How to generate a weak topology by the family of semi-norm $$ \{P_i \colon A \rightarrow B, i \in I\} $$ in which $$ P_i(a)=\lVert ia\rVert+\lVert ai\rVert$$ when $A$ is a Banach algebra and $I$ is a closed ideal of $A$ and $a$ belongs to $A$.
Actually it is based on the definition of strict topology from the book: Volker Runde - lectures on amenability- 2nd chapter Amenability Banach algebra.