= Operator predual from matrix coefficients
Let $H$ be a <Hilbert space> and let $E_0$ be the space of <finite-rank operators> on $H$, equipped with the <trace norm>. A rank-one operator $R_{x,y}z=\langle z,y\rangle x$ gives the matrix coefficient $T\mapsto\operatorname{Tr}(TR_{x,y})=\langle Tx,y\rangle$, with the <inner product> linear in its first argument. <Trace-class duality> identifies $E_0^*$ isometrically with $B(H)$. However, $E_0$ is generally incomplete: the usual <Banach space> predual is its completion, the <trace-class operators> $\mathcal S_1(H)$.
The <weak operator topology> tests only these matrix coefficients, equivalently all pairings with $E_0$. The <weak-star topology> supplied by the complete predual tests every <trace-class operator>. These two <topologies> agree on each operator-norm-bounded subset. Indeed, if $\|T_i\|,\|T\|\leq M$ and $A_N$ approximates $A$ in <trace norm>, then
$$
|\operatorname{Tr}((T_i-T)(A-A_N))|\leq2M\|A-A_N\|_1.
$$
First choose $N$ and then use matrix-coefficient convergence for $A_N$. This also explains why the operator unit ball is compact in the <weak operator topology>: apply the <Banach-Alaoglu theorem> to the complete predual and restrict to that ball. Completeness of a predual and agreement of <topologies> on bounded sets are distinct issues.
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