This is Exercise 3(a) on p. 157 in Takesaki's Operator algebras.
Let $A$ be a C*-algebra. Then each opeator $T_a\colon A\to A$ given by $T_ax = ax$ ($a\in A$) is weakly compact if and only if $A$ is an ideal of $A^{**}$, the universal enveloping von Neumann algebra.
Any hints how to solve it will be appreciated.