SHAI property for complemented subspaces of L_p with an unconditional basis

Determine whether every complemented subspace of L_p with an unconditional basis has the SHAI property.

Background

This is the more specific version of the unresolved question concerning complemented subspaces of L_p. It asks whether the presence of an unconditional basis is sufficient to guarantee the SHAI property, namely, injectivity of every surjective algebra homomorphism from the space's bounded-operator algebra onto the bounded-operator algebra of any nonzero Banach space.

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