The algebras of bounded operators on the Tsirelson and Baernstein spaces are not Grothendieck spaces

TL;DR

This paper demonstrates that the algebras of bounded operators on certain reflexive Banach spaces, specifically Tsirelson and Baernstein spaces, are not Grothendieck spaces, providing new examples in functional analysis.

Contribution

It introduces new examples of reflexive Banach spaces where the algebra of bounded operators fails to be a Grothendieck space, expanding understanding of operator algebra properties.

Findings

The algebra of bounded operators on Tsirelson space is not Grothendieck.

The algebra of bounded operators on Baernstein p-space is not Grothendieck.

Provides new insights into the structure of operator algebras on reflexive Banach spaces.

Abstract

We present two new examples of reflexive Banach spaces $X$ for which $\mathscr{B}(X)$ is not a Grothendieck space, namely $X = T$ (the Tsirelson space) and $X = B_p$ (the $p^{\text{th}}$ Baernstein space) for $p\in(1,\infty)$.