1.3 Other Topologies on WStarAlgebras
In this section, let \(M\) be a \(W^{*}\)–algebra.
The set of \(\sigma (M,M_{*})\)–continuous linear functionals is precisely \(M_{*}\).
Standard result in lctvs.
The locally convex topology on \(M\) generated by the seminorms \(p_{K}(x)=\sup _{\varphi \in K}|\varphi (x)|\), ranging over \(K\) the \(\sigma (M_{*},M)\)–compact subsets of \(M_{*}\), is called the Mackey topology on \(M\). We denote this topology by \(\tau (M,M_{*})\).
The Mackey topology is the strongest topology on \(M\) for which the functionals in \(M_{*}\) are all continuous.
The locally convex topology on \(M\) generated by the seminorms \(x\mapsto \varphi (x^{*}x)^{1/2}\) ranging over \(\varphi \in M_{*}\) is called the strong topology, or \(s\)–topology, on \(M\). We denote it by \(s(M,M_{*})\).
The following result additionally uses the Mackey-Arens Theorem, so we may need to prove this as well. By \(A\le B\) on topologies below, we mean \(A\) is weaker than \(B\).
We have \(\sigma (M,M_{*})\le s(M,M_{*})\le \tau (M,M_{*})\).
The \(\sigma \)–topology is equivalent to the \(s\)–topology on the unit ball of \(M\), and hence on bounded sets.