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

By a result of Johnson, the Banach space F = ( ⨁ n = 1 ∞ ℓ 1 n ) ℓ ∞ contains a complemented copy of ℓ 1 . We identify F with a complemented subspace of the space of (bounded, linear) operators on the reflexive space ( ⨁ n = 1 ∞ ℓ 1 n ) ℓ p ( p ∈ ( 1 , ∞ ) ), thus solving negatively the problem posed in the monograph of Diestel and Uhl which asks whether the space of operators on a reflexive Banach space is Grothendieck.