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

TL;DR

This paper constructs a reflexive Banach space whose algebra of bounded operators is not a Grothendieck space, providing a counterexample to a longstanding open problem in functional analysis.

Contribution

It identifies a specific reflexive Banach space with an operator algebra that is not Grothendieck, answering a question posed by Diestel and Uhl.

Findings

The space of operators on the constructed reflexive space is not Grothendieck.

A complemented subspace of operators contains a copy of .

Counterexample to the assumption that operator spaces on reflexive spaces are Grothendieck.

Abstract

By a result of Johnson, the Banach space $F=(\bigoplus_{n=1}^\infty \ell_1^n)_{\ell_\infty}$ 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 $(\bigoplus_{n=1}^\infty \ell_1^n)_{\ell_p}$ ($p\in (1,\infty))$, 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.