The problem is already answered above. Perhaps it is worth to note the natural "next steps" one would take after this.

If $K$ is a compact topological space and $M(K) = C(K)^{\ast}$, then the following are equivalent.

- $M(K)$ has RNP.
- $C(K)$ does not contain a copy of $\ell^1$.
- $M(K)$ has the Schur property. (weakly convergent sequences are norm convergent)

$(1\Leftrightarrow 2)$ is already answered above. $(2\Leftrightarrow 3)$ for any Banach space $X$, $X^{\ast}$ has Schur property iff $X$ has Dunford-Pettis property (DPP) and contains no copy of $\ell^1$. $C(K)$ has DPP.

Next step is to generalize this from $C(K)$ to a general $C^{\ast}$-algebra $A$.

(a) the following are equivalent for $A$: (see
Chu,
Huruya, and
Jensen for the implication $(4\Rightarrow 1)$)

- $A^{\ast}$ has RNP.
- $A$ does not contain a copy of $\ell^1$.
- $A$ does not contain a copy of $C([0,1])$.
- If $x\in A$ is self-adjoint, then its spectrum $\sigma(x)$ is countable. (for otherwise $C(\sigma(x))$ and $C([0,1])$ are isomorphic.)

(b) the following are equivalent for $A$ by Hamana:

- $A^{\ast}$ has DPP.
- $A$ has DPP
- Every irreducible representation of $A$ is finite dimensional.

(c) Thus by (a)&(b), the following are equivalent for a $C^{\ast}$-algebra $A$.

- $A^{\ast}$ has the Schur property.
- $A^{\ast}$ has DPP and RNP.