We can get this by Gelfand duality (cf. Sakai 1.2.3) and Lemma 29.

By Lemma 28, if \(p-q \ge 0\) then \(q(p-q)q=qpq-q^3=qpq-q \ge 0\). By Lemma 30, we have \(p\le 1\) and employing Lemma 28 we obtain \(qpq\le q1q=q\). Since \(qpq\ge q\) and \(qpq\le q\) we have \(qpq=q\), which implies that \(q(p-q)q=0\). By the \(C^{*}\)-property of the norm, we have \(\| (p-q)^{1/2}q\| ^2=\| q(p-q)q\| =0\) and so \((p-q)^{1/2}q=0\) and therefore \((p-q)q=(p-q)^{1/2}(p-q)^{1/2}q=0\). It follows that \(pq=q\), and taking adjoints, \(qp=q\). Conversely, if \(qp=pq=q\), one easily checks that \(p-q\) is a projection and so its spectrum is contained in \(\{ 0,1\} \) and it is positive.

Suppose \(\mathcal{L}\) is a \(\sigma \)–closed left ideal. By Lemma 19, \(\mathcal{L^{*}}\) is also \(\sigma \)–closed and \(\mathcal{N}=\mathcal{L}\cap \mathcal{L}^{*}\) is a \(\sigma \)–closed \(*\)–subalgebra of \(M\), and therefore by Corollary 20 is a \(W^{*}\)–subalgebra of \(M\). Let \(p\) be the identity of \(\mathcal{N}\), which exists by Corollary 15. Since \(\mathcal{L}\) is a left ideal, \(xp\in \mathcal{L}\) for any \(x\in M\), thus \(Mp\subseteq \mathcal{L}\) conversely, \(xp=x\)for any \(x\in \mathcal{L}\), since \(p\) is the identity of \(\mathcal{N}\), thus \(\mathcal{L}\subseteq Mp\). If \(p_1\) is another projection in \(M\) such that \(\mathcal{L}=Mp_1\) then \(Mp=Mp_{1}\) and \(p_1=xp\) for some \(x\in M\). By the selfadjointness of \(p_1\) we have \(p_1=px^{*}xp\) and therefore \(p_1p=pp_1=p_1\) and therefore \(p_1\le p\) by Lemma 31. We get \(p\le p_1\) using the same argument. The claim for right ideals follows mutatis mutandis.

Let \(\{ e_{i}\} _{i\in \mathbb {I}}\) be any set of projections in \(M\). Consider \(\mathcal{L}_1\), the \(\sigma \)–closed left ideal generated by \(\{ Me_{i}\} _{i\in \mathbb {I}}\), and \(\mathcal{L}_{2}=\bigcap _{i\in \mathbb {I}}Me_{i}\). By Proposition 33 there exist projections \(e_1,e_2\) in \(M\) so that \(\mathcal{L}_1=Me_1\) and \(\mathcal{L}_2=Me_{2}\). By Lemma 31, we know \(e_i\le e_1\) and \(e_2\le e_i\) for any \(i\in \mathbb {I}\). (To prove the first of these, since \(e_i=xe_1\) for some \(x\in M\), we have by the self-adjointness of \(e_i\) that \(e_i=e_1x^{*}xe_1\) and so \(e_1e_i=e_ie_1=e_i\) and so \(e_i\le e_1\) by Lemma 31. The other direction is analogous.) Furthermore, if \(p\in M^{p}\) is another projection that is an upper bound for all the \(e_i\) then \(Me_1\subseteq Mp\) since \(\mathcal{L}_1=Me_1\) is the left ideal generated by the \(e_i\). It follows \(e_1\le p\) by the same calculation we just did. If \(p\in M^{p}\) is a lower bound for all the \(e_i\) then \(Mp\subseteq M_{e_i}\) for every \(i\in \mathbb {I}\) and so \(Mp\subseteq \bigcap _{i}Me_{i}=Me_{2}\) and we obtain \(p\le e_2\). It follows that \(e_1=\sup (e_i)\) and \(e_2=\inf (e_i)\).