Other Properties
The adjoint operation T → T*, as an immediate consequence of its definition, is continuous in WOT.
Multiplication is not jointly continuous in WOT: again let T be the unilateral shift. Appealing to Cauchy-Schwarz, one has that both Tn and T*n converges to 0 in WOT. But T*nTn is the identity operator for all n. (Because WOT coincides with the σ-weak topology on bounded sets, multiplication is not jointly continuous in the σ-weak topology.)
However, a weaker claim can be made: multiplication is separately continuous in WOT. If a net Ti → T in WOT, then STi → ST and TiS → TS in WOT.
Read more about this topic: Weak Operator Topology
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