Let $\mathcal H$ be a Hilbert space. Define a topology $\tau\_1$ on $B(\mathcal H)$ by the family of seminorms $x\mapsto |Tr(xa)|,$ $a\in L^1(B(\mathcal H)).$ Here $B(\mathcal H)$ denotes the set of all bounded linear maps on $\mathcal H$ and $L^1(B(\mathcal H))$ denotes the trace class operators.
 Again define the $\sigma$-WOT topology $\tau\_2$ on $B(\mathcal H)$ by pulling back the weak operator topology of $B(\mathcal H\otimes\ell\_2)$ to $B(\mathcal H)$ via the map $x\mapsto x\otimes 1.$ How to show that $\tau\_1=\tau\_2$? In many books and lecture notes in von Neumann algebras they have just mentioned that this is true. But I could not find a solid proof.



Equivalence of $\sigma$-weak topology to another topology
I think this is just a definition chase, and use of the standard representation of a trace-class operator. Starting with the later, following Chapter II in the first volume of Takesaki, for example, we have that any trace-class operator $a$ can be written as
$$ a(\xi) = \sum\_n a\_n (\xi|\xi\_n) \eta\_n \qquad (\xi\in H), $$
where $(\xi\_n),(\eta\_n)$ are orthogonal sequences (perhaps finite) in $H$ and $(a\_n)$ is a sequence of positive reals with $\sum\_n a\_n<\infty$. Further, the trace-class norm of $a$ is $\sum\_n a\_n$.


Given such a representative, let $\alpha\_n = a\_n^{1/2}\xi\_n$ and $\beta\_n = a\_n^{1/2}\eta\_n$ we find that $a(\xi) = \sum\_n (\xi|\alpha\_n)\beta\_n$ and $\sum\_n \|\alpha\_n\| \|\beta\_n\| = \sum\_n a\_n < \infty$. Conversely, if $(\alpha\_n), (\beta\_n)$ are any sequences in $H$ with $\sum\_n \|\alpha\_n\| \|\beta\_n\| < \infty$ we have
$$ B\_0(H)\ni x \mapsto \sum\_n (x\alpha\_n|\beta\_n) $$
defines a bounded linear functional on the compact operators $B\_0(H)$, and hence there is a trace-class operator $a:\xi\mapsto \sum\_n (\xi|\alpha\_n)\beta\_n$ with trace-class norm at most $\sum\_n \|\alpha\_n\| \|\beta\_n\|$.


We conclude that the $\sigma$-weak topology, $\tau\_1$, is given by the seminorms
$$ B(H)\ni x \mapsto \Big|\sum\_n (x(\alpha\_n)|\beta\_n)\Big| \text{ where }
\sum\_n \|\alpha\_n\| \|\beta\_n\|<\infty. $$


The WOT is defined by seminorms $x\mapsto |(x\xi|\eta)|$. If $B(H)$ acts on $H\otimes\ell\_2$ via $x\mapsto x\otimes 1$ then consider the respresentation of vectors in $H\otimes\ell\_2$ which is $\xi=\sum\_n \xi\_n\otimes e\_n$ and $\eta=\sum\_n \eta\_n\otimes e\_n$ where $(e\_n)$ is the standard unit vector basis of $\ell\_2$ and $\sum\_n \|\xi\_n\|^2, \sum\_n \|\eta\_n\|^2<\infty$. Hence the induced seminorm on $B(H)$ is $\big| \sum\_n (x\xi\_n|\eta\_n)\big|$. By Cauchy-Schwarz, $\sum\_n \|\xi\_n\| \|\eta\_n\| \leq \Big(\sum\_n \|\xi\_n\|^2, \sum\_n \|\eta\_n\|^2\Big)^{1/2}<\infty$. Conversely, if we have $(\alpha\_n),(\beta\_n)$ with $\sum\_n \|\alpha\_n\| \|\beta\_n\|<\infty$ then by rescaling, we may suppose that $\|\alpha\_n\|=\|\beta\_n\|$ for each $n$; notice that this does not change the seminorm $x\mapsto \big|\sum\_n (x(\alpha\_n)|\beta\_n)\big|$. Then $\sum\_n \|\alpha\_n\|^2 = \sum\_n \|\beta\_n\|^2 < \infty$.


The equivalent between $\tau\_1$ and $\tau\_2$ follows.


