Explanation of $\sigma$-weak topology von a von Neumann algebra [closed]

Question

Let $A$ be a von Neumann algebra. I want to understand the precise meaning of the $\sigma$-weak topology on $A$. What I understand so far is the following: The $\sigma$-weak topology, which we will denote by $\tau$, is the weak$^*$-topology on $A$ (which can be defined since every von Neumann algebra has a (unique) predual $A_*$). It seems that $A_*$ is the Banach space of ultraweakly continuous linear functionals on $A$. But how to understand the ultraweak topology on $A$? Do one use the fact that $A$ is continuously embedded into some $B(H)$?

In addition I have also some other questions regarding the structure of the $\sigma$-weak topology: Is the $\sigma$-weak topology a Hausdorff topology on $A$ and possibly sequentially complete on norm-bounded sets? Is this topology even norming on $A$, i.e., for all $x\in A$ one has that $$ ||x||=\sup_{\substack{\varphi\in(X,\tau)'\\||\varphi||\leq1}}{|\varphi(x)|}. $$ Probably these questions are all trivial to answer, however it would be great to have some references. Could you help me with this? Thank you very much in advance.

Answer 1

This is not really research level, and is probably better suited to math.stackexchange, but here's an answer anyway. I will take as given that you know the notation $\sigma(E^*,E)$ for the weak-* topology on $E^*$, or more generally $\sigma(F,E)$ on a Banach space $F$ isomorphic to the dual space of a Banach space $E$.

For any Hilbert space $\newcommand{\Hil}{\mathcal{H}}\Hil$, the space of bounded operators $B(\Hil)$ is isomorphic to the dual space of the space of trace-class operators $B_*(\Hil)$ by the pairing $\langle T, \rho \rangle = \mathrm{tr}(T\rho)$, for $T \in B(\Hil)$ and $\rho \in B_*(\Hil)$. The weak-* topology $\sigma(B(\Hil),B_*(\Hil))$ on $B(\Hil)$ can be called the $\sigma$-weak or ultraweak topology, according to the author's preference, i.e. these are two names for the same thing.

When we have a C$^*$-algebra $A$ that is isometrically isomorphic to the dual space of a Banach space $A_*$, then we say it is a W$^*$-algebra, and the weak-* topology $\sigma(A,A_*)$ is also called the $\sigma$-weak or ultraweak topology.

If $A \subseteq B(\Hil)$ is a von Neumann algebra, then we can consider the restriction of $\sigma(B(\Hil),B_*(\Hil))$ to $A$. We can then define the space $A_*$ to be the set of linear functionals continuous in this topology. It is then a (nontrivial) theorem, that $A_*$ is a closed subspace of $A^*$, and $A$ is isomorphic to the dual space of $A_*$ under the restriction of the double dual embedding, and the topologies $\sigma(B(\Hil),B_*(\Hil))$ and $\sigma(A,A_*)$ agree on $A$, justifying why they're both called the $\sigma$-weak/ultraweak topology.

The questions in your second paragraph are all consequences of being a weak-* topology (the Banach-Alaoglu theorem implies that weak-* closed norm-bounded sets are complete, because they're compact).

I hope this helps you to understand standard references like Takesaki's Theory of Operator Algebras, Dixmier's Von Neumann Algebras, Kadison and Ringrose's Fundamentals of the Theory of Operator Algebras.