A reflexive Banach space whose algebra of operators is not a Grothendieck space

By a result of Johnson, the Banach space $F=(\bigoplus_{n=1}^\infty \ell_1^n){\ell\infty}$containsacomplementedcopyof$\ell_1$\mathrm{.}Weidentify$F$ with a complemented subspace of the space of (bounded, linear) operators on the reflexive space $({⨁}_{n=1}^{\mathrm{\infty }}{\mathrm{\ell }}_1^n{)}_{{\mathrm{\ell }}_p}$ ($p\in (1,\infty))$, thus giving a negative answer to the problem posed in the monograph of Diestel and Uhl which asks whether the space of operators on a reflexive Banach space is Grothendieck.