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

Abstract

By a result of Johnson, the Banach space F=(⨁n=1∞ℓ1n)ℓ∞ contains a complemented copy of ℓ1. We identify F with a complemented subspace of the space of (bounded, linear) operators on the reflexive space (⨁n=1∞ℓ1n)ℓp (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.