6 Normality and Ultraweak Continuity for Positive Functionals

In what follows, let \(M\) be a (nonzero) \(W^{*}\)–algebra. Let \(\mathcal{P}(M)\) denote the projection lattice of \(M\). Let \(\varphi \) be a positive linear functional \(\varphi \) on \(M\). We say \(\varphi \) is normal if whenever \((p_{\alpha })\) is an increasing net of projections in \(M\) with supremum \(p\), we have \(\varphi (p_{\alpha })\to \varphi (p)\). In this section we show that this property is equivalent to \(\sigma (M, M_{*})\)–continuity. We say that a linear functional \(\varphi \) on \(M\) is positive if \(\varphi (x)\ge 0\) whenever \(x\ge 0\).