1.2 The Ultraweak (Sigma) Topology
The weak-\(*\) topology \(\sigma (M,M_{*})\) on a \(W^{*}\)–algebra \(M\) is called the \(\sigma \)–topology.
Every \(W^{*}\)–algebra is unital.
The norm unit ball of \(M=(M_{*})^{*}\) is closed in the \(\sigma \)–topology, is convex, and so has an extreme point by the Krein-Milman Theorem. By Theorem 9, the result follows.
(Sakai 1.7.6) Let \(p\) be any projection in \(M\). Then the subalgebra \(pMp\) is \(\sigma \)–closed and the mapping \(x\mapsto pxp\) is \(\sigma \)–continuous.
If \(p\in M\) is a projection, \(pMp\) is also a \(W^{*}\)–algebra with identity \(p\).
It is clear that \(eMe\) is an algebra, since any sum or product of elements preserves the pre- and post- multiplication by \(e\). It is also clear that \(e\) is the multiplicative identity of this algebra. The product reversing property of \(*\) and the fact that \(e\) is a self-adjoint projection implies that \(eMe\) is a \(*\)–algebra. By Lemma 13, if this \(*\)–subalgebra is \(\sigma \)–closed, then it is a \(W^{*}\)–subalgebra of \(M\). But this is the result of 16.
(Sak 1.7.7) If \(p\) is any projection of \(M\), then the mappings \(x\mapsto px\) and \(x\mapsto xp\) are \(\sigma \)–continuous.
(Sak 1.7.8) The mappings \(x\mapsto x^{*},ax\), and \(xa\) are \(\sigma \)–continuous for \(x,a \in M\).
(Sak 1.7.9) Let \(H\) be a subset of \(M\) and \(\mathcal{A}\) be the \(C^{*}\)–subalgebra of \(M\) generated by \(H\), and let \(\overline{\mathcal{A}}\) be the \(\sigma (M,M_{*})\)–closure of \(\mathcal{A}\). Then \(\overline{\mathcal{A}}\) is a \(W^{*}\)–subalgebra of \(M\). \(\overline{\mathcal{A}}\) is called a \(W^{*}\)–subalgebra of \(M\) generated by \(H\). Moreover, \(\overline{\mathcal{A}}\) is commutative if \(\mathcal{A}\) is commutative.
A maximal abelian \(*\)–subalgebra \(\mathcal{A}\) of a \(W^{*}\)–algebra \(M\) is also a \(W^{*}\)–algebra.
By maximality, \(\mathcal{A} =\overline{\mathcal{A}}\), and so Corollary 20 implies \(\mathcal{A}\) is a \(W^{*}\)–subalgebra of \(M\).