SHAI property for complemented subspaces of L_p

Determine whether every complemented subspace of L_p has the SHAI property.

Background

The SHAI property requires every surjective algebra homomorphism from the bounded-operator algebra of a Banach space onto the bounded-operator algebra of any nonzero Banach space to be injective. The paper places the question of whether this property holds for all complemented subspaces of L_p in the context of prior work on complemented subspaces of L_p and explicitly states that the question remains unresolved.

References

In the same paper, they asked whether every complemented subspace of $L_p$, or, more specifically, every complemented subspace of $L_p$ with an unconditional basis, has the SHAI property. Both questions remain open.

— On the SHAI property of the Bourgain-Rosenthal-Schechtman spaces  (2608.21297 - Konstantos, 21 Aug 2026) in Introduction