra of \(pMp\) generated by \(x^{*}x\) and \(p\) is commutative, and is hence contained in a maximal abelian \(*\)–subalgebra \(A\) of \(pMp\). Now \(A\) is a \(W^{*}\)–subalgebra of \(pMp\) by Lemma 21 and hence is a maximal commutative \(C^{*}\)–subalgebra of \(pMp\). Via the Gelfand Transform, \(A\) is star isomorphic to \(C(K)\), where \(K\) is Stonean by Lemmas 17 and 42. By Lemma 39 it follows that \(\varphi (a)\le \psi (a)\) for every \(a\ge 0\) in \(A\), which holds a fortiori for \(a\ge 0\) in \(C^{*}(x^{*}x,p)\). In particular, \(\varphi (px^{*}xp)\le \psi (px^{*}xp)\). Therefore,

Since \(x\mapsto \varphi (xp_0)\) is \(\sigma \)–continuous, it is \(s\)-continuous by Lemma 49. The seminorm \(x \mapsto \psi (px^{*}xp)^{1/2}\) is a defining seminorm for the \(s\)–topology on \(M\). It follows that \(x\mapsto \varphi (x(p_0+p))\) is \(s\)-continuous and therefore \(\sigma \)–continuous, by Corollary 26. This contradicts the maximality of \(p_0\), and therefore \(p_0=1\) and the result follows.