Show the algebra of all bouned operators is a $W^*$-algebra

Question

Given that the algebra of all bounded operators $B(H)$ on Hilbert space is a $C^*$-algebra such that $\| A^*A \| = \|A\|^2$. Then how can we show that it is also a $W^*$-algebra, where a $W^*$-algebra $X$ is a $C^*$-algebra such that $X$ is a dual space as a Banach space.

Answer 1

The usual definition of a $W^*$-algebra is a weak operator closed $*$-subalgebra of $B(H)$, that it's equivalent to the one you gave is a theorem of Sakai. More directly, the predual to $B(H)$ is the space of trace class operators.