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

Abstract

By a result of Johnson, the Banach space $F=({⨁}_{n=1}^{\infty }{\ell }_1^n{)}_{{\ell }_{\infty }}$ contains a complemented copy of ${\ell }_1$. We identify $F$ with a complemented subspace of the space of (bounded, linear) operators on the reflexive space $({⨁}_{n=1}^{\infty }{\ell }_1^n{)}_{{\ell }_p}$ ($p\in (1,\infty ))$, 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.