On the SHAI property of the Bourgain-Rosenthal-Schechtman spaces

A Banach space $X$ has the SHAI property if, for every non-zero Banach space $Y$, every surjective algebra homomorphism from the algebra $\mathcal{L}(X)$ of bounded linear operators on $X$ onto $\mathcal{L}(Y)$ is injective. In this work, we prove that the Bourgain-Rosenthal-Schechtman spaces have the SHAI property, thereby answering a question posed by Johnson, Phillips, and Schechtman in 2022.