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.



Explanation of $\sigma$-weak topology von a von Neumann algebra
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*.


